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Intended as supplementary material for undergraduate physics students, this wide-ranging collection of problems in applied mathematics and physics features complete solutions. The problems were specially chosen for the inventiveness and resourcefulness their solutions demand, and they offer students ... read more
Methods of Applied Mathematics by Francis B. Hildebrand Offering a number of mathematical facts and techniques not commonly treated in courses in advanced calculus, this book explores linear algebraic equations, quadratic and Hermitian forms, the calculus of variations, more.
Challenging Problems in Algebra by Alfred S. Posamentier, Charles T. Salkind Over 300 unusual problems, ranging from easy to difficult, involving equations and inequalities, Diophantine equations, number theory, quadratic equations, logarithms, and more. Detailed solutions, as well as brief answers, for all problems are provided.Problems in Quantum Mechanics by I. I. Gol'dman, V. D. Krivchenkov A comprehensive collection of problems of varying degrees of difficulty in nonrelativistic quantum mechanics, with answers and completely worked-out solutions. An ideal adjunct to any textbook in quantum mechanics.
The Red Book of Mathematical Problems by Kenneth S. Williams, Kenneth Hardy Handy compilation of 100 practice problems, hints, and solutions indispensable for students preparing for the William Lowell Putnam and other mathematical competitions. Preface to the First Edition. Sources. 1988Fifty Challenging Problems in Probability with Solutions by Frederick Mosteller Remarkable puzzlers, graded in difficulty, illustrate elementary and advanced aspects of probability. These problems were selected for originality, general interest, or because they demonstrate valuable techniques. Also includes detailed solutions.
Product Description:
Intended as supplementary material for undergraduate physics students, this wide-ranging collection of problems in applied mathematics and physics features complete solutions. The problems were specially chosen for the inventiveness and resourcefulness their solutions demand, and they offer students the opportunity to apply their general knowledge to specific areas. Numerous problems, many of them illustrated with figures, cover a diverse array of fields: kinematics; the dynamics of motion in a straight line; statics; work, power, and energy; the dynamics of motion in a circle; and the universal theory of gravitation. Additional topics include oscillation, waves, and sound; the mechanics of liquids and gases; heat and capillary phenomena; electricity; and optics |
Practical Problems in Mathematics: For Welders
Book Description: Practical Problems in Mathematics for Welders, 5E, takes the same straightforward and practical approach to mathematics that made previous editions so highly effective, and combines it with the latest procedures and practices in the welding industry. With this applications oriented book, readers will learn how to solve the types of math problems faced regularly by welders. Each unit begins with a review of the basic mathematical procedures used in standard operations and progresses to more advanced formulas. With real-world welding examples and clear, uncomplicated explanations, this book will provide readers with the mathematical tools needed to be successful in their welding careers |
Second year of ESOFunctions Graphing data from a tableand graphs Describing phenomena given by a table Interpreting monotony, continuity and extremes in graphsStatistics and Discrete variablesProbability Absolute, relative and cumulative frecuencies Graphics: bars and pie charts Mean, median and mode for a few discrete data
Third year of ESONumbers Rational numbers. Comparing, ordering and graphing on a number line Decimals and fractions. Exact and periodic decimals Operations with fractions and decimals Order of operations. Brackets. Integers exponents powers. Laws. Operating in standard form. Using the calculator Rounding and errors Time and angles measuring. The sexagesimal system. Word problems involving direct and inverse proportionality. Proportional distributions Simple interest. Composed percentages |
Advanced Mathematical Concepts - 06 edition
Summary: Advanced Mathematical Concepts, 2006 provides comprehensive coverage of all the topics covered in a full-year Pre-calculus course. Its unique unit organization readily allows for semester courses in Trigonometry, Discrete Mathematics, Analytic Geometry, and Algebra and Elementary Functions. Pacing and Chapter Charts for Semester Courses are conveniently located in the Teacher Wraparound Edition.
Advanced Mathematical Concepts lessons develop mathematics us...show moreing numerous examples, real-world applications, and an engaging narrative. Graphs, diagrams, and illustrations are used throughout to help students visualize concepts. Directions clearly indicate which problems may require the use of a graphing calculator.
New Features: " A full-color design, a wide range of exercise sets, relevant special features, and an emphasis on graphing and technology invite your students to experience the excitement of understanding and applying higher-level mathematics skills. " Graphing calculator instructions is provided in the Graphing Calculator Appendix. Each Graphing Calculator Exploration provides a unique problem-solving situation. " SAT/ACT Preparation is a feature of the chapter end matter. The Glencoe Web site offers additional practice: amc.glencoe.com " Applications immediately engage your students; interest. Concepts are reinforced through a variety of examples and exercise sets that encourage students to write, read, practice, think logically, and review. " Calculus concepts and skills are integrated throughout the course. ...show less10.88 +$3.99 s/h
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The Triola Statistics TI-83/TI-84 Plus Reference
9780321399670
ISBN:
0321399676
Pub Date: 2005 Publisher: Addison-Wesley
Summary: Organized by topic, the Graphing Calculator Study Card guides students through the keystrokes needed to most efficiently use their graphing calculator. This study card includes instruction on using and entering formulas, and instructions on how to construct various types of graphs such as histograms, scatterplots, and box plots.
Addison Wesley is the author of The Triola Statistics TI-83/TI-84 Plus Reference..., published 2005 under ISBN 9780321399670 and 0321399676. Nine The Triola Statistics TI-83/TI-84 Plus Reference textbooks are available for sale on ValoreBooks.com, five used from the cheapest price of $34.88, or buy new starting at $175.71 |
Available.
Three components contribute to a theme sustained throughout the Coburn Series: that of laying a firm foundation, building a solid framework, and providing strong connections. Not only does Coburn present a sound problem-solving process to teach students to recognize a problem, organize a procedure, and formulate a solution, the text... more
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Table of Contents Exponential and Logarithmic Series Complex Quantities. De Moivre-s Theorem Expansions of sin n, and cos n.Series for sin , and cos in Powers of Expansions of Sines and Cosines of Multiple Angles and of Powers of Sines and Cosines Exponential Series for Complex Quantities Circular Functions for... more
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Check your work-and your understanding-with this manual, which provides solutions for all of the odd-numbered exercises in the text. You will also find strategies for solving additional exercises and many helpful hints and warnings. The classic in the series of highly respected Swokowski/Cole mathematics texts retains the elements that have... more
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Boiled-down essentials of the top-selling Schaum's Outline series for the student with limited time What,... more
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This brand-new review book offers high school students in New York State advance preparation for the all-new Regents Exam in Algebra 2/Trigonometry, which will be administered starting in 2010. Fourteen chapters review all exam topics and include practice exercises in each chapter. The book concludes with a sample Regents-style exam... more
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The goal of this book is to provide a solid mathematical foundation via visualization of real world data. This book uses technology as a tool to solve problems, motivate concepts, explore and preview mathematical concepts and to find curves of best fit to the data. Most mathematical concepts are developed... more
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This text uses the graphing utility to enhance the study of mathematics. Technology is used as a tool to solve problems, motivate concepts, and explore mathematical ideas. Sullivan's Series "Enhanced with Graphing Utilities" provides clear and focused coverage. Many of the problems are solved using both algebra and a graphing... more
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The Dugopolski Precalculus series for 1999 is technology optional. With this approach, teachers will be able to choose to offer either a strong technology-oriented course, or a course that does not make use of technology. For departments requiring both options, this text provides the advantage of flexibility. College Algebra and... more
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Strong Algebra and Trig skills are crucial to success in calculus. This text is designed to bolster these skills while students study calculus. As students make their way through the calculus course, this supplemental text shows them the relevant algebra or trigonometry topics and points out potential problem spots. The... more
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Over the years, the text has been shaped and adapted to meet the changing needs of both students and educators. As always, special care was taken to respond to the specific suggestions of users and reviewers through enhanced discussions, new and updated examples and exercises, helpful features, and an extensive... more
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Represents mathematics as it appears in life, providing understandable, realistic applications consistent with the abilities of any reader. This book develops trigonometric functions using a right triangle approach and progresses to the unit circle approach. Graphing techniques are emphasized, including a thorough discussion of polar coordinates, parametric equations, and conics... more
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This intermediate algebra text, based on standards in the AMATYC Crossroads document, motivates college math students to develop mathematical literacy and a solid foundation for future study in mathematics and other disciplines. This third book of a three-book series presents mathematical concepts and skills through relevant activities derived from real-life... more
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Primarily designed as a textbook, Trigonometry is a unique treatise on vectors, aiming at providing a fairly complete account of the basic concepts required to build a strong foundation for a student endeavouring to study this subject. The analytical approach to the major theories Inverse Circular Functions, De-Moivre s Theorem,... more |
Books
Geometry & Topology
The first book to discuss fractals solely from the point of view of computer graphics, this work includes an introduction to the basic axioms of fractals and their applications in the natural sciences, a survey of random fractals together with many pseudocodes for selected algorithms, an introduction into fantastic fractals such as the Mandelbrot set and the Julia sets, together with a detailed discussion of algorithms and fractal modeling of real world objects. 142 illustrations in 277 parts. 39 color plates.
Mathematics education in schools has seen a revolution in recent years. Students everywhere expect the subject to be well-motivated, relevant and practical. When such students reach higher education, the traditional development of analysis, often divorced from the calculus they learned at school, seems highly inappropriate. Shouldn't every step in a first course in analysis arise naturally from the student's experience of functions and calculus in school? And shouldn't such a course take every opportunity to endorse and extend the student's basic knowledge of functions? In Yet Another Introduction to Analysis, the author steers a simple and well-motivated path through the central ideas of real analysis. Each concept is introduced only after its need has become clear and after it has already been used informally. Wherever appropriate, new ideas are related to common topics in math curricula and are used to extend the reader's understanding of those topics. In this book the readers are led carefully through every step in such a way that they will soon be predicting the next step for themselves. In this way students will not only understand analysis, but also enjoy it.
The first monograph to treat topological, group-theoretic, and geometric problems of ideal hydrodynamics and magnetohydrodynamics from a unified point of view. It describes the necessary preliminary notions both in hydrodynamics and pure mathematics with numerous examples and figures. The book is accessible to graduates as well as pure and applied mathematicians working in hydrodynamics, Lie groups, dynamical systems, and differential geometry.
This book comprises a broad selection of expository articles that were written in conjunction with an international conference held to honor F.W. Gehring on the occasion of his 70th birthday. The objective of both the symposium and the present volume was to survey a wide array of topics related to Gehring's fundamental research in the field of quasiconformal mappings, emphasizing the relation of these mappings to other areas of analysis. The book begins with a short biographical sketch and an overview of Gehring's mathematical achievements, including a complete list of his publications. This is followed by Olli Lehto's account of Gehring's career-long involvement with the Finnish mathematical community and his role in the evolution of the Finnish school of quasiconformal mapping. The remaining articles, written by prominent authorities in diverse branches of analysis, are arranged alphabetically. The principal speakers at the symposium were: Astala, Baernstein Earle, Jones, Kra, Lehto, Martin, Sullivan, and Va"isa"la". Other individuals, some unable to attend the conference, were invited to contribute articles to the volume, which should give readers new insights into numerous aspects of quasiconformal mappings and their applications to other fields of mathematical analysis. Friends and colleagues of Professor Gehring will be especially interested in the personal accounts of his mathematical career and the descriptions of his many important research contributions.
Complex Analysis with Mathematica offers a new way of learning and teaching a subject that lies at the heart of many areas of pure and applied mathematics, physics, engineering and even art. This book offers teachers and students an opportunity to learn about complex numbers in a state-of-the-art computational environment. The innovative approach also offers insights into many areas too often neglected in a student treatment, including complex chaos and mathematical art. Thus readers can also use the book for self-study and for enrichment. The use of Mathematica enables the author to cover several topics that are often absent from a traditional treatment. Students are also led, optionally, into cubic or quartic equations, investigations of symmetric chaos, and advanced conformal mapping. A CD is included which contains a live version of the book, and the Mathematica code enables the user to run computer experiments.
The recent introduction of the Seiberg-Witten invariants of smooth four-manifolds has revolutionized the study of those manifolds. The invariants are gauge-theoretic in nature and are close cousins of the much-studied SU(2)-invariants defined over fifteen years ago by Donaldson. On a practical level, the new invariants have proved to be more powerful and have led to a vast generalization of earlier results. This book is an introduction to the Seiberg-Witten invariants.
The work begins with a review of the classical material on Spin c structures and their associated Dirac operators. Next comes a discussion of the Seiberg-Witten equations, which is set in the context of nonlinear elliptic operators on an appropriate infinite dimensional space of configurations. It is demonstrated that the space of solutions to these equations, called the Seiberg-Witten moduli space, is finite dimensional, and its dimension is then computed. In contrast to the SU(2)-case, the Seiberg-Witten moduli spaces are shown to be compact. The Seiberg-Witten invariant is then essentially the homology class in the space of configurations represented by the Seiberg-Witten moduli space. The last chapter gives a flavor for the applications of these new invariants by computing the invariants for most Kahler surfaces and then deriving some basic toological consequences for these surfaces.
The object of this book is two-fold -- on the one hand it conveys to mathematical readers a rigorous presentation and exploration of the important applications of analysis leading to numerical calculations. On the other hand, it presents physics readers with a body of theory in which the well-known formulae find their justification. The basic study of fundamental notions, such as Lebesgue integration and theory of distribution, allow the establishment of the following areas: Fourier analysis and convolution Filters and signal analysis time-frequency analysis (gabor transforms and wavelets). The whole is rounded off with a large number of exercises as well as selected worked-out solutions. |
Detailed Course Information
When the term course schedule is available the "Schedule Types" are hyperlinks. Clicking on the link will then display individual section information.
MATH 112 - Intermediate Algebra
Graphing calculator required. This is an extension of beginning algebra with emphasis on graphing and diverse, real-life applications. Also emphasized are functions & relations, polynomial, rational, radical, and quadratic expressions and equations, rational exponents, and absolute value equations and inequalities with an introduction to complex numbers. (F,Sp,Su)
Course Note: MATH 112 is usually offered in 4 different delivery methods: Online, Online/Hybrid, Lecture, Learning Lab. For an explanation of the Learning Lab delivery method, refer to and select What is a Learning Lab?, or call (517) 483-1073 and press 4. The course fee provides all enrolled students online access to a full electronic textbook (e-text), online course homework system, and additional online resources. These course materials will be available to students the first day of class. Students are encouraged to speak with their instructor BEFORE purchasing any optional course materials. A graphing calculator from the TI-83/84/Nspire family is required for course work. |
Ripples in Mathematics The Discrete Wavelet Transform
9783540416623
ISBN:
3540416625
Pub Date: 2001 Publisher: Springer Verlag
Summary: This book gives an introduction to the discrete wavelet transform and some of its applications. It is based on a novel approach to discrete wavelets called lifting. The first part is a completely elementary introduction to the subject, and the prerequisites for this part are knowledge of basic calculus and linear algebra. The second part requires some knowledge of Fourier series and digital signal analysis. The conne...ctions between lifting and filter theory are presented and the wavelet packet transforms are defined. The time-frequency plane is used for interpretation of signals. The problems with finite length signals are treated in detail. MATLAB is used as the computational environment for examples and implementation of transforms. The book is well suited for undergraduate mathematics and electrical engineering students and engineers in industry.
Jensen, A. is the author of Ripples in Mathematics The Discrete Wavelet Transform, published 2001 under ISBN 9783540416623 and 3540416625. Six hundred sixty eight Ripples in Mathematics The Discrete Wavelet Transform textbooks are available for sale on ValoreBooks.com, one hundred eight used from the cheapest price of $47.24, or buy new starting at $48 |
MA 125 Intermediate Algebra Diaz, Richard My educational philosophy is based on the active involvement of the student in all aspects of the learning process. Engaging students through lecture, classroom discussions, group work, reading, writing and assessments are important components of the learning process. A student is required to learn and practice basic reasoning and problem solving skills. The textbook must be read and examples worked out in order to be able to accomplish the homework. Lectures and in class work should not be the only thing that the student strictly relies on. While working on homework problems, the student should try to grasp the concepts in small sections at a time for deep learning. Each student will be responsible for the comprehension, application, analysis, synthesis and evaluation of presented information and materials. Feedback will be provided for each chapter as the course progressesSolve equations involving radicals
Apply the method of completing the square
Apply the quadratic formula
Graph algebraic equations and inequalities of one and two variables.
Instructor Learning Outcomes
Explain and specify methods used to manipulate equations algebraically.
Classify and differentiate between Linear, Quadratic functional relationships from non-functions.
Apply and evaluate methods for the quadratic formula.
Organize and construct graphs of equations and inequalities of one and two variables.
Late Submission of Course Materials: Students must contact Instructor for submission of any late work, usually only when emergencies come up.
Classroom Rules of Conduct: Students will act appropriate and professional. Students will create and maintain a learning environment. Cell phones should be turned on vibrate before entering the classroom. If you must take a call, I expect you to leave the classroom while you do so. I consider it academic misconduct for you to be using your cell phone or other electronic equipment while in class. Respect others at all times.
Course Topic/Dates/Assignments:
Week 1
Introduction
Chapter 1: Review of the Real Number System
Chapter 2: Linear Equations, Inequalities, and Applications
HW 1
Week 2
Chapter 3: Graphs, Linear Equations, and Functions
Chapter 4: Systems of Linear Equations
HW 2
Quiz 1 Ch 1-2
Week 3
Chapter 5: Exponents, Polynomials and Polynomial Functions
Quiz 2 Ch 3-4
Week 4
Midterm Chapters 1-5
Week 5
Chapter 6 Factoring
Chapter 7 Rational Expressions and Functions
Chapter 8 Roots, Radicals, and Root Functions
HW 3
Week 6
Chapter 8: Roots, Radicals, and Root Functions (continued)
Quiz 3 Ch 6-8
HW 4
Week 7
Chapter 9: Quadratic Equations, Inequalities, and Functions
Week 8
Final Chapters 1-9 operations |
Number Theory
Number Theory, which deals with properties of the positive integers, is one of the oldest branches of mathematics. Many of its problems are very easy to understand, but some such as Fermat's famous "Last Theorem" are devilishly hard to solve. In recent years, old ideas have found practical applications.
This course provides an introduction to the important basic topics of number theory: prime numbers, factorisation, congruence and diophantine equations. These topics are treated from a modern point of view, emphasising the underlying algebraic structure. They provide the necessary background for a brief introduction to modern cryptography.
Available in 2014
Callaghan Campus
Semester 2
Previously offered in 2013, 2012, 2011, 2010, 2009, 2007, 2005
Objectives
On successful completion of this course, students will be able to:
1. hold an in-depth knowledge of a primary branch of mathematics 2. demonstrate how many of the abstract ideas they have previously studied can be used 3. develop problem-solving and communication skills |
More About
This Textbook
Overview
The first student-centred guide on how to write projects and case studies in mathematics, with particular attention given to working in groups (something maths undergraduates have not traditionally done). With half of all universities in the UK including major project work of significant importance, this book will be essential reading for all students on the second or final year of a mathematics degree, or on courses with a high mathematical content, for example, physics and engineering.
Editorial Reviews
From the Publisher
From the reviews:
"More than 20 years of the author's experience in supervising and assessing mathematical projects have resulted in this book, the title of which can hardly be more eloquent. … The features of individual projects, group projects and case studies are carefully described … . It is a sheer joy reading it and appreciating the authors' refined style and humor. From the viewpoint of students, supervisors, as well as of assessors, this book must be unanimously marked as excellent." (EMS Newsletter, September, 2005)
"The text does discuss fully the supervision and assessment of both pure and applied mathematics projects, and, for me personally, the level of detail on this aspect has been the most useful feature of the book. … Academic staff will certainly find this book relevant. For departments considering introducing final year projects based on mathematics, Managing mathematical projects gives invaluable guidelines. For departments already running such projects, the book is both useful and stimulating." (David Hood, The Mathematical Gazette, Vol. 91 (520 |
Directories
LaCurts, Carvel
MATH 130-001 130-0022014 Fall003 230-001
MATH 230-002 |
books.google.com - Every...
Mathematics
Every of the NJ Department of Education,this fourth edition of our popular test prep provides the up-to-date instruction and practice that eighth grade students need to improve their math skills and pass this important high-stakes exam.The comprehensive review features student-friendly, easy-to-follow lessons and examples that reinforce the key concepts tested on the NJ ASK8, including:
Focused lessons explain math concepts in easy-to-understand language that's suitable for eighth grade students at any learning level. Our tutorials and targeted drills increase comprehension while enhancing math skills. Color icons and graphics throughout the book highlight practice problems, charts, and figures.The book includes a full-length diagnostic test and a full-length practice test based on official exam questions.Each test comes complete with detailed explanations of answers, allowing you to focus on areas in need of further study.Our interactive TestWare CD features both of the book's tests in a timed format with automatic scoring, on-screen answer explanations, and diagnostic feedback.The TestWare CD contains the most powerful scoring and diagnostic tools available today. Automatic scoring and instant reports help you zero in on the topics and types of questions that give you trouble now, so you'll succeed when it counts! REA's test-taking tips and strategies offer an added boost of confidence and ease anxiety before the exam.Whether used in a classroom, at home for self-study, or as a textbook supplement, teachers, parents, and students will consider this book a "must-have" prep for the NJ ASK 8 Mathematics exam.
REA test preps have proven to be the extra support students need to pass their challenging state-required tests. Our comprehensive test preps are teacher-recommended and written by experienced educators.
About the author (2010)
Mel Friedman is Lead Mathematics Editor at REA. After teaching mathematics at the high school level for twelve years, he went on to teach at a number of colleges and universities, the most recent being Kutztown University in Kutztown, Pa. and West Chester University in West Chester, Pa. In addition to his work as a math consultant, he served as a test-item writer for Educational Testing Service and ACT, Inc. He received a Bachelor of Arts degree in math from Rutgers University, and an M.S. in math from Fairleigh Dickinson University.
Stephen Hearne is a professor at Skyline College in San Bruno, California, where he teaches Quantitative Reasoning. He earned his Ph.D. degree from the University of Mississippi in 1998, specializing in Quantitative Psychology. For the past twenty years, Dr. Hearne has tutored students of all ages in math, algebra, statistics, and test preparation. He prides himself in being able to make the complex simple.
Penny Luczak received her B.A. in Mathematics from Rutgers University and her M.A. in Mathematics from Villanova University. She is currently a full-time faculty member at Camden County College in New Jersey. She previously worked as an adjunct instructor at Camden County College as well as Burlington County College and Rutgers University. |
Learn how to think the way mathematicians do - a powerful cognitive process developed over thousands of years.
Course requirements
High school mathematics. Specific requirements are familiarity with elementary symbolic algebra, the concept of a number system (in particular, the characteristics of, and distinctions between, the natural numbers, the integers, the rational numbers, and the real numbers), and some elementary set theory (including inequalities and intervals of the real line). Students whose familiarity with these topics is somewhat rusty typically find that with a little extra effort they can pick up what is required along the way. The only heavy use of these topics is in the (optional) final two weeks of the Extended Course. A good way to assess if your basic school background is adequate (even if currently rusty) is to glance at the topics in the book Adding It Up: Helping Children Learn Mathematics (free download), published by the US National Academies Press in 2001. Though aimed at K-8 mathematics teachers and teacher educators, it provides an excellent coverage of what constitutes a good basic mathematics education for life in the Twenty-First Century (which was the National Academies' aim in producing it).
27 Reviews for Introduction to Mathematical Thinking
I graduated from 2 of the most prestigious universities in the US and over the years I've watched a few dozens online classes and I have to say Intro to Math Thinking is one of the best classes anywhere, online or off. Devlin is imaginative as a teacher. The way he approaches math is so creative that it's a joy to watch his lectures. This class is more than about math, it's about proper thinking and reasoning. In addition, Devlin cares about teaching (and your learning) and it shows throughout his lectures, and especially when he explains the solutions to the exercises.
The first and still best course I have taken online. Keith Devlin inspires beyond belief. His little companion book was amazingly helpful as well. If you are willing to work and think hard, then this course has a ton to offer. Thank you, Prof. Devlin!
The pros: Learned a lot about the construction of proofs (especially while taking the final). There were a lot of ungraded sample problems to work from.
the cons: Some video clips were particularly long (40 minutes if not more). In addition, it seemed fast paced at times (good thing they are considering making a 10 weeks version a second time round). it would have been helpful if the Notes that the Instructor had written were provided in pdf form (The Cons are mostly concerned with logistics)
The Instructor: Simply great; Has high charisma and knows how to make the course very engaging.(still would have preferred if the vids were shorter though) |
Discrete Mathemetics
9780618415380
0618415386
Summary: Discrete Mathematics combines a balance of theory and applications with mathematical rigor and an accessible writing style. The author uses a range of examples to teach core concepts, while corresponding exercises allow students to apply what they learn. Throughout the text, engaging anecdotes and topics of interest inform as well as motivate learners. The text is ideal for one- or two-semester courses and for studen...ts who are typically mathematics, mathematics education, or computer science majors. Part I teaches student how to write proofs; Part II focuses on computation and problem solving. The second half of the book may also be suitable for introductory courses in combinatorics and graph theory.
Ferland is the author of Discrete Mathemetics, published 2008 under ISBN 9780618415380 and 0618415386. Three hundred ninety eight Discrete Mathemetics textbooks are available for sale on ValoreBooks.com, one hundred twenty one used from the cheapest price of $65.57, or buy new starting at $184.60 |
Building on introductory calculus courses, this text provides a sound foundation in the underlying principles of ordinary differential equations. Important concepts, including uniqueness and existence theorems, are worked through in detail and the student is encouraged to develop much of the routine material themselves, thus helping to ensure a solid understanding of the fundamentals required.
Exact solutions of differential equations continue to play an important role in the understanding of many phenomena and processes throughout the natural sciences in that they can verify the correctness of or estimate errors in solutions reached by numerical, asymptotic, and approximate analytical methods. The new edition of this bestselling handbook now contains the exact solutions to more than 6200 ordinary differential equations. |
Precalculus : Funcations and Graphs - 4th edition
Summary: Dugopolski'sPrecalculus: Functions and Graphs, Fourth Edition, gives you the essential strategies you need to make the transition to calculus. Throughout this book, you will find carefully placed learning aids and review tools to help them learn the math without getting distracted. The new edition includes over 900 additional exercises that are specifically designed to increase student understanding and retention of the concepts. Along the way, you'll see how the algebra conne...show morects to your future calculus course, with tools like Foreshadowing Calculus and Concepts of Calculus. Dugopolski's emphasis on problem solving and critical thinking helps you be successful in this course, as well as in future calculus courses new instructor's edition. May contain answers and/or notes in margins. Ships same day or next business day. Free USPS Tracking Number. Excellent Customer Service. Ships from TN
$115116.35 +$3.99 s/h
VeryGood
BookCellar-NH Nashua, NH
0321789431119120157177 |
Mathematics : a very short introduction
".再讀一些...
摘要:
This book aims to explain, in clear non-technical language,what it is that mathematicians do, and how that differs from and builds on the mathematics that most people are familiar with from school. It is the ideal introduction for anyone who wishes to deepen their understanding of mathematics.再讀一些...
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49936189
0192853619 9780192853615 0199504687 9780199504688
Mathematics : a very short introduction/Timothy Gowers; Oxford ; New York : Oxford University Press, 2002. |
Schaum's Outlines present all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills.
Suitable for an introductory combinatorics course lasting one or two semesters, this book includes an extensive list of problems, ranging from routine exercises to research questions. It walks the reader |
Algebra
You are here
Algebra is full expressions and equations and of course, all the cookies the will turn out perfectly when you know how to add up the ingredients. We can help you with your polynomials, offer reminders about the order of operations and walk you through making a graph with the website and videos here. If story problems worry you or you've wondered when you will ever use math in real life, look no furtherWorld Book publishes reference books on many subjects. Examples are World Book Online Reference Center, World Book Online for Kids, and Enciclopedia Estudiantil Hallazgos. This resource lets you access all of them with one click. You can also access them one at a time if you like.
One-to-one help writing resumes and finding a job, and high-quality after-school help from expert tutors in more than 20 subjects to prepare students for college or just get through the evening's assignment. Live one-to-one help is available 2 pm to 10 pm daily and in English, Spanish, and Vietnamese. |
The Secondary Mathematics programme at YCIS Shanghai is designed to prepare students for the IGCSE Mathematics and International Baccalaureate Mathematics exams.
The Key Stage 3 Programme (Years 7 to 9) develops skills in a spiral, each year repeating numeric, algebraic, geometric and trigonometric topics by extending and enriching the foundation set in the previous years. Statistics and probability are also introduced. Various projects are explored which give the students the opportunity to view how the mathematics they have learned in the classroom is applicable to the real world, bringing to light its significance to them. Often such activities extend from the topics of geometry and statistics by means of investigating tessellations and volume and by performing surveys that generate data for statistical analysis.
In Years 10 and 11, students build on the knowledge they have acquired in the previous stage to prepare for the IGCSE exams taken at the end of the two-year programme. Knowing that students will continue on to a more rigorous IB programme, students extend beyond the IGCSE curriculum with problem-solving that demand a more in-depth exploration of mathematics. The topics previously studied are those from Key Stage 3: numeracy, algebra, geometry, trigonometry, statistics, and probability.
In Years 12 and 13, students choose from three IB Mathematics courses, each challenging the student at the appropriate level: Higher Level, Mathematical Methods, and Mathematical Studies. Each course, in addition to providing exposure to many advanced topics, requires students to explore projects and activities that require an advanced mathematics level, which will be evaluated by an international examiner. IB courses reflect the topics mentioned in the earlier stages with the inclusion of calculus for the Higher Level and Mathematical Methods courses. |
Calculus II –
mth320
(4 credits)
This course examines integral calculus topics. Students are presented with integration techniques for functions of one variable and more applications of definite integrals. Students explore numerical techniques of integration. Students also examine the area function, Riemann sums and indefinite integrals, and apply these to real-life problems. The course concludes with the fundamental theorem of calculus.
Use integral calculus to determine volumes, lengths of plane curves, and surface areas.
Solve indefinite integrals.
Describe the relationship between derivatives and integrals (Fundamental Theorem of Calculus).
Integration
Solve improper integrals.
Estimate integrals with the Trapezoidal rule and Simpson's rule.
Use integral tables to solve problems.
Use partial fractions to simplify integration of rational functions |
Math Center Mission
The Math Center is designed specifically to support and encourage students enrolled in mathematics courses. Our goal is not only to help the students learn the mathematics as it is found in their textbooks, but also to delve deeper into the underlying mathematical concepts. The deeper understanding of the material will help develop deep critical thinking skills, in addition to enhanced understand of logic and algorithms. An additional focus of the Math Center is to help students overcome math anxiety. See the Resources page for a description of the tools used in the Math Center to reach these goals. |
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Introduction to the Foundations of Mathematics: Second Edition by Raymond L. Wilder Classic undergraduate text acquaints students with fundamental concepts and methods of mathematics. Topics include axiomatic method, set theory, infinite sets, groups, intuitionism, formal systems, mathematical logic, and much more. 1965 second editionThe Elements of Mathematical Logic by Paul C. Rosenbloom This excellent introduction to mathematical logic provides a sound knowledge of the most important approaches, stressing the use of logical methods. "Reliable." — The Mathematical Gazette. 1950 editionProduct Description:
history of the axiom in English, and is much more complete than the two other books on the subject, one in French and the other in Russian. This book covers the Axiom's prehistory of implicit uses in the 19th century, its explicit formulation by Zermelo in 1904, the firestorm of controversy that it caused — in England, France, Germany, Italy, and the U.S. — its role in stimulating his axiomatization of set theory in 1908, and its proliferating uses all over mathematics throughout the 20th century. The book is written so as to be accessible to the advanced mathematics undergraduate, but equally to be informative and stimulating to the professional mathematician. Most technical terms are defined in footnotes, making it accessible by students of the philosophy of mathematics as well. This new edition has an expanded bibliography and a new preface examining developments since its original 1982 publication |
Secondary Solutions
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Our unique approach involves splitting instructional time between student-centered instruction in the classroom and self-paced learning using our adaptive software. Students spend 60% of the time in the classroom using our textbooks in an approach that involves task-based lessons, collaborative learning, and real-world problems and contexts. The remaining 40% of students' time is spent learning via our Cognitive Tutor software, which offers the most precise method for differentiating instruction available.
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Mathematics for Elementary Teachers: A Conceptual Approach
9780073519579
ISBN:
007351957X
Publisher: McGraw-Hill
Summary: Would you like to rent Mathematics for Elementary Teachers: A Conceptual Approach online from Valore Books now? If you would like to take advantage of discounted prices on pre-owned copies of this book published by McGraw-Hill, look at our selection now. Written by Albert B Bennett, Laurie J Burton and Leonard T Nelson, you can find the cheapest copies of this text book by using our site now. Buy Mathematics for Elem...entary Teachers: A Conceptual Approach online from us today and find out why so many people rent and buy books for college from us. Try our website now for the cheapest deals.
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Mathematical Ideas
9780321168085
ISBN:
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Summary: Covering a variety of mathematical ideas, this text includes information on: the art of problem solving; the basic concepts of set theory; logic; numeration and mathematical systems; and number theory.
Miller, Charles David is the author of Mathematical Ideas, published 2003 under ISBN 9780321168085 and 0321168089. One hundred twenty Mathematical Ideas textbooks are available for sale on ValoreBooks.com, nin...ety nine used from the cheapest price of $0.01, or buy new starting at $42 |
Mathematics Of Voting And Elections A Hands-on Approach
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Pub Date: 2005 Publisher: American Mathematical Society
Summary: The results of an election depend not just upon the wishes of the electorate, but also upon the mathematics used to calculate the result. Using numerous case studies & featuring discussions of actual elections from the perspectives of both politics & popular culture, this text explores vote counting systems.
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College Algebra - 9th edition
Mike Sullivan'stime-tested approach focuses students on the fundamental skills they need for the course:preparingfor class,practicingwith homework, andreviewingthe concepts. In theNinth Edition,College Algebrahas evolved to meet today's course needs, building on these hallmarks by integrating projects and other interactive learning tools for use in the classroom or online109110.55114.40 +$3.99 s/h
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Prerequisite: Part of Junior block courses; application and admission to School of Education
A study of teaching techniques and materials used in contemporary Elementary mathematics programs. Focus on skills of inquiry and deductive reasoning through hands-on work with experiments and manipulatives. Content strands: Data and Chance; Geometry; Measurement and Reference Frames; Numeration; Operations and Computation; and Patterns, Functions, and Algebra. Themes: Algorithmic and Procedural Thinking; Estimation Skills and Number Sense; Mental Arithmetic Skills; Reasoning and Proof; Communication; Connections; and problem-solving. Study of issues related to assessment, student diversity, and organizing for teaching mathematics in a K-6 setting. Integration with reading, language arts, social sciences, health, nutrition, and the visual and performing arts. |
8 of the Best Free Linux Geometry Software
In the field of mathematical software packages, applications
such as Wolfram Research's Mathematica, and Maplesoft's Maple
system instantly spring to mind. These are both highly popular,
proprietary,
commercial, integrated mathematical software environments. Other
types of mathematical software packages generally receive much less
publicity.
One such area is interactive geometry software, which combines
three branches of mathematics: geometry, calculus and algebra. This
type of software allows users to create and modify constructions, which
are generally in plane geometry. Construction involves
building mathematical shapes out of points, lines, conic sections,
hyperbola, ellipses, and circles. These diagrams can then be altered
and the effects of the mathematical properties of the shapes can be
observed.
Typically geometry software covers a wide range of application
areas, including pure Euclidean and non-Euclidean geometry,
computer-aided design, and computational kinematics. It is
often found being used for learning and teaching mathematics in schools
and colleges and for research purposes.
To provide an insight into the quality of software that is
available, we have compiled a list of 8 free high quality Linux
interactive geometry software. We have included 2D and 3D software.
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students and teachers alike.
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Book Description:PRENTICE HALL, 2008. Hardcover. Book Condition: New. MULT. COPIES! This is the NJ edition! Same as the national except for the first few pages with state specific material in the beginning of the book. The national edition begins thereafter - all page#'s match up. Brand new, never used. We ship daily!. Bookseller Inventory # AMAN0005763Hardcover. Book Condition: New. Hardcover. Algebra success for allBasic concepts and properties of algebra are introduced early to prepare students for equation solving. Abundant exercises graded by difficulty level address a wide range of student abilities. The Basic Algebra Planning Guide assures that even the at-risk student can acquire course content. Multiple representations of conceptsConcepts and skills are introduced algebraically, graphically, numerically, and verbally-often in the same lesson to help students make the connection and to address diverse learning styles. Focused on developing algebra concepts and skillsKey algebraic concepts are introduced early and opportunities to develop conceptual understanding appear throughout the text, including in Activity Labs. Frequent and varied skill practice ensures student proficiency and success. This item ships from multiple locations. Your book may arrive from Roseburg,OR, La Vergne,TN, Momence,IL, Commerce,GA. book. Bookseller Inventory # 97801336594 |
chaum's Easy Outlines of College Mathematics
If you are looking for a quick nuts-and-bolts overview, turn to Schaum's Easy Outlines! "Schaum's Easy Outline of College Mathematics" is a pared ...Show synopsisIf you are looking for a quick nuts-and-bolts overview, turn to Schaum's Easy Outlines! "Schaum's Easy Outline of College Mathematics" is a pared-down, simplified, and tightly focused review of the topic. With an emphasis on clarity and brevity, it features a streamlined and updated format and the absolute essence of the subject, presented in a concise and readily understandable form. Graphic elements such as sidebars, reader-alert icons, and boxed highlights stress selected points from the text, illuminate keys to learning, and give you quick pointers to the essentials. Expert tips for mastering college mathematics Last-minute essentials to pass the course Easily understood review of college mathematics Supports all the major textbooks for college mathematics courses Appropriate for the following courses: Introduction to Mathematical Modeling, Pre-Calculus, Discrete Mathematics, Trigonometry, College Algebra, Calculus Schaum's Easy Outline of College Mathematics, Revised...Good. Schaum's Easy Outline of College Mathematics, Revised Edition (Schaum's Easy Outlines |
CSUSM Math Club
The purpose of the Math Club is promote math enthusiasm and to be a tool to inspire mathematical interest amongst students. The Math Club will also be an opportunity to meet and interact with fellow students interested in mathematics. |
I'm far from the level of mathematical knowledge every user on this website posseses, however I am very much determined to get there as my love for mathematics increases. These are the topics:
Thanks, Im not from US so I don't know what grade level this stuff is. However it is a further maths course here are resources for these topics are either scarce or of poor quality. I would like to improve my knowledge of mathematics, and not just learn to pass an exam.
Simarly, is there a solid text I can use to self teach statistics? I have to cover basic topics like deviation/variance all the way to harder probability.
2 Answers
Try looking first at Khan Academy for help with stats and linear algebra (matrices), as well as to review on the fundamentals, which you'll need to fully grasp in order to successfully master more advanced topics. Another helpful resource for learning linear algebra and differential equations, and for reviewing the "basics" is Paul's Online Math Notes and tutorials.
Then perhaps you'd like to explore MIT's Open Courseware - Mathematics for access to classes and topics of interest and where you'll also learn which texts are used for the available classes. Often, course notes, videos of lectures, and exams are available to assist learners, all free of charge.
All great references, so I'll just add something on this answer. Schaum's collection is a really good way to learn too, and it covers from basic topics to advanced ones in a lot of areas of math.
–
Ivan LernerNov 26 '12 at 18:44 |
Precalculus : Funcations and Graphs - 4th edition
Summary: Dugopolski'sPrecalculus: Functions and Graphs, Fourth Edition, gives you the essential strategies you need to make the transition to calculus. Throughout this book, you will find carefully placed learning aids and review tools to help them learn the math without getting distracted. The new edition includes over 900 additional exercises that are specifically designed to increase student understanding and retention of the concepts. Along the way, you'll see how the algebra conne...show morects to your future calculus course, with tools like Foreshadowing Calculus and Concepts of Calculus. Dugopolski's emphasis on problem solving and critical thinking helps you be successful in this course, as well as in future calculus |
ALEX Lesson Plans
Title: Exponential Growth and Decay
Description:
This ALC (9-12) 3: Use formulas or equations of functions to calculate outcomes of exponential growth or decay. (Alabama) [MA2013] ALT (9-12) 12: Interpret expressions that represent a quantity in terms of its context.* [A-SSE 25: Compare effects of parameter changes on graphs of transcendental functions. (Alabama)
Subject: Mathematics (9 - 12) Title: Exponential Growth and Decay Description: This
Thinkfinity Lesson Plans
Title: Building Bridges
Description:
In
Standard(s):.* 45: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. [S-ID6]
Subject: Mathematics,Professional Development Title: Building Bridges Description: In Thinkfinity Partner: Illuminations Grade Span: 6,7,8
Title: Graphing What
Description:
This reproducible activity sheet, from an Illuminations lesson, is used by students to record independent and dependent variables as well as the function and symbolic function rule for a set of graphs.
Standard(s): [MA2013] (6) 17: Use variables to represent numbers, and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number or, depending on the purpose at hand, any number in a specified set. [6-EE6] [MA2013] (6) 20: [6-EE9 7: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. [8-EE5] [MA2013] (8) 11: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.) [8-F1] [MA2013] (8) 13: Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear. [8-F3] 25: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). [F-IF1] [MA2013] AL1 (9-12) 26: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. [F-IF Graphing What Description: This reproducible activity sheet, from an Illuminations lesson, is used by students to record independent and dependent variables as well as the function and symbolic function rule for a set of graphs. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Title: Shedding the Light
Description:
In
Standard(s): DM1 (9-12) 3: Use the recursive process and difference equations to create fractals, population growth models, sequences, series, and compound interest models. (Alabama)
Subject: Mathematics,Science Title: Shedding the Light moreTitle: Exact Ratio
Description:
This
Standard(s): [MA2013] AL1 (9-12) 2: Rewrite expressions involving radicals and rational exponents using the properties of exponents. [N-RN Mathematics Title: Exact Ratio Description: This Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12 |
Every year, thousands of students go to university to study mathematics (single honours or combined with another subject). Many of these students are extremely intelligent and hardworking, but even the best will, at some point, struggle with the demands of making the transition to advanced mathematics. Some have difficulty adjusting to independent... more...,... more...
Welcome back to Ian Stewart's magical world of mathematics! This is a strange world of never-ending chess games, empires on the moon, furious fireflies, and, of course, disputes over how best to cut a cake. Each quirky tale presents a fascinating mathematical puzzle -- challenging, fun, and also introducing the reader to a significant mathematical... more...
An accessible text introducing algebraic geometries and algebraic groups at advanced undergraduate and early graduate level, this book develops the language of algebraic geometry from scratch and uses it to set up the theory of affine algebraic groups from first principles.Building on the background material from algebraic geometry and algebraic groups,... more...
Telerik Reporting is a lightweight reporting solution for all .NET cloud, web, and desktop platforms (Azure, Silverlight, WPF, ASP.NET, and Windows Forms) which targets developers and end users alike. Rich interactive and reusable reports can be created by developers in Visual Studio and by end users in the desktop-based Report Designer. This book |
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The Elements of Non-Euclidean Geometry by D. M.Y. Sommerville Renowned for its lucid yet meticulous exposition, this classic allows students to follow the development of non-Euclidean geometry from a fundamental analysis of the concept of parallelism to more advanced topics. 1914 edition. Includes 133 figures.
Undergraduate Topology by Robert H. Kasriel This introductory treatment is essentially self-contained and features explanations and proofs that relate to every practical aspect of point set topology. Hundreds of exercises appear throughout the text. 1971 edition.
A Concept of Limits by Donald W. Hight An exploration of conceptual foundations and the practical applications of limits in mathematics, this text offers a concise introduction to the theoretical study of calculus. Many exercises with solutions. 1966 edition.
Problems and Solutions in Euclidean Geometry by M. N. Aref, William Wernick Based on classical principles, this book is intended for a second course in Euclidean geometry and can be used as a refresher. More than 200 problems include hints and solutions. 1968 edition.
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focus on congruence classes defined by transformations in real Euclidean space. As the number of permitted transformations increases, these classes become larger, and their common topological properties become intuitively clear. Chapters 4–12 give a largely intuitive presentation of selected topics. In the remaining five chapters, the author moves to a more conventional presentation of continuity, sets, functions, metric spaces, and topological spaces. Exercises and Problems. 101 black-and-white illustrations. 1974 |
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Description:
Vector Independence, Span, Basis, and Dimension -- Lecture 9. Learn when vectors are independent or dependent, what a vector span is, what a basis is, and how to find the dimension.
Additional Resources:
Questions answered by this video:
What does it mean for vectors to be independent?
When are vectors dependent?
What does it mean for vectors to span a space?
What is a basis for a space?
How do you find the dimension in linear algebra?
Staff Review:
This video is a great video for learning about vector independence and dependence, and how to know whether vectors are independent or dependent. It also discusses what a vector span is, a basis, and how to find their dimension. A very diversified and full lesson. There is a small error in this video that is corrected in lecture 10. |
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From the kitchen table to the playground, children are intrigued by their world. The TI-15 is a pedagogically sound tool that helps students make connections between classroom learning and real-world situations.The TI-15 combines the fraction capabilities of the Math Explorer with a two-line display, problem solving, place value and more. When the TI-15 is combined with traditional learning ...Ideal for the algebra classroom. Lets students graph and compare functions, as well as perform data plotting and analysis. Horizontal and vertical split screen options. Advanced functions accessed through pull-down display menus. Includes tools for finance. I/O port for communication with other TI products.Helps students develop skills in addition, subtraction, powers, and answer format. Makes connections between classroom learning and real world situations. Backspace and edit capabilities. Operates in well-lighted areas using solar and in other light settings using battery. Color: Blue |
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Publications
Major Publications
Busynessgirl.comcontains more than a thousand short articles about teaching ideas, resources for learning, tutorials for using technology, teaching in the digital age, game design, and general adventures. This might not be considered a "scholarly" publication, but it was a heck of a lot of work and gets viewed by thousands of people every week.
Algebra Activities, a 1000-page Instructor resource binder of activities and teaching guides for algebra, published January 2010 by Cengage Learning. This is accompanied by approximately 30 individual Algebra Activities student workbooks for the entire Cengage Learning Algebra textbook line. Visit the Cengage Community Site for Algebra Activities.
Instructor Resource Binders and Workbooksfor Tussy/Gustafson Algebra Series, Andersen, M. H., Cengage Learning, 2009.
An extensive collection of teaching guides, assessments, and concept-oriented activities for the algebra classroom. This set of publications includes three workbooks and three Instructor Resource Binders. Sample Chapter or Teasers to Try (these may be used in your classrooms free of charge) |
FAQ - Math
Q:What can I do with a degree in math?
A: As you know, many students major in mathematics because they want to become math teachers, either at a high school level or at the college or university level. What you may not know is that SJSU math alumni also work in industries ranging from aviation safety to risk management to financial planning to satellite design. In fact, one of the three co-founders of Oracle, Edward Oates ( ), graduated from SJSU with a BA Math degree.
You will find more information on math careers the MAA website, .
Q:Are there any requirements if I want to change my major to math ? A:Yes. If you are interested in changing your major to math, please fill out the online "Contact Us" form and request to see an advisor. After you have talked to an advisor,
you need to fill out the change of major form found at the registrar's website and get department approval from Dr. Blockus.
Please note, the requirements to change your major to math are as follows:
A C- or better in Calculus I (Math 30 or Math 30P) or AP credit for Calculus I;
A C- or better in Discrete Math (Math 42);
A 2.0 GPA or higher in all mathematics courses taken, Precalculus (Math 19) and above.
Q: What do discrete and continuous math mean?
A:Discrete mathematics deals with the mathematics of countable things. A countable set is one whose elements can be put in one-to-one correspondence with the positive integers. They can be counted - hence the term countable. Finite sets are countable. For example, if you are dealing with a variable, say X, which takes on the values of the positive even integers {2, 4, 6, 8, ....}, then it is a discrete variable. However, if your variable X can take on any value between 0 and 1, then it is considered a continuous variable. They are related but not the same.) Why don't you "google it" to see what others have to say about it.
Q: What is pure and applied mathematics?
A: Pure mathematics generally refers to the study of mathematics for its own sake without any regard to its applications. It is generally associated with rigor and abstraction and in some sense, it can be considered art. Mathematics can be beautiful and exquisite, just like a painting. Examples of areas which are usually considered pure mathematics are abstract algebra, topology, and number theory. But each of these areas have been applied to real world problems. Lie Algebras are used in physics, knot theory is being applied to protein folding, and number theory is used in cryptography. Unlike art, beautiful mathematics can be useful.
Applied mathematics is motivated by real world problems. It still requires rigor and abstraction. There are still theorems to be proven, algorithms to be developed and evaluated, errors to be estimated. Examples of areas of applied mathematics are differential equations, numerical analysis, operations research, statistics, actuarial science. Differential equations is sometimes referred to as the mathematical language of science and engineering because many laws and principles of science can be expressed as a differential equation. For example, in calculus you learn that Newton's second law, F = ma, as applied to a free falling object, can written as a differential equation, x''(t)=-g.
But applied mathematics is not as simple as "plugging it in". In high school, you learned how to solve a system of 2 linear equations with 2 unknowns. What if you had 1,000,000 equations and 1,000,000 unknowns? Can you still use the same method? Assuming there is a solution to the problem, the answer is "Sure. Why not?". You can use a method called Gaussian elimination to find the answer to the system of equations. Yes, in theory, you can do that. We are talking about 1,000,001,000,000 coefficients. That's a LOT of numbers to remember. And it would require 1,000,000,999,996,500,002 arithmetic operations. (How long are you willing to work on this?) And if you use a computer, there are other issues like memory, efficiency, accuracy, and stability. The problem of solving a large or ill-behaved system of equations comes up in so many applications that we offer an entire course on the subject, Math 143M. |
Synopses & Reviews
Publisher Comments:
Combining three books into a single volume, this text comprises Multicolor Problems, dealing with several of the classical map-coloring problems; Problems in the Theory of Numbers, an elementary introduction to algebraic number theory; and Random Walks, addressing basic problems in probability theory. High-school algebra is the only prerequisite. 1963 |
quinta-feira, 6 de março de 2014
University of Alberta Mathematical Sciences Society | 2005 | 56 páginas | pdf | 174 kb online: math.ualberta.ca Cambridge University Press | 1967 | 80 páginas online: archive.org G. H. Hardy was one of this century's finest mathematical thinkers, renowned among his contemporaries as a 'real mathematician ... the purest of the pure'. He was also, as C. P. Snow recounts in his Foreword, 'unorthodox, eccentric, radical, ready to talk about anything'. This 'apology', written in 1940 as his mathematical powers were declining, offers a brilliant and engaging account of mathematics as very much more than a science; when it was first published, Graham Greene hailed it alongside Henry James's notebooks as 'the best account of what it was like to be a creative artist'. C. P. Snow's Foreword gives sympathetic and witty insights into Hardy's life, with its rich store of anecdotes concerning his collaboration with the brilliant Indian mathematician Ramanujan, his aphorisms and idiosyncrasies, and his passion for cricket. This is a unique account of the fascination of mathematics and of one of its most compelling exponents in modern times. Para mais livros sobre filosofia da matemática procure em: link
Descrição: This volume examines the teaching of statistics in the whole range of education, but concentrates on primary and secondary schools. It is based upon selected topics from the Second International Congress on Teaching Statistics (ICOTS 2), convened in Canada in August 1986. The contents of this volume divide broadly into four parts: statistics in primary education; statistics in secondary education; theoretical concepts; and two case studies. Part 1 comprises four contributions, three of them based on discovery. The fourth is a comparative study of what is currently taught to children in the age range of 5 to 11 years in Canada, the United Kingdom, and the United States. Part 2 provides an account of recent developments in the teaching of statistics in Australia, the Federal Republic of Germany, the Netherlands, and the United States. Part 3 of the volume is a collection of four contributions for the consideration of teachers in the collection and representation of data. Two case studies make up Part 4. The first describes the competition for the annual statistics prize in the United Kingdom and the second is a wide-ranging account of the growth of the teaching of probability and statistics in Italian schools. There is also a personal view by Ed Jacobsen called "Why in the World Should We Teach Statistics?" and a history of the teaching of statistics by Maria Gabriella Ottaviani, devoted to the growth of statistics in the universities of Europe and the Americas
Descrição:Reports on seven specific programs for improving mathematics education in the schools of Hungary, Indonesia, Japan, the Philippines, the Union of Soviet Socialist Republics, the United Kingdom, and the United Republic of Tanzania are presented. The report from the United Kingdom deals with the Continuing Mathematics Project that focuses on innumeracy in the 16 to 19 age group and new materials designed to deal with this problem. The remaining six articles describe developments in secondary mathematics as a part of the general secondary program. Biographical notes about the contributing authors are appended.
Descrição:This volume was geared to answering the question, does the teaching of mathematics correspond to the needs of the majority of pupils and the society. There are three types of chapters: (1) descriptions of goals reflecting some need of society; (2) case studies of national goal setting; and (3) a summary of the May 1980 meeting of the United Nations Educational, Scientific, and Cultural Organization (UNESCO), which undertook a review of the goals of mathematics teaching. Individual chapters are: (1) Goals as a Reflection of the Needs of Society; (2) Goals as a Reflection of the Needs of the Learner; (3) Goals of Mathematics for Rural Development; (4) School Mathematics-Links with Commerce and Industry; (5) Goals of Mathematics as a Reflection of the Requirements of Production; (6) Educational Objectives for Mathematics Compatible with its Development as a Discipline; (7) New Goals for Old: An Analysis of Reactions to Recent Reforms in Several Countries; (8) The NCTM PRISM Project: An Attempt to Make Curriculum Change More Rational and Systematic; (9) The Evolution of Mathematics Curricula in the Arab States; (10) Goals of the Mathematics Curriculum in British Columbia: Intended, Implemented, and Realized; and (11) Report of a Meeting on the Goals of Mathematics Education
Descrição: This is the sixth volume in a series designed to improve mathematics instruction by providing resource materials for those responsible for mathematics teaching. Focusing on out-of-school mathematics education, this volume presents a panorama of current practices around the world and suggests future trends. Subjects considered include: (1) "Activities Arranged for the Younger Learner"; (2) "Mathematics and the Media"; (3) "Other Sources"; and (4) a case study. The 11 chapters include: "Mathematics Clubs" (Rada-Aranda); "Mathematical Camps" (Rabijewska and Trad); "Mathematical Contests and Olympiads" (Greitzer); "National Mathematical Olympiads in Vietnam" (Le Hai Chau); "Broadcasting and the Open University of the United Kingdom" (Lovis); "Distance Education in Mathematics" (Knight); "The Education of Talented Children in Mathematics in Hungary" (Genzwein); "Mathematics in Literacy Classes" (Zepp); "Mathematics Training for Work" (Straesser); "Family Math" (Stenmark); and "Out-of-school Mathematics in Colombia" (de Losada and Marquez)
Descrição: This is the fifth volume in a series designed to improve mathematics instruction by providing resource materials for those responsible for mathematics teaching. Focused on geometry in schools, it presents a panorama of current practices around the world and suggests future trends. The 14 chapters consider: "Developments in Geometry Teaching in Three Arab States" (Bannout and Hussain); "Geometry for 13-year-olds in Canada and the United States" (Robitaille and Travers); "Geometry Teaching in Latin America" (Lluis); "Geometry in Southeast Asia" (Peng-yee and Chong-keang); "Transformation Geometry in Retrospect" (Sinha); "Geometry at Secondary School Level in Sierra Leone" (Labor); "Geometry in the Primary School: What is Possible and Desirable" (Jzn); "Some Problems Concerning Teaching Geometry to Pupils Aged 10 to 14" (Koman, Kurina, and Ticha); "Teaching Geometry in the USSR" (Chernysheva, Firsov, and Teljakovskii); "The Crisis of Geometry Teaching" (Glaeser); "An Analysis of Geometry Teaching in the United Kingdom" (Fielker); "What are Some Obstacles to Learning Geometry?" (Bishop); "Teacher Education and the Teaching of Geometry" (Meserve and Meserve); and "Microcomputer-based Courses for Secondary School Plane. Geometry |
title promises, this helpful volume offers easy access to the abstract principles common to science and mathematics. It eschews technical terms and omits troublesome details in favor of straightforward explanations that will allow scientists to read papers in branches of science other than their own, mathematicians to appreciate papers on topics on which they have no specialized knowledge, and other readers to cultivate an improved understanding of subjects employing mathematical principles. The broad scope of topics encompasses Euclid's algorithm; congruences; polynomials; complex numbers and algebraic fields; algebraic integers, ideals, and p-adic numbers; groups; the Galois theory of equations; algebraic geometry; matrices and determinants; invariants and tensors; algebras; group algebras; and more. "It is refreshing to find a book which deals briefly but competently with a variety of concatenated algebraic topics, that is not written for the specialist," enthused the Journal of the Institute of ActuariesStudents' Society about this volume, adding "Littlewood's book can be unreservedly recommended."
{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":8.96,"ASIN":"0486425436","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":11.61,"ASIN":"0486462404","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":13.34,"ASIN":"0486478823","isPreorder":0}],"shippingId":"0486425436::4GUyrn2vuypPNI5WQkBLb52Z%2FJGkh077V8BkCQUMMjPbRcJZCif%2BpYGCmmh0y0Y0wxmAMksJqD5Nd4HP1hqjtCSlnH%2Bn1rmlAWrM7Yqb7yo%3D,0486462404::7CyMH%2BsmmOB2%2BqFxyNEBh1i4YnyQjWDp%2BdAapfy6D0cDhcbbULRT%2BT4IyoABNWl%2B6GwYFoyRB8w0qciLiwRKLig%2FC7t9Uh7ET%2BtLuU7RFTM%3D,0486478823::%2BGs44m3AMs3I%2Bhmm%2Bbyuc7I4nP9saMwmFTQUWdrMnbhPGPcuLA%2BzdCvtaMZSyuQdInl%2FlLdKmVL%2Fv3iu7kAxBpmVkNbNy2HGavFfsqs9TZ3HYDQLodYHC the chapter on Galois Theory as a reference for a course I was in because it had such a concrete explanation on how Galois Theory works. However, it didn't mention anything about field extensions & barely mentioned groups. That's why I liked this book, it gives concrete versions of things like (off the top of my head) complex variables, matrices, Galois Theory, polynomials, tensors & more. There's a chapter on each subject & treats it in a very concrete manner. Any math student not totally comfortable with abstractness (as I was, with Galois Theory of equations anyway) would find this book helpful, imo.
This book, covers just about everything important in algebra from basic operations on numbers to groups and tensors, both pure and applied, and all in less than 150 pages!
Given, the above scope, it moves quickly, but should be completely comprehensible to anyone with a strong High School math background, as long as the student reads slowly and carefully. The advantage of the compression is that it is possible to see the whole field and its connections without getting mired in the details.
Fantastic for a broad overview of algebra, prior to studying in more detail (or afterwards, to see how it all fits together). Don't be put off by the age of the book and, for the money, unbeatable.
This is an old, elegantly presented, reminder of algebraic theories which should be general knowledge of the mathematics and physics students now. Disgracefully, they are not, and a short, concise review is something which could be readen in a pair of afternoons for the mere pleasure of adquiring a general idea of many theories. This book provides that, if one is prepared to cope with a little antique names. Dr. Littlewoods "Simple Account" should be amply recommended. |
world of mathematics is probably one of the most fascinating creations of mankind. The world of mathematics with a Computer Algebra System, like MuPAD, is even more fascinating. With MuPAD, we can develop mathematical concepts, explore them and visualize them with just a few simple commands. This book is a gentle introduction to MuPAD - a modern Computer Algebra System. The author introduces MuPAD step-by-step and shows how we can use it in various areas of mathematics. A large chapter of the book is devoted to the graphical visualization of mathematical concepts, and MuPAD graphics are also used extensively throughout the rest of the book. Each chapter of the book should be considered as a single workshop for MuPAD beginners. The whole book is a perfect resource for conducting workshops on using Computer Algebra Systems to explore, experiment with, and visualize mathematical concepts. less |
Maple by Example, Third Edition, is a reference/text with CD for beginning and experienced students, professional engineers, and other Maple users. This new edition has been updated to be compatible with the most recent release of the Maple software. Coverage includes built-in Maple commands used in courses and practices that involve calculus, linear algebra, business mathematics, ordinary and partial differential equations, numerical methods, graphics and more. The CD-ROM provides updated Maple input and all text from the book.
* Updated coverage of Maple features and functions
* New applications from a variety of fields, including biology, physics and engineering
* Backwards compatible for all previous Maple version
* More detail in its step-by-step examples"
Amazon Editorial review :
""Overall, I found the book very nice to read and easy to follow. All major aspects of Maple that a novice to intermediate user would come across are covered."
- Laurent Bernard, Maplesoft, Inc.
"The book is eminently readable with a good narrative style and a good blend of didactic exposition followed by the relevant Maple commands. What is especially instructive is the numerous times alternative ways are used to produce equivalent answers showing the versatility of Maple and leaving the choice of method to the reader who attempts similar problems."
- David Hodgkinson, University of Liverpool
Maple is a very powerful computer algebra system used by students, educators, mathematicians, statisticians, scientists, and engineers for doing numerical and symbolic computations. Greatly expanded and updated from the author's MAPLE V Primer, The MAPLE Book offers extensive coverage of the latest version of this outstanding software package, MAPLE 7.0 The MAPLE Book serves both as an introduction to Maple and as a reference. Organized according to level and subject area of mathematics, it first covers the basics of high school algebra and graphing, continues with calculus and differential equations then moves on to more advanced topics, such as linear algebra, vector calculus, complex analysis, special functions, group theory, number theory and combinatorics. The MAPLE Book includes a tutorial for learning the Maple programming language. Once readers have learned how to program, they will appreciate the real power of Maple. The convenient format and straightforward style of The MAPLE Book let users proceed at their own pace, practice with the examples, experiment with graphics, and learn new functions as they need them. All of the Maple commands used in the book are available on the Internet, as are links to various other files referred to in the book. Whatever your level of expertise, you'll want to keep The MAPLE Book next to your computer. |
College Algebra and Trigonometry - 5th edition
Summary: This text provides a supportive environment to help students successfully learn the content of a standard algebra and trigonometry course. By incorporating interactive learning techniques, the Aufmann team helps students to better understand concepts, focus their studying habits, and obtain greater mathematical success.
Many new components added to this edition of College Algebra and Trigonometry have been designed to help students diagnose and review weak ...show morealgebra skills. Prerequisite review is include in the textbook (and supporting materials) so that instructors can spend less time covering review material and students can still fill in the gaps in their mathematical knowledge. ...show less
Section 7.1 The Law of Sines Section 7.2 The Law of Cosines and Area Section 7.3 Vectors Section 7.4 Trigonometric Form of Complex Numbers Section 7.5 De Moivre's Theorem Exploring Concepts with Technology: Optimal Branching of Arteries0618386807 Hardcover. 5 |
introduction to Laplace transforms and Fourier series is aimed at second year students in applied mathematics. It is unusual in treating Laplace transforms at a relatively simple level with many examples. Mathematics students do not usually meet this material until later in their degree course but applied mathematicians and engineers need an early introduction. Suitable as a course text, it will also be of interest to physicists and engineers as supplementary material. |
Suwanee ACT textbooks as well as supplementary material when necessary to promote students? understanding of concepts. I am very comfortable with the subject and have taken several advanced Math classes in college. I have tutored Geometry for the last eleven years to middle school and high school students.
...The Product Rule resolves (f.g)' - the differential of the product of two functions, f(x) times g(x) which becomes
f'.g + g'.f i.e. the prime of f(x) times g(x) plus the prime of g(x) times f(x). Many say they would rather die than speak in public - but it is easy with practice and knowledge. ... |
Summary: Math 3B/3C Syllabus
SIMS Program, Grace Kennedy
SIMS Website: programs/sims.html
Course Website:
Expectations :
· Please be on time or a few minutes early. We will spend the first few
minutes working on a problem I'll have on the board.
· Please turn off cell phones. If a cell goes off, you will be expected to
lead us in a round of the quadratic formula song. (Don't worry, we'll sing
back-up.)
· Take notes. Not everything I cover will be in the course reader or in
handouts I provide, and you will wish you could remember what I said
about such-and such.
· Work with your classmates. You will get your homework finished faster,
and you will gain deeper understanding of the concepts by discussing them.
Make sure you are explaining as well as listening, even if it is explaining
back something that was explained to you.
· Turn in your own work. Definitely work with others, but the work you
should submit should be your own. Once you discuss problems with your
classmates, make sure that you do your own original write-up of the so- |
More About
This Textbook
Overview
Exciting, hands-on approach to understanding fundamental underpinnings of modern arithmetic, algebra, geometry and number systems, by examining their origins in early Egyptian, Babylonian and Greek sources. Students can do division like the ancient Egyptians, solve quadratic equations like the Babylonians |
MATLAB Tutorial
to accompany
Partial Differential Equations: Analytical and Numerical
Methods
by
Mark S. Gockenbach
(SIAM, 2002)
Introduction .................................................................................................................................................. 3
About this tutorial ....................................................................................................................................... 3
About MATLAB ........................................................................................................................................ 3
MATLAB notebooks .................................................................................................................................. 3
Getting help with MATLAB commands ..................................................................................................... 4
Getting started with MATLAB ................................................................................................................... 4
More about M-Books ................................................................................................................................. 9
Simple graphics in MATLAB ..................................................................................................................... 9
Symbolic computation in MATLAB ........................................................................................................ 15
Manipulating functions in MATLAB ....................................................................................................... 18
Saving your MATLAB session ................................................................................................................. 20
About the rest of this tutorial .................................................................................................................... 20
Chapter 1: Classification of differential equations .................................................................................. 20
Chapter 2: Models in one dimension......................................................................................................... 23
Section 2.1: Heat flow in a bar; Fourier's Law.......................................................................................... 23
Chapter 3: Essential linear algebra........................................................................................................... 29
Section 3.1 Linear systems as linear operator equations ........................................................................... 29
Section 3.2: Existence and uniqueness of solutions to Ax=b .................................................................... 30
Section 3.3: Basis and dimension ............................................................................................................. 32
Programming in MATLAB, part I ............................................................................................................ 35
Section 3.4: Orthogonal bases and projection .......................................................................................... 42
Section 3.5: Eigenvalues and eigenvectors of a symmetric matrix ........................................................... 49
Review: Functions in MATLAB............................................................................................................... 53
Chapter 4: Essential ordinary differential equations .............................................................................. 56
Section 4.2: Solutions to some simple ODEs ........................................................................................... 56
Section 4.3: Linear systems with constant coefficients ............................................................................. 60
Programming in MATLAB, Part II .......................................................................................................... 64
Section 4.4: Numerical methods for initial value problems ...................................................................... 70
Programming in MATLAB, part III ......................................................................................................... 75
Efficient MATLAB programming ............................................................................................................ 77
More about graphics in MATLAB ........................................................................................................... 78
Chapter 5: Boundary value problems in statics ....................................................................................... 79
Section 5.2: Introduction to the spectral method; eigenfunctions ............................................................ 79
Section 5.5: The Galerkin method ............................................................................................................ 84
Section 5.6: Piecewise polynomials and the finite element method ......................................................... 89
More about sparse matrices ...................................................................................................................... 98
Chapter 6: Heat flow and diffusion......................................................................................................... 101
Section 6.1: Fourier series methods for the heat equation ...................................................................... 101
Section 6.4: Finite element methods for the heat equation ..................................................................... 104
Chapter 8: Problems in multiple spatial dimensions ............................................................................. 107
Section 8.2: Fourier series on a rectangular domain ............................................................................... 107
Two-dimensional graphics in MATLAB ................................................................................................ 108
Section 8.3: Fourier series on a disk ....................................................................................................... 111
Chapter 9: More about Fourier series .................................................................................................... 114
Section 9.1: The complex Fourier series ................................................................................................ 114
1
Section 9.2: Fourier series and the FFT .................................................................................................. 117
Chapter 10: More about finite element methods ................................................................................... 118
Section 10.1 Implementation of finite element methods ......................................................................... 118
Using the code ........................................................................................................................................ 130
2
Introduction
In this introduction, I will explain the organization of this tutorial and give some basic information about
MATLAB and MATLAB notebooks. I will also give a preliminary introduction to the capabilities of
MATLAB.
About this tutorial
The purpose of this document is to explain the features of MATLAB that are useful for applying the
techniques presented in my textbook. This really is a tutorial (not a reference), meant to be read and used
in parallel with the textbook. For this reason, I have structured the tutorial to have the same chapter and
sections titles as the book. However, the purpose of the sections of this document is not to re-explain the
material in the text; rather, it is to present the capabilities of MATLAB as they are needed by someone
studying the text.
Therefore, for example, in Section 2.1, "Heat flow in a bar; Fourier's Law", I do not explain any physics or
modeling. (The physics and modeling are found in the text.) Instead, I explain the MATLAB command for
integration, because Section 2.1 is the first place in the text where the student is asked to integrate a
function. Because of this style of organization, some parts of the text have no counterpart in this tutorial.
For example, there is no Chapter 7, because, by the time you have worked through the first six chapters of
the tutorial, you have learned all of the capabilities of MATLAB that you need to address the material in
Chapter 7 of the text. For the same reason, you will see that some individual sections are missing; Chapter
5, for example, begins with Section 5.2.
I should point out that my purpose is writing this tutorial is not to show you how to solve the problems in
the text; rather, it is to give you the tools to solve them. Therefore, I do not give you a worked-out example
of every problem type---if I did, your "studying" could degenerate to simply looking for an example,
copying it, and making a few changes. At crucial points, I do provide some complete examples, since I see
no other way to illustrate the power of MATLAB than in context. However, there is still plenty for you to
figure out for yourself!
About MATLAB
MATLAB, which is short for Matrix Laboratory, incorporates numerical computation, symbolic
computation, graphics, and programming. As the name suggests, it is particularly oriented towards matrix
computations, and it provides both state-of-the-art algorithms and a simple, easy to learn interface for
manipulating matrices. In this tutorial, I will touch on all of the capabilities mentioned above: numerical
and symbolic computation, graphics, and programming.
MATLAB notebooks
This document you are reading is called an M-Book. It integrates text and MATLAB commands (with their
output, including graphics). If you are running MATLAB Version 6 under Microsoft Windows, then an M-
Book becomes an interactive document: by running the M-Book under MATLAB, you can enter new
MATLAB commands and see their output inside the M-Book itself. The MATLAB command that allows
you to do this is called notebook.
Since the notebook facility is available only under Microsoft Windows, I will not emphasize it in this
tutorial. However, Windows users should take advantage of it!
3
The most important thing to understand about a notebook is that it is interactive---at any time you can
execute a MATLAB command and see what it does. This makes a MATLAB notebook a tremendous
learning environment: when you read an explanation of a MATLAB feature, you can immediately try it out!
Getting help with MATLAB commands
Documentation about MATLAB and MATLAB commands is available from within the program itself. If
you know the name of the command and need more information about how it works, you can just type "help
<command name>" at the MATLAB prompt. In the same way, you can get information about a group of
commands with common uses by typing "help <topic name>". I will show examples of using the
command-line help feature below.
The MATLAB desktop, a new feature of MATLAB Version 6, contains a help browser covering both
reference and tutorial material. To access the browser, click on the Help menu and choose MATLAB
Help. You can then choose "Getting Started" from the table of contents for a tutorial introduction to
MATLAB, or use the index to find specific information.
Getting started with MATLAB
As mentioned above, MATLAB has many capabilities, such as the fact that one can write programs made
up of MATLAB commands. The simplest way to use MATLAB, though, is as an interactive computing
environment (essentially, a very fancy graphing calculator). You enter a command and MATLAB executes
it and returns the result. Here is an example:
2+2
ans =
4
You can assign values to variables for later use:
x=2
x =
2
The variable x can now be used in future calculations:
x^2
ans =
4
At any time, you can list the variables that are defined with the who command:
who
Your variables are:
ans x
At the current time, there are 2 variables defined. One is x, which I explicitly defined above. The other is
ans (short for "answer"), which automatically holds the most recent result that was not assigned to a
4
variable (you may have noticed how ans appeared after the first command above). You can always check
the value of a variable simply by typing it:
x
x =
2
ans
ans =
4
If you enter a variable that has not been defined, MATLAB prints an error message:
y
??? Undefined function or variable 'y'.
To clear a variable from the workspace, use the clear command:
who
Your variables are:
ans x
clear x
who
Your variables are:
ans
To clear all of the variables from the workspace, just use clear by itself:
clear
who
MATLAB knows the elementary mathematical functions: trigonometric functions, exponentials, logarithms,
square root, and so forth. Here are some examples:
sqrt(2)
ans =
1.4142
sin(pi/3)
ans =
0.8660
exp(1)
ans =
2.7183
log(ans)
ans =
1
A couple of remarks about the above examples:
MATLAB knows the number , which is called pi.
5
Computations in MATLAB are done in floating point arithmetic by default. For example,
MATLAB computes the sine of /3 to be (approximately) 0.8660 instead of exactly 3/2.
A complete list of the elementary functions can be obtained by entering "help elfun":
help elfun
Elementary math functions.
Trigonometric.
sin - Sine.
sinh - Hyperbolic sine.
asin - Inverse sine.
asinh - Inverse hyperbolic sine.
cos - Cosine.
cosh - Hyperbolic cosine.
acos - Inverse cosine.
acosh - Inverse hyperbolic cosine.
tan - Tangent.
tanh - Hyperbolic tangent.
atan - Inverse tangent.
atan2 - Four quadrant inverse tangent.
atanh - Inverse hyperbolic tangent.
sec - Secant.
sech - Hyperbolic secant.
asec - Inverse secant.
asech - Inverse hyperbolic secant.
csc - Cosecant.
csch - Hyperbolic cosecant.
acsc - Inverse cosecant.
acsch - Inverse hyperbolic cosecant.
cot - Cotangent.
coth - Hyperbolic cotangent.
acot - Inverse cotangent.
acoth - Inverse hyperbolic cotangent.
Exponential.
exp - Exponential.
log - Natural logarithm.
log10 - Common (base 10) logarithm.
log2 - Base 2 logarithm and dissect floating point number.
pow2 - Base 2 power and scale floating point number.
realpow - Power that will error out on complex result.
reallog - Natural logarithm of real number.
realsqrt - Square root of number greater than or equal to zero.
sqrt - Square root.
nextpow2 - Next higher power of 2.
Complex.
abs - Absolute value.
angle - Phase angle.
complex - Construct complex data from real and imaginary parts.
conj - Complex conjugate.
imag - Complex imaginary part.
real - Complex real part.
unwrap - Unwrap phase angle.
isreal - True for real array.
cplxpair - Sort numbers into complex conjugate pairs.
Rounding and remainder.
fix - Round towards zero.
floor - Round towards minus infinity.
6
ceil - Round towards plus infinity.
round - Round towards nearest integer.
mod - Modulus (signed remainder after division).
rem - Remainder after division.
sign - Signum.
For more information about any of these elementary functions, type "help <function_name>". For a list of
help topics like "elfun", just type "help". There are other commands that for part of the help system; to see
them, type "help help".
MATLAB does floating point arithmetic using the IEEE standard, which means that numbers have about 16
decimal digits of precision (the actual representation is in binary, so the precision is not exactly 16 digits).
However, MATLAB only displays 5 digits by default. To change the display, use the format command.
For example, "format long" changes the display to 15 digits:
format long
pi
ans =
3.14159265358979
Other options for the format command are "format short e" (scientific notation with 5 digits) and "format
long e" (scientific notation with 15 digits).
In addition to pi, other predefined variables in MATLAB include i and j, both of which represent the
imaginary unit: i=j=sqrt(-1).
clear
i^2
ans =
-1
j^2
ans =
-1
Although it is usual, in mathematical notation, to use i and j as arbitrary indices, this can easily lead to
errors in MATLAB because these symbols are predefined. For this reason, I will use ii and jj as my
standard indices when needed.
Vectors and matrices in MATLAB
The default type for any variable or quantity in MATLAB is a matrix---a two-dimensional array. Scalars
and vectors are regarded as special cases of matrices. A scalar is a 1 by 1matrix, while a vector is an n by 1
or 1 by n matrix. A matrix is entered by rows, with entries in a row separated by spaces or commas, and the
rows separated by semicolons. The entire matrix is enclosed in square brackets. For example, I can enter a
3 by 2 matrix as follows:
A=[1 2;3 4;5 6]
A =
1 2
3 4
5 6
Here is how I would enter a 2 by 1 (column) vector:
7
x=[1;-1]
x =
1
-1
A scalar, as we have seen above, requires no brackets:
a=4
a =
4
A variation of the who command, called whos, gives more information about the defined variables:
whos
Name Size Bytes Class
A 3x2 48 double array
a 1x1 8 double array
ans 1x1 8 double array
x 2x1 16 double array
Grand total is 10 elements using 80 bytes
The column labeled "size" gives the size of each array; you should notice that, as I mentioned above, a
scalar is regarded as a 1 by 1 matrix (see the entry for a, for example).
MATLAB can perform the usual matrix arithmetic. Here is a matrix-vector product:
A*x
ans =
-1
-1
-1
Here is a matrix-matrix product:
B=[-1 3 4 6;2 0 1 -2]
B =
-1 3 4 6
2 0 1 -2
A*B
ans =
3 3 6 2
5 9 16 10
7 15 26 18
MATLAB signals an error if you attempt an operation that is undefined:
B*A
??? Error using ==> *
Inner matrix dimensions must agree.
A+B
??? Error using ==> +
8
Matrix dimensions must agree.
More about M-Books
If you are reading this document using the MATLAB notebook facility, then you may wish to execute the
commands as you read them. Otherwise, the variables shown in the examples are not actually created in the
MATLAB workspace. To execute a command, click on it (or select it) and press control-enter (that is,
press the enter key while holding down the control key). While reading the tutorial, you should execute
each of my commands as you come to it. Otherwise, the state of MATLAB is not what it appears to be, and
if you try to experiment by entering your own commands, you might get unexpected results if your
calculations depend on the ones you see in this document.
Notice that the command lines in this document appear in green, and are enclosed in gray square brackets.
Output appears in blue text, also enclosed in gray square brackets. These comments do not apply if you are
reading a version of this document that has been printed or converted to another format (such as PostScript
or PDF).
If you are reading this using MATLAB's notebook command, then, as I mentioned above, you can try your
own MATLAB commands at any time. Just move the cursor to a new line, type the command, and then
type control-enter. You should definitely take advantage of this facility, as it will make learning MATLAB
much easier.
Simple graphics in MATLAB
Two-dimensional graphics are particularly easy to understand: If you define vectors x and y of equal length
(each with n components, say), then MATLAB's plot command will graph the points (x1,y1), (x2,y2), …,
(xn,yn) in the plane and connect them with line segments. Here is an example:
format short
x=[0,0.25,0.5,0.75,1]
x =
Columns 1 through 4
0 0.2500 0.5000 0.7500
Column 5
1.0000
y=[1,0,1,0,1]
y =
1 0 1 0 1
plot(x,y)
9Two features of MATLAB make it easy to generate graphs. First of all, the command linspace creates a
vector with linearly spaced components---essentially, a regular grid on an interval. (Mathematically,
linspace creates a finite arithmetic sequence.) To be specific, linspace(a,b,n) creates the (row) vector
whose components are a,a+h,a+ 2h,…,a+(n-1)h, where h=1/(n-1).
x=linspace(0,1,6)
x =
Columns 1 through 4
0 0.2000 0.4000 0.6000
Columns 5 through 6
0.8000 1.0000
The second feature that makes it easy to create graphs is the fact that all standard functions in MATLAB,
such as sine, cosine, exp, and so forth, are vectorized. A vectorized function f, when applied to a vector x,
returns a vector y (of the same size as x) with ithcomponent equal to f(xi). Here is an example:
y=sin(pi*x)
y =
Columns 1 through 4
0 0.5878 0.9511 0.9511
Columns 5 through 6
0.5878 0.0000
I can now plot the function:
plot(x,y)
10Of course, this is not a very good plot of sin(x), since the grid is too coarse. Below I create a finer grid
and thereby obtain a better graph of the function. Often when I create a new vector or matrix, I do not want
MATLAB to display it on the screen. (The following example is a case in point: I do not need to see the 41
components of vector x or vector y.) When a MATLAB command is followed by a semicolon, MATLAB
will not display the output.
x=linspace(0,1,41);
y=sin(pi*x);
plot(x,y11
The basic arithmetic operations have vectorized versions in MATLAB. For example, I can multiply two
vectors component-by-component using the ".*" operator. That is, z=x.*y sets zi equal to xiyi. Here is an
example:
x=[1;2;3]
x =
1
2
3
y=[2;4;6]
y =
2
4
6
z=x.*y
z =
2
8
18
The "./" operator works in an analogous fashion. There are no ".+" or ".-" operators, since addition and
subtraction of vectors are defined componentwise already. However, there is a ".^" operator that applies an
exponent to each component of a vector.
x
x =
1
2
3
x.^2
ans =
1
4
9
Finally, scalar addition is automatically vectorized in the sense that a+x, where a is a scalar and x is a
vector, adds a to every component of x.
The vectorized operators make it easy to graph a function such as f(x)=x/(1+x2). Here is how it is done:
x=linspace(-5,5,41);
y=x./(1+x.^2);
plot(x,y)
12If I prefer, I can just graph the points themselves, or connect them with dashed line segments. Here are
examples:
plot(x,y,'.')o')
13--')The string following the vectors x and y in the plot command ('.', 'o', and '--' in the above examples)
specifies the linetype. For other linetypes, see "help plot".
It is not much harder to plot two curves on the same graph. For example, I plot y=x2 and y=x3 together:
x=linspace(-1,1,101);
plot(x,x.^2,x,x.^3)
14I can also give the lines different linetypes:
plot(x,x.^2,'-',x,x.^3,'--')Symbolic computation in MATLAB
In addition to numerical computation, MATLAB can also perform symbolic computations. However, since,
by default, variables are floating point, you must explicitly indicate that a variable is intended to be
15
symbolic. One way to do this is using the syms command, which tells MATLAB that one or more variables
are symbolic. For example, the following command defines a and b to be symbolic variables:
syms a b
I can now form symbolic expressions involving these variables:
2*a*b
ans =
2*a*b
Notice how the result is symbolic, not numeric as it would be if the variables were floating point variables.
Also, the above calculation does not result in an error, even though a and b do not have values.
Another way to create a symbolic variable is to assign a symbolic value to a new symbol. Since numbers
are, by default, floating point, it is necessary to use the sym function to tell MATLAB that a number is
symbolic:
c=sym(2)
c =
2
whos
Name Size Bytes Class
A 3x2 48 double array
B 2x4 64 double array
a 1x1 126 sym object
ans 1x1 134 sym object
b 1x1 126 sym object
c 1x1 126 sym object
x 1x101 808 double array
y 1x41 328 double array
z 3x1 24 double array
Grand total is 171 elements using 1784 bytes
I can do symbolic computations:
a=sqrt(c)
a =
2^(1/2)
You should notice the difference between the above result and the following:
a=sqrt(2)
a =
1.4142
whos
Name Size Bytes Class
A 3x2 48 double array
B 2x4 64 double array
a 1x1 8 double array
ans 1x1 134 sym object
b 1x1 126 sym object
c 1x1 126 sym object
16
x 1x101 808 double array
y 1x41 328 double array
z 3x1 24 double array
Grand total is 170 elements using 1666 bytes
Even though a was declared to be a symbolic variable, once I assign a floating point value to it, it becomes
numeric. This example also emphasizes that sym must be used with literal constants if they are to
interpreted as symbolic and not numeric:
a=sqrt(sym(2))
a =
2^(1/2)
As a further elementary example, consider the following two commands:
sin(sym(pi))
ans =
0
sin(pi)
ans =
1.2246e-016
In the first case, since π is symbolic, MATLAB notices that the result is exactly zero; in the second, both π
and the result are represented in floating point, so the result is not exactly zero (the error is just roundoff
error).
Using symbolic variables, I can perform algebraic manipulations.
syms x
p=(x-1)*(x-2)*(x-3)
p =
(x-1)*(x-2)*(x-3)
expand(p)
ans =
x^3-6*x^2+11*x-6
factor(ans)
ans =
(x-1)*(x-2)*(x-3)
Integers can also be factored.
factor(144)
ans =
2 2 2 2 3 3
The result is a list of the factors, repeated according to multiplicity.
An important command for working with symbolic expressions is simplify, which tries to reduce an
expression to a simpler one equal to the original. Here is an example:
syms x a b c
17
p=(x-1)*(a*x^2+b*x+c)+x^2+3*x+a*x^2+b*x
p =
(x-1)*(a*x^2+b*x+c)+x^2+3*x+a*x^2+b*x
simplify(p)
ans =
a*x^3+b*x^2+x*c-c+x^2+3*x
Since the concept of simplification is not precisely defined (which is simpler, a polynomial in factored form
or in expanded form a+bx+cx2+…?), MATLAB has a number of specialized simplification commands. I
have already used two of them, factor and expand. Another is collect, which "gathers like terms":
p
p =
(x-1)*(a*x^2+b*x+c)+x^2+3*x+a*x^2+b*x
collect(p,x)
ans =
a*x^3+(b+1)*x^2+(c+3)*x-c
By the way, the display of symbolic output can be made more mathematical using the pretty command:
pretty(ans)
3 2
a x + (b + 1) x + (c + 3) x - c
Note: MATLAB's symbolic computation is based on Maple, a computer algebra system originally
developed at the University of Waterloo, Canada, and marketed by Waterloo Maple, Inc. If you have the
Extended Symbolic Math Toolbox with your installation of MATLAB, then you have access to all
nongraphical Maple functions; see "help maple" for more details. However, these capabilities are not
included in the standard Student Version of MATLAB, and so I will not emphasize them in this tutorial.
Manipulating functions in MATLAB
Symbolic expressions can be treated as functions. Here is an example:
syms x
p=x/(1+x^2)
p =
x/(1+x^2)
Using the subs command, I can evaluate the function p for a given value of x. The following command
substitutes 3 for every occurrence of x in p:
subs(p,x,3)
ans =
0.3000
The calculation is automatically vectorized:
y=linspace(-5,5,101);
z=subs(p,x,y);
plot(y,z)
18One of the most powerful features of symbolic computation in MATLAB is that certain calculus operations,
notably integration and differentiation, can be performed symbolically. These capabilities will be explained
later in the tutorial.
Another method for manipulating functions in MATLAB is the use of the inline command, which takes an
expression and one or more independent variables, and creates from it a function that can be evaluated
directly (without using subs). Here is an example:
f=inline('x^2','x')
f =
Inline function:
f(x) = x^2
Notice how both the expression defining the function and the name of the independent variable must be
enclosed in quotation marks. The function f can be evaluated in the expected fashion, and the input can be
either floating point or symbolic:
f(1)
ans =
1
f(sqrt(pi))
ans =
3.1416
f(sqrt(sym(pi)))
ans =
pi
The inline command can also create a function of several variables:
g=inline('x^2*y','x','y')
g =
Inline function:
g(x,y) = x^2*y
19
g(1,2)
ans =
2
g(pi,14)
ans =
138.1745
There is a third way to define a function in MATLAB, namely, to write a program that evaluates the
function. I will defer an explanation of this technique until Chapter 3, where I first discuss programming in
MATLAB.
Saving your MATLAB session
When using MATLAB, you will frequently wish to end your session and return to it later. Using the save
command, you can save all of the variables in the MATLAB workspace to a file. The variables are stored
efficiently in a binary format to a file with a ".mat" extension. The next time you start MATLAB, you can
load the data using the load command. See "help save" and "help load" for details.
About the rest of this tutorial
The remainder of this tutorial is organized in parallel with my textbook. Each section in the tutorial
introduces any new MATLAB commands that would be useful in addressing the material in the
corresponding section of the textbook. As I mentioned above, some sections of the textbook have no
counterpart in the tutorial, since all of the necessary MATLAB commands have already been explained.
For this reason, the tutorial is intended to be read from beginning to end, in conjunction with the textbook.
Chapter 1: Classification of differential equations
As I mentioned above, MATLAB can perform symbolic calculus on expressions. Consider the following
example:
syms x
f=sin(x^2)
f =
sin(x^2)
I can differentiate this expression using the diff command:
diff(f,x)
ans =
2*cos(x^2)*x
The same techniques work with expressions involving two or more variables:
syms x y
q=x^2*y^3*exp(x)
q =
x^2*y^3*exp(x)
pretty(q)
2 3
20
x y exp(x)
diff(q,y)
ans =
3*x^2*y^2*exp(x)
Thus MATLAB can compute partial derivatives just as easily as ordinary derivatives.
One use of these capabilities is to test whether a certain function is a solution of a given differential
equation. For example, suppose I want to check whether the function u(t)=eat is a solution of the ODE
du
au 0.
dt
I define
syms a t
u=exp(a*t)
u =
exp(a*t)
I can then compute the left side of the differential equation, and see if it agrees with the right side (zero):
diff(u,t)-a*u
ans =
0
Thus the given function u is a solution. Is the function v(t)=at another solution? I can check it as follows:
v=a*t
v =
a*t
diff(v,t)-a*v
ans =
a-a^2*t
Since the result is not zero, the function v is not a solution.
It is no more difficult to check whether a function of several variables is the solution of a PDE. For
example, is w(x,y)=sin(πx)+sin(πy) a solution of the differential equation
2u 2u
0?
x 2 y 2
As before, I can answer this question by defining the function and substituting it into the differential
equation:
syms x y
w=sin(pi*x)+sin(pi*y)
w =
sin(pi*x)+sin(pi*y)
diff(w,x,2)+diff(w,y,2)
ans =
-sin(pi*x)*pi^2-sin(pi*y)*pi^2
simplify(ans)
ans =
21
-sin(pi*x)*pi^2-sin(pi*y)*pi^2
Since the result is not zero, the function w is not a solution of the PDE.
The above example shows how to compute higher derivatives of an expression. For example, here is the
fifth derivative of w with respect to x:
diff(w,x,5)
ans =
cos(pi*x)*pi^5
To compute a mixed partial derivative, we have to iterate the diff command. Here is the mixed partial
derivative of w(x,y)=x2+xy2 with respect to x and then y:
syms x y
w=x^2*exp(y)+x*y^2
w =
x^2*exp(y)+x*y^2
diff(diff(w,x),y)
ans =
2*x*exp(y)+2*y
Instead of using expressions in the above calculations, I can use inline functions. Consider the following:
clear
syms a x
f=inline('exp(a*x)','x','a')
f =
Inline function:
f(x,a) = exp(a*x)
diff(f(x,a),x)-a*f(x,a)
ans =
0
Notice, however, that any variable appearing in an inline function must be an input:
clear
syms a x
f=inline('exp(a*x)','x')
f =
Inline function:
f(x) = exp(a*x)
f(x)
??? Error using ==> inlineeval
Error in inline expression ==> exp(a*x)
??? Undefined function or variable 'a'.diff(f(x),x)
??? Error using ==> inlineeval
Error in inline expression ==> exp(a*x)
??? Undefined function or variable 'a'.
22This is a possible drawback to using inline functions in some instances.
Chapter 2: Models in one dimension
Section 2.1: Heat flow in a bar; Fourier's Law
MATLAB can compute both indefinite and definite integrals. The command for computing an indefinite
integral (antiderivative) is exactly analogous to that for computing a derivative:
syms x
f=x^2
f =
x^2
int(f,x)
ans =
1/3*x^3
As this example shows, MATLAB does not add the "constant of integration." It simply returns one
antiderivative (when possible). If the integrand is too complicated, MATLAB just returns the integral
unevaluated, and prints a warning message.
int(exp(cos(x)),x)
To compute a definite integral, we must specify the limits of integration:
int(x^2,x,0,1)
ans =
1/3
MATLAB has commands for estimating the value of a definite integral that cannot be computed
analytically. Consider the following example:
int(exp(cos(x)),x,0,1 = 0 .. 1)
Since MATLAB could not find an explicit antiderivative, I can use the quad function to estimate the
definite integral. The quad command takes, as input, a function rather than an expression (as does int).
Therefore, I must first create a function:
f=inline('exp(cos(x))','x');
23
Now I can invoke quad:
quad(f,0,1)
ans =
2.3416
"quad" is short for quadrature, another term for numerical integration. For more information about the
quad command, see "help quad".
As a further example of symbolic calculus, I will use the commands for integration and differentiation to
test Theorem 2.1 from the text. The theorem states that (under certain conditions) a partial derivative can
be moved under an integral sign:
F
d d
d
F ( x, y) dy x ( x, y) dy.
dx c c
Here is a specific instance of the theorem:
syms x y c d
f=x*y^3+x^2*y
f =
x*y^3+x^2*y
r1=diff(int(f,y,c,d),x)
r1 =
1/4*d^4-1/4*c^4+x*(d^2-c^2)
r2=int(diff(f,x),y,c,d)
r2 =
1/4*d^4-1/4*c^4+x*(d^2-c^2)
r1-r2
ans =
0
Solving simple boundary value problems by integration
Consider the following BVP:The solution can be found by two integrations (cf. Example 2.2 in the text). Remember, MATLAB does not
add a constant of integration, so I do it explicitly:
syms x C1 C2
int(-(1+x),x)+C1
ans =
-1/2*x^2-x+C1
int(ans,x)+C2
ans =
-1/6*x^3-1/2*x^2+C1*x+C2
24
u=ans
u =
-1/6*x^3-1/2*x^2+C1*x+C2
The above function u, with the proper choice of C1 and C2, is the desired solution. To find the constants, I
solve the (algebraic) equations implied by the boundary conditions. The MATLAB command solve can be
used for this purpose.
The MATLAB solve command
Before completing the previous example, I will explain the solve command on some simpler examples.
Suppose I wish to solve the linear equation ax+b=0 for x. I can regard this as a root-finding problem: I
want the root of the function f(x)=ax+b. The solve command finds the root of a function with respect to a
given independent variable:
syms f x a b
f=a*x+b
f =
a*x+b
solve(f,x)
ans =
-b/a
If the equation has more than one solution, solve returns the possibilities in an array:
syms f x
f=x^2-3*x+2;
solve(f,x)
ans =
[ 1]
[ 2]
As these examples show, solve is used to find solutions of equations of the form f(x)=0; only the expression
f(x) is input to solve.
solve can also be used to solve a system of equations in several variables. In this case, the equations are
listed first, followed by the unknowns. For example, suppose I wish to solve the equations
x+y=1,
2x-y=1.
Here is the command:
syms x y
s=solve(x+y-1,2*x-y-1,x,y)
s =
x: [1x1 sym]
y: [1x1 sym]
What kind of variable is the output s? If we list the variables in the workspace,
whos
Name Size Bytes Class
C1 1x1 128 sym object
C2 1x1 128 sym object
a 1x1 126 sym object
25
ans 2x1 188 sym object
b 1x1 126 sym object
c 1x1 126 sym object
d 1x1 126 sym object
f 1x1 142 sym object
r1 1x1 178 sym object
r2 1x1 178 sym object
s 1x1 508 struct array
u 1x1 172 sym object
x 1x1 126 sym object
y 1x1 126 sym object
Grand total is 123 elements using 2378 bytes
we see that s is a 1 by 1 struct array, that is, an array containing a single struct. A struct is a data type with
named fields that can be accessed using the syntax variable.field. The variable s has two fields:
s
s =
x: [1x1 sym]
y: [1x1 sym]
The two fields hold the values of the unknowns x and y:
s.x
ans =
2/3
s.y
ans =
1/3
If the system has more than one solution, then the output of solve will be a struct, each of whose fields is an
array containing the values of the unknowns. Here is an example:
s=solve(x^2+y^2-1,y-x^2,x,y)
s =
x: [4x1 sym]
y: [4x1 sym]
The first solution is
pretty(s.x(1))
1/2 1/2
1/2 (-2 + 2 5 )
pretty(s.y(1))
1/2
1/2 5 - 1/2
The second solution is
pretty(s.x(2))
1/2 1/2
- 1/2 (-2 + 2 5 )
pretty(s.y(2))
1/2
26
1/2 5 - 1/2
Actually, the above are the only real solutions. However, solve will also find complex solutions:
pretty(s.x(3))
1/2 1/2
1/2 (-2 - 2 5 )
pretty(s.y(3))
1/2
-1/2 - 1/2 5
To see the numeric value of the last solution, I can use the double command, which converts a symbolic
quantity to floating point:
double(s.x(3))
ans =
0 + 1.2720i
double(s.y(3))
ans =
-1.6180
The symbol i represents the imaginary unit in MATLAB (that is, i is the square root of –1). Thus the third
solution found by MATLAB is complex. (The fourth is also.)
Back to the example
We can now solve for the constants of integrations in the solution to the BVPRecall that the solution is of the form
u
u =
-1/6*x^3-1/2*x^2+C1*x+C2
We must use the boundary conditions to compute the constants C1 and C2. The equations are u(0)-2=0 and
u(1)=0. Notice how I use the subs command to form u(0) and u(1):
s=solve(subs(u,x,0)-2,subs(u,x,1),C1,C2)
s =
C1: [1x1 sym]
C2: [1x1 sym]
Here are the values of C1 and C2:
s.C1
ans =
-4/3
s.C2
ans =
27
2
Here is the solution:
u=subs(u,{C1,C2},{s.C1,s.C2})
u =
-1/6*x^3-1/2*x^2-4/3*x+2
Notice how, when substituting for two or more variables, the variables and the values are enclosed in curly
braces.
Let us check our solution:
-diff(u,x,2)
ans =
x+1
subs(u,x,0)
ans =
2
subs(u,x,1)
ans =
0
The differential equation and the boundary conditions are satisfied.
Another example
Now consider the BVP with a nonconstant coefficient:
d x du
1 2 dx 0, 0 x 1,
dx
u (0) 20, u (1) 25.
Integrating once yields
du C1
( x) .
dx 1 x 2
(It is easier to perform this calculation in my head than to ask MATLAB to integrate 0.) I now perform the
second integration:
u=int(C1/(1+x/2),x)+C2
u =
2*C1*log(1+1/2*x)+C2
Now I use solve to find C1 and C2:
solve(subs(u,x,0)-20,subs(u,x,1)-25,C1,C2)
ans =
C1: [1x1 sym]
C2: [1x1 sym]
Here is the solution:
28
u=subs(u,{C1,C2},{ans.C1,ans.C2})
u =
5/log(3/2)*log(1+1/2*x)+20
I will check the answer:
-diff((1+x/2)*diff(u,x),x)
ans =
0
subs(u,x,0)
ans =
20
subs(u,x,1)
ans =
25
Chapter 3: Essential linear algebra
Section 3.1 Linear systems as linear operator equations
I have already showed you how to enter matrices and vectors in MATLAB. I will now introduce a few
more elementary operations on matrices and vectors, and explain how to extract components from a vector
and entries, rows, or columns from a matrix. At the end of this section, I will describe how MATLAB can
perform symbolic linear algebra; until then, the examples will use floating point arithmetic.
clear
Consider the matrix
A=[1 2 3;4 5 6;7 8 9]
A =
1 2 3
4 5 6
7 8 9
The transpose is indicated by a single quote following the matrix name:
A'
ans =
1 4 7
2 5 8
3 6 9
Recall that, if x and y are column vectors, then the dot product of x and y can be computed as xTy:
x=[1;0;-1]
x =
1
0
-1
29
y=[1;2;3]
y =
1
2
3
x'*y
ans =
-2
Alternatively, I can use the dot function:
dot(x,y)
ans =
-2
I can extract a component of a vector,
x(1)
ans =
1
or an entry of a matrix:
A(2,3)
ans =
6
In place of an index, I can use a colon, which represents the entire range. For example, A(:,1) indicates all
of the rows in the first column of A. This yields a column vector:
A(:,1)
ans =
1
4
7
Similarly, I can extract a row:
A(2,:)
ans =
4 5 6
Section 3.2: Existence and uniqueness of solutions to Ax=b
MATLAB can find a basis for the null space of a matrix. Consider the matrix
B=[1 2 3;4 5 6;7 8 9]
B =
1 2 3
4 5 6
7 8 9
Here is a basis for the null space:
x=null(B)
30
x =
-0.4082
0.8165
-0.4082
Since MATLAB returned a single vector, this indicates that the null space is one-dimensional. Here is a
check of the result:
B*x
ans =
1.0e-015 *
0
-0.4441
0
Notice that, since the computation was done in floating point, the product Bx is not exactly zero, but just
very close.
If a matrix is nonsingular, its null space is trivial:
A=[1,2;2,1]
A =
1 2
2 1
null(A)
ans =
Empty matrix: 2-by-0
On the other hand, the null space can have dimension greater than one:
A=[1 1 1;2 2 2;3 3 3;]
A =
1 1 1
2 2 2
3 3 3
null(A)
ans =
0.8165 0
-0.4082 -0.7071
-0.4082 0.7071
The matrix A has a two-dimensional null space.
MATLAB can compute the inverse of a nonsingular matrix:
A=[1 0 -1;3 2 1;2 -1 1]
A =
1 0 -1
3 2 1
2 -1 1
The command is called inv:
Ainv=inv(A)
Ainv =
0.3000 0.1000 0.2000
31
-0.1000 0.3000 -0.4000
-0.7000 0.1000 0.2000
A*Ainv
ans =
1.0000 -0.0000 0.0000
0 1.0000 0.0000
0 -0.0000 1.0000
Using the inverse, you can solve a linear system
b=[1;1;1]
b =
1
1
1
x=Ainv*b
x =
0.6000
-0.2000
-0.4000
A*x
ans =
1.0000
1.0000
1.0000
On the other hand, computing and using the inverse of a matrix A is not the most efficient way to solve
Ax=b. It is preferable to solve the system directly using some variant of Gaussian elimination. The
backslash operator indicates to MATLAB that a linear system is to be solved:
x1=A\b
x1 =
0.6000
-0.2000
-0.4000
x-x1
ans =
1.0e-015 *
-0.1110
0
0
(To remember how the backslash operator works, just think of A\b as "dividing b on the left by A," or
multiplying b on the left by A-1. However, MATLAB does not compute the inverse.)
Section 3.3: Basis and dimension
In this section, I will further demonstrate some of the capabilities of MATLAB by repeating some of the
examples from Section 3.3 of the text.
Example 3.25
Consider the three vectors v1, v2, v3 defined as follows:
v1=[1/sqrt(3);1/sqrt(3);1/sqrt(3)]
32
v1 =
0.5774
0.5774
0.5774
v2=[1/sqrt(2);0;-1/sqrt(2)]
v2 =
0.7071
0
-0.7071
v3=[1/sqrt(6);-2/sqrt(6);1/sqrt(6)]
v3 =
0.4082
-0.8165
0.4082
I will verify that these vectors are orthogonal:
v1'*v2
ans =
0
v1'*v3
ans =
0
v2'*v3
ans =
0
Example 3.26
I will verify that the following three quadratic polynomials for a basis for P 2. Note that while I did the
previous example in floating point arithmetic, this example requires symbolic computation.
clear
syms p1 p2 p3 x
p1=1
p1 =
1
p2=x-1/2
p2 =
x-1/2
p3=x^2-x+1/2
p3 =
x^2-x+1/2
Now suppose that q(x) is an arbitrary quadratic polynomial:
syms q a b c
q=a*x^2+b*x+c
q =
a*x^2+b*x+c
I want to show that q can be written in terms of p1, p2, and p3:
syms c1 c2 c3
q-(c1*p1+c2*p2+c3*p3)
ans =
a*x^2+b*x+c-c1-c2*(x-1/2)-c3*(x^2-x+1/2)
33
I need to gather like terms in this expression, which is accomplished with the collect command:
collect(ans,x)
ans =
(a-c3)*x^2+(b-c2+c3)*x+1/2*c2+c-c1-1/2*c3
I can now set each coefficient equal to zero and solve for the coefficients:
r=solve(a-c3,b-c2+c3,1/2*c2+c-c1-1/2*c3,c1,c2,c3)
r =
c1: [1x1 sym]
c2: [1x1 sym]
c3: [1x1 sym]
r.c1
ans =
1/2*b+c
r.c2
ans =
b+a
r.c3
ans =
a
The fact that there is a unique solution to this system, regardless of the values of a, b, c, shows that every
quadratic polynomial can be uniquely written as a linear combination of p1, p2, p3, and hence that these
three polynomials form a basis for P2.
Example
Here is a final example. Consider the following three vectors in R3:
u1=[1;0;2];
u2=[0;1;1];
u3=[1;2;-1];
I will verify that {u1,u2,u3} is a basis for R3, and express the vector
x=[8;2;-4];
in terms of the basis. As discussed in the text, {u1,u2,u3} is a basis if and only if the matrix A whose
columns are u1, u2, u3 is nonsingular. It is easy to form the matrix A:
A=[u1,u2,u3]
A =
1 0 1
0 1 2
2 1 -1
One way to determine if A is nonsingular is to compute its determinant:
det(A)
ans =
-5
34
Another way to determine whether A is nonsingular is to simply try to solve a system involving A, since
MATLAB will print a warning or error message if A is singular:
c=A\x
c =
3.6000
-6.8000
4.4000
Here is a verification of the results:
x-(c(1)*u1+c(2)*u2+c(3)*u3)
ans =
0
0
0
Symbolic linear algebra
Recall that MATLAB performs its calculations in floating point arithmetic by default. However, if desired,
we can perform them symbolically. For an illustration, I will repeat the previous example.
clear
syms u1 u2 u3
u1=sym([1;0;2]);
u2=sym([0;1;1]);
u3=sym([1;2;-1]);
A=[u1,u2,u3]
A =
[ 1, 0, 1]
[ 0, 1, 2]
[ 2, 1, -1]
x=sym([8;2;-4]);
c=A\x
c =
[ 18/5]
[ -34/5]
[ 22/5]
The solution is the same as before.
Programming in MATLAB, part I
Before I continue on to Section 3.4, I want to explain simple programming in MATLAB---specifically, how
to define new MATLAB functions. I have already shown you one way to do this: using the inline
command. (Also, a symbolic expression can be used in place of a function for many purposes. For
instance, it can be evaluated using the subs command.) However, in addition to the inline command, there
is another, more powerful method for defining MATLAB functions:
Defining a MATLAB function in an M-file.
An M-file is a plain text file whose name ends in ".m" and which contains MATLAB commands. There are
two types of M-files, scripts and functions. I will explain scripts later. A function is a MATLAB
subprogram: it accepts inputs and computes outputs using local variables. The first line in a function must
be of the form
35
function [output1,output2,…]=function_name(input1,input2,…)
If there is a single output, the square brackets can be omitted. Also, a function can have zero inputs and/or
zero outputs.
Here is the simplest type of example. Suppose I wish to define the function f(x)=sin(x2). The following
lines, stored in the M-file f.m, accomplish this:
function y=f(x)
y=sin(x.^2);
(Notice how I use the ".^" operator to vectorize the computation. Predefined MATLAB functions are
always vectorized, and so user-defined functions typically should be as well.) I can now use f just as a pre-
defined function such as sin. For example:
clear
x=linspace(-3,3,101);
plot(x,f(x))
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-3 -2 -1 0 1 2 3
A few important points:
The names of user-defined functions can be listed using the what command (similar to who, but
instead of listing variables, it lists M-files in the working directory).
The contents of an M-file can be displayed using the type command.
In order for you to use an M-file, MATLAB must be able to find it, which means that the M-file
must be in a directory on the MATLAB path. The MATLAB path always includes the working
directory, which can be determined using the pwd (print working directory) command. The
MATLAB path can be listed using the path command. Other directories can be added to the
MATLAB path using the addpath command. For more information, see "help addpath".
The current path is
path
36
MATLABPATH
C:\MATLAB6p5\toolbox\matlab\general
C:\MATLAB6p5\toolbox\matlab\ops
C:\MATLAB6p5\toolbox\matlab\lang
C:\MATLAB6p5\toolbox\matlab\elmat
C:\MATLAB6p5\toolbox\matlab\elfun
C:\MATLAB6p5\toolbox\matlab\specfun
C:\MATLAB6p5\toolbox\matlab\matfun
C:\MATLAB6p5\toolbox\matlab\datafun
C:\MATLAB6p5\toolbox\matlab\audio
C:\MATLAB6p5\toolbox\matlab\polyfun
C:\MATLAB6p5\toolbox\matlab\funfun
C:\MATLAB6p5\toolbox\matlab\sparfun
C:\MATLAB6p5\toolbox\matlab\graph2d
C:\MATLAB6p5\toolbox\matlab\graph3d
C:\MATLAB6p5\toolbox\matlab\specgraph
C:\MATLAB6p5\toolbox\matlab\graphics
C:\MATLAB6p5\toolbox\matlab\uitools
C:\MATLAB6p5\toolbox\matlab\strfun
C:\MATLAB6p5\toolbox\matlab\iofun
C:\MATLAB6p5\toolbox\matlab\timefun
C:\MATLAB6p5\toolbox\matlab\datatypes
C:\MATLAB6p5\toolbox\matlab\verctrl
C:\MATLAB6p5\toolbox\matlab\winfun
C:\MATLAB6p5\toolbox\matlab\winfun\comcli
C:\MATLAB6p5\toolbox\matlab\demos
C:\MATLAB6p5\toolbox\local
C:\MATLAB6p5\toolbox\curvefit\curvefit
C:\MATLAB6p5\toolbox\curvefit\cftoolgui
C:\MATLAB6p5\toolbox\nnet\nnet
C:\MATLAB6p5\toolbox\nnet\nnutils
C:\MATLAB6p5\toolbox\nnet\nncontrol
C:\MATLAB6p5\toolbox\nnet\nndemos
C:\MATLAB6p5\toolbox\nnet\nnobsolete
C:\MATLAB6p5\toolbox\optim
C:\MATLAB6p5\toolbox\pde
C:\MATLAB6p5\toolbox\rptgen
C:\MATLAB6p5\toolbox\signal\signal
C:\MATLAB6p5\toolbox\signal\sigtools
C:\MATLAB6p5\toolbox\signal\sptoolgui
C:\MATLAB6p5\toolbox\signal\sigdemos
C:\MATLAB6p5\toolbox\splines
C:\MATLAB6p5\toolbox\stats
C:\MATLAB6p5\toolbox\symbolic
C:\MATLAB6p5\toolbox\wavelet\wavelet
C:\MATLAB6p5\toolbox\wavelet\wavedemo
C:\MATLAB6p5\work
The working directory is
pwd
ans =
C:\MATLAB6p5\bin\win32
The files associated with this tutorial are (on my computer) in C:\MATLAB6p5\work. (When you install
the tutorial on your computer, you will have to make sure that all of the files that come with the tutorial are
in an accessible directory.) Here are all of the M-files in the directory called work:
what work
37
M-files in directory C:\MATLAB6p5\work
FemMesh f2
LinFcn f2a
LoadV f6
NodalValues g
QuadTriOne1 g1
QuadTriOne2 h
RectangleMeshD l2ip
RectangleMeshM l2norm
ShowMesh mkpl
ShowPWLinFcn myplot
Stiffness myplot1
beuler mysubs
eip mysubs1
euler scriptex
euler1 testfun
euler2 testit
f testsym
f1
I can look at f.m:
type f
function y=f(x)
y=sin(x.^2);
One important feature of functions defined in M-files is that variables defined in an M-file are local;
that is, they exist only while the M-file is being executed. Moreover, variables defined in the MATLAB
workspace (that is, the variables listed when you give the who command at the MATLAB prompt) are not
accessible inside of an M-file function. Here are examples:
type g
function y=g(x)
a=3;
y=sin(a*x);
clear
g(1)
ans =
0.1411
a
??? Undefined function or variable 'a'.
The variable a is not defined in the MATLAB workspace after g executes, since a was local to the M-file
g.m.
On the other hand, consider:
type h
function y=g(x)
y=sin(a*x);
clear
38
a=3
a =
3
h(1)
??? Undefined function or variable 'a'.
Error in ==> C:\MATLAB6p5\work\h.m
On line 3 ==> y=sin(a*x);
The M-file h.m cannot use the variable a, since the MATLAB workspace is not accessible within h.m.
(In short, we say that a is not "in scope".)
Here is an example with two outputs:
type f1
function [y,dy]=f1(x)
y=3*x.^2-x+4;
dy=6*x-1;
The M-file function f1.m computes both f(x)=3x2-x+4 and its derivative. It can be used as follows:
[v1,v2]=f1(1)
v1 =
6
v2 =
5
As another example, here is a function of two variables:
type g1
function z=g1(x,y)
z=2*x.^2+y.^2+x.*y;
g1(1,2)
ans =
8
Optional inputs with default values
I now want to explain the use of optional inputs in M-files. Suppose you are going to be working with the
function f2(x)=sin(ax2), and you know that, most of the time, the parameter a will have value 1. However, a
will occasionally take other values. It would be nice if you only had to give the input a when its value is not
the typical value of 1. The following M-file has this feature:
type f2
function y=f2(x,a)
if nargin<2
a=1;
end
y=sin(a.*x.^2);
39
The first executable statement, "if nargin<2", checks to see if f2 was invoked with one input or two. The
keyword nargin is an automatic (local) variable whose value is the number of inputs. The M-file f2.m
assigns the input a to have the value 1 if the user did not provide it. Now f2 can be used with one or two
parameters:
f2(pi)
ans =
-0.4303
f2(pi,sqrt(2))
ans =
0.9839
Comments in M-files
If the percent sign % is used in a MATLAB command, the remainder of the line is considered to be a
comment and is ignored by MATLAB. Here is an example:
if sin(pi)==0 % Testing for roundoff error
disp('No roundoff error!')
else
disp('Roundoff error detected!')
end
Roundoff error detected!
(Notice the use of the disp command, which displays a string or the value of a variable. See "help disp" for
more information. Notice also the use of the if-else block, which I discuss later in the tutorial.)
The most common use of comments is for documentation in M-files. Here is a second version of the
function f2 defined above:
type f2a
function y=f2a(x,a)
%y=f2a(x,a)
%
% This function implements the function f2(x)=sin(a*x). The parameter
% a is optional; if it is not provided, it is taken to be 1.
if nargin<2
a=1;
end
y=sin(a*x);
Notice how the block of comment lines explains the purpose and usage of the function. One of the
convenient features of the MATLAB help system is that the first block of comments in an M-file is
displayed when "help" is requested for that function:
help f2a
y=f2a(x,a)
This function implements the function f2(x)=sin(a*x). The parameter
a is optional; if it is not provided, it is taken to be 1.
I will explain more about MATLAB programming in Chapter 4.
40
M-files as scripts
An M-file that does not begin with the word function is regarded as a script, which is nothing more than a
collection of MATLAB commands that are executed sequentially, just as if they had been typed at the
MATLAB prompt. Scripts do not have local variables, and do not accept inputs or return outputs. A
common use for a script is to collect the commands that define and solve a certain problem (e.g. a
homework problem!).
Here is a sample script. Its purpose is to graph the function
x
f ( x) e cos(s ) ds
0
on the interval [0,1]. (Recall that MATLAB cannot compute the integral explicitly, so this is a nontrivial
task.) (Caveat: I did not try to make the following script particularly efficient.)
type scriptex
% Define the integrand
g=inline('exp(cos(s))','s');
% Create a grid
x=linspace(0,1,21);
y=zeros(1,21);
% Evaluate the integral int(exp(cos(s)),s,0,x) for
% each value of x on the grid:
for ii=1:length(x)
y(ii)=quad(g,0,x(ii));
end
% Now plot the result
plot(x,y)
Here is the result of running scriptex.m:
clear
scriptex
Warning: Minimum step size reached; singularity possible.
(Type "warning off MATLAB:quad:MinStepSize" to suppress this warning.)
> In C:\MATLAB6p5\toolbox\matlab\funfun\quad.m at line 85
In C:\MATLAB6p5\work\scriptex.m at line 15
41
2.5
2
1.5
1
0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(The warning message above means that MATLAB's quad function could not verify the accuracy of the
computed result.) As I mentioned above, scripts do not have local variables. Any variables defined in
scriptex exist in the MATLAB workspace:
whos
Name Size Bytes Class
g 1x1 838 inline object
ii 1x1 8 double array
x 1x21 168 double array
y 1x21 168 double array
Grand total is 87 elements using 1182 bytes
Section 3.4: Orthogonal bases and projection
Consider the following three vectors:
clear
v1=[1/sqrt(3);1/sqrt(3);1/sqrt(3)];
v2=[1/sqrt(2);0;-1/sqrt(2)];
v3=[1/sqrt(6);-2/sqrt(6);1/sqrt(6)];
These vectors are orthonormal, as can easily be checked. Therefore, I can easily express any vector as a
linear combination of the three vectors, which form an orthonormal basis. To test this, I will use the randn
command to generate a random vector (for more information, see "help randn"):
42
x=randn(3,1)
x =
0.1139
1.0668
0.0593
y=(x'*v1)*v1+(x'*v2)*v2+(x'*v3)*v3
y =
0.1139
1.0668
0.0593
y-x
ans =
1.0e-015 *
-0.0278
0.2220
-0.0139
Notice that the difference between y and x (which should be equal) is due to roundoff error, and is very
small.
Working with the L2 inner product
Since MATLAB can compute integrals symbolically, we can use it to compute the L2 inner product and
norm. For example:
clear
syms x
f=x
f =
x
g=x^2
g =
x^2
int(f*g,x,0,1)
ans =
1/4
If you are going to perform such calculations repeatedly, it is convenient to define a function to compute the
L2 inner product. The M-file l2ip.m does this:
help l2ip
I=l2ip(f,g,a,b,x)
This function computes the L^2 inner product of two functions
f(x) and g(x), that is, it computes the integral from a to b of
f(x)*g(x). The two functions must be defined by symbolic
expressions f and g.
The variable of integration is assumed to be x. A different
variable can passed in as the (optional) fifth input.
The inputs a and b, defining the interval [a,b] of integration,
are optional. The default values are a=0 and b=1.
43
Notice that I assigned the default interval to [0,1]. Here is an example:
syms x
l2ip(x,x^2)
ans =
1/4
For convenience, I also define a function computing the L2 norm:
help l2norm
I=l2norm(f,a,b,x)
This function computes the L^2 inner product of the function f(x)
The functions must be defined by the symbolic expressions f.
The variable of integration is assumed to be x. A different
variable can passed in as the (optional) fourth input.
The inputs a and b, defining the interval [a,b] of integration,
are optional. The default values are a=0 and b=1.
l2norm(x)
ans =
1/3*3^(1/2)
double(ans)
ans =
0.5774
Example 3.35
Now consider the following two functions:
clear
syms x pi
f=x*(1-x)
f =
x*(1-x)
g=8/pi^3*sin(pi*x)
g =
8/pi^3*sin(pi*x)
(I must tell MATLAB that pi is to be regarded as symbolic.) The following graph shows that the two
functions are quite similar on the interval [0,1]:
t=linspace(0,1,51);
f1=subs(f,x,t);
g1=subs(g,x,t);
plot(t,f1,'-',t,g1,'--')
44By how much do the two functions differ? One way to answer this question is to compute the relative
difference in the L2 norm:
l2norm(f-g)/l2norm(f)
ans =
1/pi^3*(pi^6-960)^(1/2)
double(ans)
ans =
0.0380
The difference is less than 4%.
Here are two more examples from Section 3.4 that illustrate some of the capabilities of MATLAB.
Example 3.37
The purpose of this example to compute the first-degree polynomial f(x)=mx+b best fitting given data points
(xi,yi). The data given in this example can be stored in two vectors:
clear
x=[0.1;0.3;0.4;0.75;0.9];
y=[1.7805;2.2285;2.3941;3.2226;3.5697];
Here is a plot of the data:
plot(x,y,'o')
45
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
To compute the first-order polynomial f(x)=mx+b that best fits this data, I first form the Gram matrix. The
ones command creates a vector of all ones:
e=ones(5,1)
e =
1
1
1
1
1
G=[x'*x,x'*e;e'*x,e'*e]
G =
1.6325 2.4500
2.4500 5.0000
Next, I compute the right hand side of the normal equations:
z=[x'*y;e'*y]
z =
7.4339
13.1954
Now I can solve for the coefficients c=[m;b]:
c=G\z
c =
2.2411
1.5409
I will now define the solution as an inline function:
46
l=inline('c(1)*x+c(2)','x','c')
l =
Inline function:
l(x,c) = c(1)*x+c(2)
(Any variables that appear in the definition of an inline function are local to that function, so the vector c
must be passed as an input.) Here is a plot of the best fit line, together with the data:
t=linspace(0,1,11);
plot(t,l(t,c),x,y,'o')
4
3.5
3
2.5
2
1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The fit is not bad.
Example 3.38
In this example, I will compute the best quadratic approximation to the function ex on the interval [0,1].
Here are the basis functions for the subspace P2:
clear
syms x
p1=1;
p2=x;
p3=x^2;
I now compute the Gram matrix and the right hand side of the normal equations:
G=[l2ip(p1,p1),l2ip(p1,p2),l2ip(p1,p3)
l2ip(p2,p1),l2ip(p2,p2),l2ip(p2,p3)
l2ip(p3,p1),l2ip(p3,p2),l2ip(p3,p3)]
G =
[ 1, 1/2, 1/3]
[ 1/2, 1/3, 1/4]
[ 1/3, 1/4, 1/5]
47
b=[l2ip(p1,exp(x));l2ip(p2,exp(x));l2ip(p3,exp(x))]
b =
[ exp(1)-1]
[ 1]
[ exp(1)-2]
Now I solve the normal equations and find the best fit quadratic:
c=G\b
c =
[ -105+39*exp(1)]
[ 588-216*exp(1)]
[ -570+210*exp(1)]
Here is the solution:
q=c(1)*p1+c(2)*p2+c(3)*p3
q =
-105+39*exp(1)+(588-216*exp(1))*x+(-570+210*exp(1))*x^2
Here is the graph of y=ex and the quadratic approximation:
t=linspace(0,1,41)';
plot(t,exp(t),'-',t,subs(q,x,t),'--')
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Since the approximation is so accurate, it is more informative to graph the error:
plot(t,exp(t)-subs(q,x,t))
48
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
We can also judge the fit by computing the relative error in the L2 norm:
l2norm(exp(x)-q)/l2norm(exp(x))
ans =
(-7334-994*exp(1)^2+5400*exp(1))^(1/2)/(2*exp(1)^2-2)^(1/2)
double(ans)
ans =
0.0030
The error is less than 0.3%.
Section 3.5: Eigenvalues and eigenvectors of a symmetric matrix
MATLAB can compute eigenvalues and eigenvectors of a square matrix, either numerically or
symbolically.
Numerical eigenvalues and eigenvectors
Recall that a matrix or any other quantity is stored in floating point by default:
clear
A=[1,2,-1;4,0,1;-7,-2,3]
A =
1 2 -1
4 0 1
-7 -2 3
The eig command computes the eigenvalues and eigenvectors:
49
[V,D]=eig(A)
V =
-0.4986 0.2554 0.0000
0.8056 0.0113 0.4472
-0.3200 -0.9668 0.8944
D =
-2.8730 0 0
0 4.8730 0
0 0 2.0000
The eig command returns two matrices. The first contains the eigenvectors as the columns of the matrix,
while the second is a diagonal matrix with the eigenvalues on the diagonal. The eigenvectors and
eigenvalues are given in the same order. For example:
A*V(:,1)-D(1,1)*V(:,1)
ans =
1.0e-014 *
-0.1776
0.1776
0.0555
(Notice how the colon is used to extract the first column of A.) The eigenvalue-eigenvector equation holds
up to roundoff error.
It is also possible to call the eig command with a single output, in which case only the eigenvalues are
returned, and in a vector instead of a matrix:
ev=eig(A)
ev =
-2.8730
4.8730
2.0000
Symbolic eigenvalues and eigenvectors:
To obtain symbolic (exact) eigenvalues and eigenvectors, it is only necessary to define the matrix to be
symbolic:
clear
A=sym([1,2,-1;4,0,1;-7,-2,3])
A =
[ 1, 2, -1]
[ 4, 0, 1]
[ -7, -2, 3]
The computation then proceeds exactly as before:
[V,D]=eig(A)
V =
[ 0, 11/17-4/17*15^(1/2), 11/17+4/17*15^(1/2)]
[ 1, -43/34+11/34*15^(1/2), -43/34-11/34*15^(1/2)]
[ 2, 1, 1]
D =
[ 2, 0, 0]
[ 0, 1+15^(1/2), 0]
[ 0, 0, 1-15^(1/2)]
Recall that you can use the pretty command to make output easier to read. For example, here is the second
eigenvector:
50
pretty(V(:,2))
[11 1/2]
[-- - 4/17 15 ]
[17 ]
[ ]
[ 43 11 1/2 ]
[--- + -- 15 ]
[ 34 34 ]
[ ]
[ 1 ]
It is not always possible to compute eigenvalues and eigenvectors exactly. Indeed, the eigenvalues of an n
by n matrix are the roots of the nth-degree characteristic polynomial. One of the most famous results of 19 th
century mathematics is that it is impossible to find a formula (analogous to the quadratic formula)
expressing the roots of an arbitrary polynomial of degree 5 or more. For this reason, MATLAB cannot
always find the eigenvalues of a symbolic matrix exactly. When it cannot, it automatically computes them
approximately, using high precision arithmetic.
Here is a famous matrix, the Hilbert matrix:
clear
H=sym(hilb(5))
H =
[ 1, 1/2, 1/3, 1/4, 1/5]
[ 1/2, 1/3, 1/4, 1/5, 1/6]
[ 1/3, 1/4, 1/5, 1/6, 1/7]
[ 1/4, 1/5, 1/6, 1/7, 1/8]
[ 1/5, 1/6, 1/7, 1/8, 1/9]
I will now try to compute the eigenvalues and eigenvectors symbolically:
[V,D]=eig(H);
Here is the first computed eigenvector:
V(:,1)
ans =
[ -.52907002564424155158755766298139e-1]
[ 1]
[ -4.3375760021578551686331547179714]
[ 6.5744548295382247175008321367533]
[ -3.2242410569431814113972564982240]
Notice that the result appears to be in floating point, but with a large number of digits. In fact, MATLAB
has returned a symbolic quantity, as the command whos shows:
whos
Name Size Bytes Class
D 5x5 1952 sym object
H 5x5 1710 sym object
V 5x5 2862 sym object
ans 5x1 642 sym object
Grand total is 1135 elements using 7166 bytes
51
The results are highly accurate, as the number of digits would suggest:
H*V(:,1)-D(1,1)*V(:,1)
ans =
[ -.5279857e-31]
[ -.305448e-31]
[ -.30122e-31]
[ -.4237e-32]
[ -.18583e-31]
The explanation of these results is that when MATLAB could not compute the eigenpairs symbolically, it
automatically switched to variable precision arithmetic, which, by default, uses 32 digits in all calculations.
I will not have any need for variable precision arithmetic in this tutorial, but I wanted to mention it in case
you encounter it unexpectedly, as in the above example. You may wish to explore this capability of
MATLAB further for your own personal use. You can start with "help vpa".
Since it is not usually possible to find eigenpairs symbolically (except for small matrices), it is typical to
perform a floating point computation from the very beginning.
Example 3.49
I will now use the spectral method to solve Ax=b, where A and b are as defined below:
clear
A=[11,-4,-1;-4,14,-4;-1,-4,11]
A =
11 -4 -1
-4 14 -4
-1 -4 11
b=[1;2;1]
b =
1
2
1
The matrix A is symmetric, so its eigenvalues are necessarily real and its eigenvectors orthogonal.
Moreover, MATLAB's eig returns normalized eigenvectors when the input is a floating point matrix.
[V,D]=eig(A)
V =
0.5774 0.7071 -0.4082
0.5774 -0.0000 0.8165
0.5774 -0.7071 -0.4082
D =
6.0000 0 0
0 12.0000 0
0 0 18.0000
The solution of Ax=b is then
x=(V(:,1)'*b/D(1,1))*V(:,1)+(V(:,2)'*b/D(2,2))*V(:,2)+(V(:,3)'*b/D(3,3))
*V(:,3)
x =
0.2037
0.2593
0.2037
52
Check:
A*x-b
ans =
1.0e-015 *
0.6661
0.8882
-0.2220
Review: Functions in MATLAB
I have presented three ways to define new functions in MATLAB. I will now review and compare these
three mechanisms.
First of all, an expression in one or more variables can be used to represent a function. For example, to
work with the function f(x)=sin(x2), I define
clear
syms x
f=x^2
f =
x^2
Now I can do whatever I wish with this function, including evaluate it, differentiate it, and integrate it. To
evaluate f, I use the subs command:
subs(f,x,3)
ans =
9
Differentiation and integration are performed with diff and int, respectively:
diff(f,x)
ans =
2*x
int(f,x)
ans =
1/3*x^3
As I have shown in earlier examples, a symbolic expression can be evaluated at a vector argument; that is, it
is automatically vectorized:
t=linspace(0,1,21);
plot(t,subs(f,x,t))
53 second way to define a function is using the inline command:
clear
f=inline('x^2','x')
f =
Inline function:
f(x) = x^2
An advantage of using an inline function is that functional notation is used for evaluation (instead of the
subs command).
f(3)
ans =
9
Inline functions can be evaluated with symbolic inputs:
syms a
f(a)
ans =
a^2
Symbolic calculus operations can be performed on inline functions indirectly, bearing in mind that diff and
int operate on expressions, not functions. Here is the right way and wrong way to differentiate f:
syms x
diff(f(x),x)
ans =
2*x
diff(f,x)
??? Error using ==> diff
Function 'diff' is not defined for values of class 'inline'.
54
Here is an important fact to notice: inline functions are not automatically vectorized:
t=linspace(0,1,41);
plot(t,f(t))
??? Error using ==> inlineeval
Error in inline expression ==> x^2
??? Error using ==> ^
Matrix must be square.You can vectorize an inline function when you write it by using ".^", ".*", and "./". You can also change a
nonvectorized inline function into a vectorized one using the vectorize command:
g=vectorize(f)
g =
Inline function:
g(x) = x.^2
plot(t,g(t)) third way to define a new function in MATLAB is to write an M-file. The major advantage of this
method is that very complicated functions can be defined, in particular, functions that require several
MATLAB commands to evaluate. You can review the section "Programming in MATLAB, part I" at this
point, if necessary.
55
Chapter 4: Essential ordinary differential equations
Section 4.2: Solutions to some simple ODEs
Second-order linear homogeneous ODEs with constant coefficients
Suppose we wish to solve the following IVP:
d 2u du
2
4 3u 0, t 0,
dt dt
u (0) 1,
du
(0) 0.
dt
The characteristic polynomial is r2+4r-3, which has the following roots:
clear
syms r
l=solve(r^2+4*r-3,r)
l =
[ -2+7^(1/2)]
[ -2-7^(1/2)]
The general solution of the ODE is then
syms t c1 c2
u=c1*exp(l(1)*t)+c2*exp(l(2)*t)
u =
c1*exp((-2+7^(1/2))*t)+c2*exp((-2-7^(1/2))*t)
We can now solve for the unknown coefficients c1, c2:
c=solve(subs(u,t,0)-1,subs(diff(u,t),t,0),c1,c2)
c =
c1: [1x1 sym]
c2: [1x1 sym]
The solution is found by substituting the correct values for c1 and c2:
subs(u,c1,c.c1)
ans =
(1/7*7^(1/2)+1/2)*exp((-2+7^(1/2))*t)+c2*exp((-2-7^(1/2))*t)
u=subs(ans,c2,c.c2)
u =
(1/7*7^(1/2)+1/2)*exp((-2+7^(1/2))*t)+1/14*(-2+7^(1/2))*7^(1/2)*exp((-2-
7^(1/2))*t)
Here is a better view of the solution:
pretty(u)
56
1/2 1/2
(1/7 7 + 1/2) exp((-2 + 7 ) t)
1/2 1/2 1/2
+ 1/14 (-2 + 7 ) 7 exp((-2 - 7 ) t)
I can now plot the solution:
tt=linspace(0,5,21);
plot(tt,subs(u,t,tt))
25
20
15
10
5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
A special inhomogeneous second-order linear ODE
Consider the IVP
d 2u
4u sin(t ), t 0,
dt 2
u (0) 0,
du
(0) 0.
dt
The solution, as given in Section 4.2.2 of the text, is
clear
syms t s pi
(1/2)*int(sin(2*(t-s))*sin(pi*s),s,0,t)
ans =
(-sin(t*pi)+pi*sin(t)*cos(t))/(-2+pi)/(2+pi)
u=simplify(ans)
u =
57
(-sin(t*pi)+pi*sin(t)*cos(t))/(-4+pi^2)
pretty(u)
-sin(t pi) + pi sin(t) cos(t)
-----------------------------
2
-4 + pi
Let us check the solution:
diff(diff(u,t),t)+4*u
ans =
(sin(t*pi)*pi^2-4*pi*sin(t)*cos(t))/(-4+pi^2)+4*(-
sin(t*pi)+pi*sin(t)*cos(t))/(-4+pi^2)
simplify(ans)
ans =
sin(t*pi)
The ODE is satisfied. How about the initial conditions?
subs(u,t,0)
ans =
0
subs(diff(u,t),t,0)
ans =
0
The initial conditions are also satisfied.
First-order linear ODEs
Now consider the following IVP:
du 1
u t , t 0,
dt 2
u ( 0 ) 1.
Section 4.2.3 contains an explicit formula for the solution:
clear
syms t s
exp(t/2)+int(exp((t-s)/2)*(-s),s,0,t)
ans =
-3*exp(1/2*t)+2*t+4
u=ans
u =
-3*exp(1/2*t)+2*t+4
Here is a graph of the solution:
tt=linspace(0,3,41);
plot(tt,subs(u,t,tt))
58Just out of curiosity, let us determine the value of t for which the solution is zero.
solve(u,t)
ans =
[ -2*lambertw(-3/4*exp(-1))-2]
[ -2*lambertw(-1,-3/4*exp(-1))-2]
The solve command finds two solutions, expressed in terms of the Lambert W function (see "help
lambertw" for details). We can convert the result to floating point:
double(ans)
ans =
-1.1603
1.9226
The second root is the one visible on the above graph.
As an alternative to solve, MATLAB provides a command, fzero, that looks for a single root numerically.
To use it, you must have an estimate of the desired root. To make it easier to find such an estimate, I will
add a grid to the previous graph using the grid command:
grid
59From the graph, we see that the desired root is a little less than 2. (Of course, we already knew that, having
done the above calculation.) fzero requires a function, not just an expression, so I convert the expression u
into an inline function:
u=inline(char(u),'t')
u =
Inline function:
u(t) = -3*exp(1/2*t)+2*t+4
(The char function converts a symbolic expression to a string, which is required by inline.) Now I call
fzero, using 2.0 as the initial estimate of the root:
fzero(u,2.0)
ans =
1.9226
The result is the same as before.
Section 4.3: Linear systems with constant coefficients
Since MATLAB can compute eigenvalues and eigenvectors (either numerically or, when possible,
symbolically), it can be used to solve linear systems with constant coefficients. I will begin with a simple
example, solving the homogeneous IVP
dx
Ax , t 0,
dt
x ( 0) x 0 ,
60
where iA and x0 have the values given below. Notice that I define A and x0 to be symbolic, and hence use
symbolic computations throughout this example.
clear
A=sym([1 2;3 4])
A =
[ 1, 2]
[ 3, 4]
x0=sym([4;1])
x0 =
[ 4]
[ 1]
The first step is to find the eigenpairs of A:
[V,D]=eig(A)
V =
[ -1/2+1/6*33^(1/2), -1/2-1/6*33^(1/2)]
[ 1, 1]
D =
[ 5/2+1/2*33^(1/2), 0]
[ 0, 5/2-1/2*33^(1/2)]
Notice that the matrix A is not symmetric, and so the eigenvectors are not orthogonal. However, they are
linearly independent, and so I can express the initial vector as a linear combination of the eigenvectors. The
coefficients are found by solving the linear system Vc=x0:
c=V\x0
c =
[ 9/22*33^(1/2)+1/2]
[ -9/22*33^(1/2)+1/2]
Now I can write down the solution:
syms t
x=c(1)*exp(D(1,1)*t)*V(:,1)+c(2)*exp(D(2,2)*t)*V(:,2)
x =
[ (9/22*33^(1/2)+1/2)*exp((5/2+1/2*33^(1/2))*t)*(-1/2+1/6*33^(1/2))+(-
9/22*33^(1/2)+1/2)*exp((5/2-1/2*33^(1/2))*t)*(-1/2-1/6*33^(1/2))]
[
(9/22*33^(1/2)+1/2)*exp((5/2+1/2*33^(1/2))*t)+(-
9/22*33^(1/2)+1/2)*exp((5/2-1/2*33^(1/2))*t)]
Here are the graphs of the two components of the solutions:
tt=linspace(0,1,21);
plot(tt,subs(x(1),t,tt),'-',tt,subs(x(2),t,tt),'--')
61
700
600
500
400
300
200
100
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Both components are dominated by a rapidly growing exponential, as a careful examination of the formulas
confirms.
Inhomogeneous systems and variation of parameters
I will now show how to use MATLAB to solve the inhomogeneous system
dx
Ax f (t ), t 0,
dt
x ( 0) x 0 .
Consider the matrix
clear
A=sym([1 2;2 1])
A =
[ 1, 2]
[ 2, 1]
and let
syms t
f=[sin(t);0]
f =
[ sin(t)]
[ 0]
x0=sym([0;1])
x0 =
[ 0]
[ 1]
Notice that the matrix A is symmetric. First, I find the eigenvalues and eigenvectors of A:
62
[V,D]=eig(A)
V =
[ 1, 1]
[ 1, -1]
D =
[ 3, 0]
[ 0, -1]
When operating on a symbolic matrix, MATLAB does not normalize the eigenvectors, so I normalize them
and call them v1 and v2:
v1=V(:,1)/sqrt(V(:,1)'*V(:,1))
v1 =
[ 1/2*2^(1/2)]
[ 1/2*2^(1/2)]
v2=V(:,2)/sqrt(V(:,2)'*V(:,2))
v2 =
[ 1/2*2^(1/2)]
[ -1/2*2^(1/2)]
For convenience, I will call the eigenvalues l1 and l2:
l1=D(1,1)
l1 =
3
l2=D(2,2)
l2 =
-1
We have f(t)=c1(t)v1+c2(t)v2 and x0=b1*v1+b2*v2, where
c1=v1'*f
c1 =
1/2*2^(1/2)*sin(t)
c2=v2'*f
c2 =
1/2*2^(1/2)*sin(t)
b1=v1'*x0
b1 =
1/2*2^(1/2)
b2=v2'*x0
b2 =
-1/2*2^(1/2)
I now solve the two decoupled IVPs
da1
l1 a1 c1 (t ), a1 (0) b1 ,
dt
da 2
l 2 a 2 c 2 (t ), a 2 (0) b2
dt
using the methods of Section 4.2. The solution to the first is
63
syms s
a1=b1*exp(l1*t)+int(exp(l1*(t-s))*subs(c1,t,s),s,0,t)
a1 =
11/20*2^(1/2)*exp(3*t)-1/20*2^(1/2)*cos(t)-3/20*2^(1/2)*sin(t)
(Notice how I used the subs command to change the variable in the expression for c1 from t to s.)
The solution to the second IVP is
a2=b2*exp(l2*t)+int(exp(l2*(t-s))*subs(c2,t,s),s,0,t)
a2 =
-1/4*2^(1/2)*exp(-t)-1/4*2^(1/2)*cos(t)+1/4*2^(1/2)*sin(t)
The solution to the original system is then
x=simplify(a1*v1+a2*v2)
x =
[ -1/20*(5*exp(-4*t)+6*cos(t)*exp(-3*t)-2*sin(t)*exp(-3*t)-11)*exp(3*t)]
[ 1/20*(5*exp(-4*t)+4*cos(t)*exp(-3*t)-8*sin(t)*exp(-3*t)+11)*exp(3*t)]
Let us check this result:
diff(x,t)-A*x-f
ans =
[ -1/20*(-20*exp(-4*t)-20*cos(t)*exp(-3*t))*exp(3*t)-
1/10*(5*exp(-4*t)+6*cos(t)*exp(-3*t)-2*sin(t)*exp(-3*t)-11)*exp(3*t)-
1/10*(5*exp(-4*t)+4*cos(t)*exp(-3*t)-8*sin(t)*exp(-3*t)+11)*exp(3*t)-
sin(t)]
[ 1/20*(-20*exp(-4*t)+20*sin(t)*exp(-3*t)-20*cos(t)*exp(-
3*t))*exp(3*t)+1/10*(5*exp(-4*t)+4*cos(t)*exp(-3*t)-8*sin(t)*exp(-
3*t)+11)*exp(3*t)+1/10*(5*exp(-4*t)+6*cos(t)*exp(-3*t)-2*sin(t)*exp(-
3*t)-11)*exp(3*t)]
The result ought to be the zero vector, if x really satisfies the differential equation. Before concluding that I
(or MATLAB) made an error, let us simplify the result:
simplify(ans)
ans =
[ 0]
[ 0]
Thus we see that the ODE is satisfied. We also have
subs(x,t,0)-x0
ans =
[ 0]
[ 0]
so the initial condition is satisfied as well.
Programming in MATLAB, Part II
In preparation for the next section, in which I discuss the implementation of numerical methods for IVPs in
MATLAB, I want to present more of the mechanisms for programming in MATLAB. In an earlier section
of this tutorial, I explained how to create a new function in an M-file. An M-file function need not
implement a simple mathematical function f(x). Indeed, a common use of M-files is to implement
algorithms, in which case the M-file should be thought of as a subprogram.
64
To program any nontrivial algorithm requires the common program control mechanisms for implementing
loops and conditionals.
Loops in MATLAB
Here is an example illustrating an indexed loop:
for jj=1:4
disp(jj^2)
end
1
4
9
16
This example illustrates the for loop in MATLAB, which has the form
for j=j1:j2
statement1
statement2
…
statementn
end
The sequence of statements statement1, statement2,…, statementn is executed j2-j1+1 times, first with j=j1,
then with j=j1+1, and so forth until j=j2.
MATLAB also has a while loop, which executes as long as a given logical condition is true. I will not need
the while loop in this tutorial, so I will not discuss it. The interested reader can consult "help while".
Conditional execution
The other common program control mechanism that I will often use in this tutorial is the if-elseif-else block.
I already used it once (see the M-file f2.m above). The general form is
if condition1
statement(s)
elseif condition2
statements(s)
.
.
.
elseif conditionk
statement(s)
else
statement(s)
end
When an if-elseif-else block is executed, MATLAB checks whether condition1 is true; if it is, the first block
of statements is executed, and then the program proceeds after the end statement. If condition1 is false,
MATLAB checks condition2, condition3, and so on until one of the conditions is true. If conditions 1
through k are all false, then the else branch executes.
65
Logical conditions in MATLAB evaluate to 0 or 1:
abs(0.5)<1
ans =
1
abs(1.5)<1
ans =
0
Therefore, the condition in an if or elseif statement can be any expression with a numerical value.
MATLAB regards a nonzero value as true, and zero as false.
To form complicated conditions, one can use the logical operators == (equality), & (logical and), and |
(logical or). Here are some examples:
clear
x=0.5
x =
0.5000
if x<-1|x>1 % true if x is less than –1 or x is greater than 1
disp('True!')
else
disp('False!')
end
False!
if x>-1 & x<1 % True if x is between –1 and 1
disp('True!')
else
disp('False')
end
True!
Passing one function into another
Often a program (which in MATLAB is a function) takes as input a function. For example, suppose I wish
to write a program that will plot a given function f on a given interval [a,b]. (This is just for the purpose of
illustration, since the program I produce below will not be much easier to use than the plot command itself.)
For the sake of this example, I will assume that we will pass either an inline function or an M-file function,
and not an expression, to the program.
Here is the desired program, which I will call myplot:
function myplot(f,a,b)
x=linspace(a,b,51);
% HERE I NEED TO CREATE A VECTOR y, WITH y(i)=f(x(i)).
plot(x,y)
To finish the program, I need to know how to evaluate a function that has been passed into myplot as the
first argument. The difficulty is that an M-file function must be passed into myplot by name, that is, as a
string, while an inline function can be passed in directly. The obvious statement, y=f(x), when put in the
above program in place of the comment line, will work if f is an inline function but not if f is an M-file
function.
66
This point is difficult to understand, so I will illustrate the different possibilities. Here is a version of
myplot that tries to use the statement y=f(x).
type myplot1
function myplot1(f,a,b)
x=linspace(a,b,51);
y=f(x);
plot(x,y)
Let us try myplot1 with the following inline function:
clear
g=inline('sin(x.^2)','x')
g =
Inline function:
g(x) = sin(x.^2)
myplot1(g,It works fine.
Now recall the function f that I defined earlier:
type f
function y=f(x)
y=sin(x.^2);
Here is the obvious way to call myplot1:
myplot(f,0,pi)
??? Input argument 'x' is undefined.
Error in ==> C:\MATLAB6p5\work\f.m
67
On line 3 ==> y=sin(x.^2);
The above error occurs before MATLAB reaches the executable statements in myplot1.m. In parsing the
command "myplot(f,0,pi)", MATLAB must evaluate all of the inputs, so that it can pass their values to the
M-file myplot1.m. However, in trying to evaluate f, MATLAB calls the M-file f.m and encounters an error
since the input to f has not been given.
What we really want to do is pass the name of f, as a string, to myplot1. This way we can tell the
subprogram which M-file to invoke, but MATLAB will not try to call f before myplot1 executes. So we try
this:
myplot1('f',0,pi)
??? Subscript indices must either be real positive integers or logicals.
Error in ==> C:\MATLAB6p5\work\myplot1.m
On line 4 ==> y=f(x);
This fails because, inside of myplot1, the variable f is a string, not a function, and so the statement
"y=f(x)" does not make any sense.
To get out of this impasse, we use the feval function, which evaluates a function, given its name and the
input. Thus
feval('f',1)
ans =
0.8415
is equivalent to
f(1)
ans =
0.8415
The feval command also accepts an inline function in place of a string:
feval(g,1)
ans =
0.8415
Here is the new version of myplot:
type myplot
function myplot(f,a,b)
x=linspace(a,b,51);
y=feval(f,x);
plot(x,y)
This version works equally well for either an inline function or an M-file function:
myplot(g,0,pi)
68myplot('f',Moral: To evaluate one function, that was passed as an argument to a second function, inside that second
function, use the feval command!
69
Section 4.4: Numerical methods for initial value problems
Now I will show how to implement numerical methods, such as those discussed in the text, in MATLAB
programs. I will focus on Euler's method; you should easily be able to extend these techniques to
implement a Runge-Kutta method.
Consider the IVP
du
f (t , u ), t t 0 ,
dt
u (t 0 ) u 0 .
A program that applies Euler's method to this problem should take, as input, f, t0, and u0, as well as
information that allows it to determine the step size and the number of steps, and return, as output, the
approximate solution on the resulting grid. I will assume that the user will specify the time interval [t0,tn]
and the number of steps n (then the program can compute the time step).
As an example, I will solve the IVP
du u
, t 0,
dt 1 t 2
u ( 0) 1.
on the interval [0,10]. I will do it first interactively, and then collect the commands in an M-file.
The first step is to define the grid and allocate space to save the computed solution. I will use 100 steps, so
the grid and the solution vector will each contain 101 components (t0,t1,t2,…,t100 and u0,u1,u2,…,u100). (It is
convenient to store the times t0,t1,t2,…,t100 in a vector for later use, such as graphing the solution.)
clear
t=linspace(0,10,101);
u=zeros(1,101);
Now I assign the initial value and compute the time step:
u(1)=1;
dt=10/100;
Next I define the function that forms the right-hand side of the ODE:
f=inline('u/(1+t^2)','t','u')
f =
Inline function:
f(t,u) = u/(1+t^2)
Euler's method is now implemented in a single loop:
for ii=1:100
u(ii+1)=u(ii)+dt*f(t(ii),u(ii));
end
70
The exact solution is u(t)=exp(tan-1(t)):
U=inline('exp(atan(t))','t')
U =
Inline function:
U(t) = exp(atan(t))
Here is a graph of the exact solution and the computed solution:
plot(t,U(t),t,u)
4.5
4
3.5
3
2.5
2
1.5
1
0 1 2 3 4 5 6 7 8 9 10
The computed solution is not too different from the exact solution. As is often the case, it is more
informative to graph the error:
plot(t,U(t)-u)
71
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
0 1 2 3 4 5 6 7 8 9 10
It is now easy to gather the above steps in an M-file that implements Euler's method:
type euler1
function [u,t]=euler1(f,t0,tf,u0,n)
%[u,t]=euler1 Compute the grid and allocate space for the solution
t=linspace(t0,tf,n+1);
u=zeros(1,n+1);
% Assign the initial value and compute the time step
u(1)=u0;
dt=(tf-t0)/n;
% Now do the computations in a loop
for ii=1:n
u(ii+1)=u(ii)+dt*feval(f,t(ii),u(ii));
end
The entire computation I did above now requires a single line:
72
[u,t]=euler(f,0,10,1,100);
plot(t,u)
4.5
4
3.5
3
2.5
2
1.5
1
0 1 2 3 4 5 6 7 8 9 10
Vectorizing the euler program
Euler's method applies equally well to systems of ODEs, and the euler1 program requires only minor
changes to handle a system. Mainly we just need to take into account that, at each time ti, the computed
solution ui is a vector. Here is the program, which works for either scalar or vector systems (since a scalar
can be viewed as a vector of length 1):
type euler2
function [u,t]=euler2(f,t0,tf,u0,n)
%[u,t]=euler2 The solution u can be a scalar or a vector. In the vector case, the
% initial value must be a kx1 vector, and the function f must return
% a kx1 vector73));
end
You should notice the use of the length command to determine the number of components in the initial
value u0. A related command is size, which returns the dimensions of an array:
A=randn(4,3);
size(A)
ans =
4 3
The length of an array is defined simply to be the maximum dimension:
length(A)
ans =
4
max(size(A))
ans =
4
The only other difference between euler1 and euler2 is that the vector version stores each computes value
ui as a column in a matrix.
As an example of solving a system, I will apply Euler's method to the system
du1
u 2 , u1 (0) 0,
dt
du 2
u1 , u 2 (0) 1.
dt
The right-hand side of the system is defined by the vector-valued function
u
f (t , u ) 2 ,
u1
which I define as an inline function:
f=inline('[u(2);-u(1)]','t','u')
f =
Inline function:
f(t,u) = [u(2);-u(1)]
Notice that the function must return a column vector, not a row vector. Notice also that, even though this
particular function f is independent of t, I wrote euler2 to expect a function f(t,u) of two variables.
Therefore, I had to define f above as a function of t and u, even though it is constant with respect to t.
74
The initial value is
u0=[0;1];
I will solve the system on the interval [0,6].
[u,t]=euler2(f,0,6,u0,100);
Here I plot both components of the solution:
plot(t,u(1,:),t,u(2,:))
1.5
1
0.5
0
-0.5
-1
-1.5
0 1 2 3 4 5 6
Programming in MATLAB, part III
There is a subtle improvement we can make in the above program implementing Euler's method. Often a
function, which is destined to be input to a program like euler2, depends not only on independent variables,
but also on one or more parameters. For example, suppose I wish to solve the following IVP for a several
different values of a:
du
au, u (0) 1.
dt
One way to handle this is to define a different function f for each different value of a:
f1=inline('u','t','u')
f1 =
Inline function:
f1(t,u) = u
f2=inline('2*u','t','u')
f2 =
Inline function:
75
f2(t,u) = 2*u
(and so forth). However, this is obviously tedious and inelegant. A better technique is to make f depend
directly on the parameter a; in effect, make f a function of three variables:
clear
f=inline('a*u','t','u','a')
f =
Inline function:
f(t,u,a) = a*u
The question is: How can a program implementing Euler's method recognize when the function f depends
on one or more parameters? The answer lies in taking advantage of MATLAB's ability to count the number
of arguments to a function. Recall that, inside an M-file function, the nargin variable records the number
of inputs. The program can then do different things, depending on the number of inputs. (I already showed
an example of the use of nargin above in the first section on MATLAB programming.)
Here is an M-file function implementing Euler's method and allowing for optional parameters:
type euler
function [u,t]=euler(f,t0,tf,u0,n,varargin)
%[u,t]=euler(f,t0,tf,u0,n,p1,p2,...) If the optional arguments p1,p2,... are given, they are passed
% to the function f),varargin{:});
end
There are only two differences between euler2 and euler. The new program accepts an additional input,
varargin, and it passes this argument, in the form varargin{:}, to f. The symbol varargin is a special
variable in MATLAB (like nargin) that is used only in M-file functions. It is initialized, when the M-file
is invoked, to contain any additional arguments beyond those explicitly listed in the function statement.
These additional arguments, if they are provided, are stored in a cell array---an indexed array of possibly
different types.
76
I do not wish to explain cell arrays in any detail, since I do not need them in this tutorial except in this one
context. It is enough to know that varargin{:} turns the fields of varargin into a list that can be passed to
another function. Moreover, if there were no additional parameters when the M-file was invoked, the
varargin is empty and passing varargin{:} to another function has absolutely no effect.
I will now do an example, solving
du
au, u (0) 1
dt
on the interval [0,1] for two different values of a.
clear
f=inline('a*u','t','u','a');
[u1,t]=euler(f,0,1,1,20,1);
[u2,t]=euler(f,0,1,1,20,2);
Now I plot the two solutions:
plot(t,u1,'-',t,u2,'--')
7
6
5
4
3
2
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Efficient MATLAB programming
Programs written in the MATLAB programming language are interpreted, not compiled. This means that
you pay a certain performance penalty for using the MATLAB interactive programming environment
instead of programming in a high-level programming language like C or Fortran. However, you can make
your MATLAB programs run very fast, in many cases, by adhering to the following simple rule: whenever
possible, use MATLAB commands, especially vectorization, rather then user-defined code.
Here is a simple example. Suppose I wish to evaluate the sine function on the grid
77
0, 0.00001, 0.00002, ... ,1.0
(100001 points). Below are two ways to accomplish this; I time each using the "tic-toc" commands (see
"help tic" for more information).
tic
x=linspace(0,1,100001);
y=sin(x);
toc
elapsed_time =
0.0310
tic
x=zeros(100001,1);
y=zeros(100001,1);
for i=1:100001,y(i)=sin(x(i));end
toc
elapsed_time =
0.6250
The version using an explicit loop is much more time-consuming than the version using MATLAB
vectorization. This is because the vectorized commands use compiled code, while the explicit loop must be
interpreted.
MATLAB commands use state-of-the-art algorithms and execute efficiently. Therefore, when you solve
linear systems, evaluate functions, sort vectors, and perform other operations using MATLAB commands,
your code will be very efficient. On the other hand, when you execute explicit loops, you pay a
performance penalty. For most academic exercises, the convenience of MATLAB far outweighs any
increase in computation time.
More about graphics in MATLAB
Adding a legend to a plot
When graphing two curves on the same plot, it is often helpful to label them so they can be distinguished.
MATLAB has a command called legend, which adds a key with user-defined descriptions of the curves.
The format is simple: just list the descriptions (as strings) in the same order as the curves were specified to
the plot command:
legend('a=1','a=2')
78
7
a=1
a=2
6
5
4
3
2
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
It is possible to tell MATLAB where on the plot to place the legend; see "help legend" for more
information.
You can also add a title and labels for the axes to a graph using title, xlabel, and ylabel. Use help to get
more information.
Changing the font size, line width, and other properties of graphs
It is possible to increase the size of fonts, the widths of curves, and otherwise modify the default properties
of MATLAB graphics objects. I do not want to explain how to do this here, but I wanted to make you
aware of the fact that this can be done. You can find out how to modify default properties from the
documentation on graphics, available through the help browser on the MATLAB desktop. Alternatively,
you can consult "help set" and "help plot" to get started.
Chapter 5: Boundary value problems in statics
Section 5.2: Introduction to the spectral method; eigenfunctions
I will begin by verifying that the eigenfunctions of the negative second derivative operator (under Dirichlet
conditions), sin(nπx/l), are mutually orthogonal:
clear
syms m n pi x l
int(sin(n*pi*x/l)*sin(m*pi*x/l),x,0,l)
79
ans =
-l*(n*cos(n*pi)*sin(m*pi)-m*sin(n*pi)*cos(m*pi))/pi/(n-m)/(n+m)
simplify(ans)
ans =
-l*(n*cos(n*pi)*sin(m*pi)-m*sin(n*pi)*cos(m*pi))/pi/(n^2-m^2)
At first glance, this result is surprising: Why did MATLAB not obtain the expected result, 0? However, a
moment's thought reveals the reason: The integral is not necessarily zero unless m and n are integers, and
MATLAB has no way of knowing that the symbols m and n are intended to represent integers. When m is
an integer, then sin(mπ)=0 and cos(mπ)=(-1)m. We can use the subs command to replace each instance of
sin(mπ) by zero, and similarly for cos(mπ), sin(nπ), and cos(nπ):
subs(ans,sin(m*pi),0)
ans =
l*m*sin(n*pi)*cos(m*pi)/pi/(n^2-m^2)
subs(ans,cos(m*pi),(-1)^m)
ans =
l*m*sin(n*pi)*(-1)^m/pi/(n^2-m^2)
subs(ans,sin(n*pi),0)
ans =
0
These substitutions are rather tedious to apply, but they are often needed when doing Fourier series
computations. Therefore, I wrote a simple program to apply them:
type mysubs
function expr=mysubs(expr,varargin)
%expr=mysubs(expr,m,n,...)
%
% This function substitutes 0 for sin(m*pi) and (-1)^m
% for cos(m*pi), and similarly for sin(n*pi) and cos(n*pi),
% and any other symbols given as inputs.
syms pi
k=length(varargin);
for j=1:k
expr=subs(expr,sin(varargin{j}*pi),0);
expr=subs(expr,cos(varargin{j}*pi),(-1)^varargin{j});
end
expr=simplify(expr);
I will use mysubs often to simplify Fourier coefficient calculations.
Example 5.7
Let f(x) be defined as follows:
clear
syms x
f=x*(1-x)
f =
x*(1-x)
I can easily compute the Fourier sine coefficients of f on the interval [0,1]:
syms n pi
2*int(f*sin(n*pi*x),x,0,1)
80
ans =
-2*(n*pi*sin(n*pi)+2*cos(n*pi)-2)/n^3/pi^3
I simplify the result using mysubs:
a=mysubs(ans,n)
a =
-4*((-1)^n-1)/n^3/pi^3
Here is the coefficient:
pretty(a)
n
(-1) - 1
-4 ---------
3 3
n pi
Using the symsum (symbolic summation) command, I can now create a partial Fourier sine series with a
given number of terms. For example, suppose I want the Fourier sine series with n=1,2,…,10. Here it is:
S=symsum(a*sin(n*pi*x),n,1,10)
S =
8/pi^3*sin(pi*x)+8/27/pi^3*sin(3*pi*x)+8/125/pi^3*sin(5*pi*x)+8/343/pi^3
*sin(7*pi*x)+8/729/pi^3*sin(9*pi*x)
I can plot the partial sum:
t=linspace(0,1,21);
plot(t,subs(S,x,t))I can also plot the error in S as an approximation to f:
plot(t,subs(f,x,t)-subs(S,x,t))
81
-4
x 10The error looks jagged because I chose a coarse grid on which to perform the computations. Here is a more
accurate graph:
t=linspace(0,1,101);
plot(t,subs(f,x,t)-subs(S,x,t))
-4
x 10
6I can also investigate how the error decreases as the number of terms in the Fourier series is increased. For
example, here is the partial Fourier series with 20 terms:
S=symsum(a*sin(n*pi*x),n,1,20);
82
Here is the error in S as an approximation to f:
plot(t,subs(f,x,t)-subs(S,x,t))
-5
x 10
12
10
8
6
4
2
0
-2
-4
-6
-8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
A few notes about symsum
The symsum command has the form "symsum(expr,ii,m,n)", where ii must be a symbolic variable without
an assigned value. If ii has previously been assigned a value, then the above command will not work. Here
are some errors to be avoided:
This does not work because ii is not symbolic:
ii=1;
symsum(ii,ii,1,10)
??? No appropriate methods for function symsum.
This does not work because ii has a value:
syms ii
ii=sym(1);
symsum(ii,ii,1,10)
??? Error using ==> sym/maple
Error, (in sum) summation variable previously assigned,
second argument evaluates to, 1 = 1 .. 10
Error in ==> C:\MATLAB6p5\toolbox\symbolic\@sym\symsum.m
On line 43 ==> r = maple('map','sum',f,[x.s '=' a.s '..' b.s]);
This is correct:
clear ii
syms ii
symsum(ii,ii,1,10)
83
ans =
55
When symsum(expr,ii,m,n) is executed, the values m, m+1, ... ,n are substituted for ii in expr, and the
results are summed. This substitution is automatic, and is part of the function of the symsum command. By
contrast, consider the following loop:
clear
syms ii
expr=ii^2;
s=sym(0);
for ii=1:10
s=s+expr;
end
s
s =
10*ii^2
Even though ii has a value when the line "s=s+expr" is executed, this value is not substituted into expr
unless we specifically direct that it be. (The fact that the analogous substitution takes place during the
execution of symsum is a special feature of symsum.) The subs command can be used to force this
substitution:
clear
syms ii
expr=ii^2;
s=sym(0);
for ii=1:10
s=s+subs(expr);
end
s
s =
385
This use of the subs command, subs(expr), tells MATLAB that if a variable appears in expr, and that
variable has a value in the MATLAB workspace, then the value should be substituted into expr.
Section 5.5: The Galerkin method
Since MATLAB can compute integrals that would be tedious to compute by hand, it is fairly easy to apply
the Galerkin method with polynomial basis functions. (In the next section, I will discuss the finite element
method, which uses piecewise polynomial basis functions.) Suppose we wish to approximate the solution tousing the subspace spanned by the following four polynomials:
clear
syms x
p1=x*(1-x)
p1 =
x*(1-x)
84
p2=x*(1/2-x)*(1-x)
p2 =
x*(1/2-x)*(1-x)
p3=x*(1/3-x)*(2/3-x)*(1-x)
p3 =
x*(1/3-x)*(2/3-x)*(1-x)
p4=x*(1/4-x)*(1/2-x)*(3/4-x)*(1-x)
p4 =
x*(1/4-x)*(1/2-x)*(3/4-x)*(1-x)
I have already defined the L2 inner product in an M-file (l2ip.m). Here is an M-file function implementing
the energy inner product:
type eip
function I=eip(f,g,k,a,b,x)
%I=eip(f,g,k,a,b,x)
%
% This function computes the energy inner product of two functions
% f(x) and g(x), that is, it computes the integral from a to b of
% k(x)*f'(x)*g'(x). The three functions must be defined by symbolic
% expressions f, g, and k.
%
% The variable of integration is assumed to be x. A different
% variable can passed in as the (optional) sixth input.
%
% The inputs a and b, defining the interval [a,b] of integration,
% are optional. The default values are a=0 and b=1.
% Assign the default values to optional inputs, if necessary
if nargin<6
syms x
end
if nargin<5
b=1;
end
if nargin<4
a=0;
end
% Compute the integral
I=int(k*diff(f,x)*diff(g,x),x,a,b);
Now the calculation is simple, although there is some repetitive typing required. (Below I will show how to
eliminate most of the typing.) We just need to compute the stiffness matrix and the load vector, and solve
the linear system. The stiffness matrix is
k=1+x;
K=[eip(p1,p1,k),eip(p1,p2,k),eip(p1,p3,k),eip(p1,p4,k)
eip(p2,p1,k),eip(p2,p2,k),eip(p2,p3,k),eip(p2,p4,k)
eip(p3,p1,k),eip(p3,p2,k),eip(p3,p3,k),eip(p3,p4,k)
eip(p4,p1,k),eip(p4,p2,k),eip(p4,p3,k),eip(p4,p4,k)]
K =
[ 1/2, -1/30, 1/90, -1/672]
85and the load vector is
f=x^2
f =
x^2
F=[l2ip(p1,f);l2ip(p2,f);l2ip(p3,f);l2ip(p4,f)]
F =
[ 1/20]
[ -1/120]
[ 1/630]
[ -1/2688]
I can now solve for the coefficients defining the (approximate) solution:
c=K\F
c =
[ 3325/34997]
[ -9507/139988]
[ 1575/69994]
[ 420/34997]
The approximate solution is
p=c(1)*p1+c(2)*p2+c(3)*p3+c(4)*p4
p =
3325/34997*x*(1-x)-9507/139988*x*(1/2-x)*(1-x)+1575/69994*x*(1/3-
x)*(2/3-x)*(1-x)+420/34997*x*(1/4-x)*(1/2-x)*(3/4-x)*(1-x)
Here is a graph:
t=linspace(0,1,41);
plot(t,subs(p,x,t))86
The exact solution can be found by integration:
syms c1 c2
int(-x^2,x)+c1
ans =
-1/3*x^3+c1
u=int(ans/k,x)+c2
u =
-1/9*x^3+1/6*x^2-1/3*x+log(3*x+3)*c1+1/3*log(3*x+3)+c2
Now I solve for the constants c1 and c2:
c=solve(subs(u,x,0),subs(u,x,1),c1,c2)
c =
c1: [1x1 sym]
c2: [1x1 sym]
subs(u,c1,c.c1)+c2
subs(ans,c2,c.c2)-5/18*log(3)/(log(6)-log(3))
u=simplify(ans)
u =
-1/18*(2*x^3*log(2)-3*x^2*log(2)+6*x*log(2)-5*log(x+1))/log(2)
Let us check the solution:
-diff(k*diff(u,x),x)
ans =
1/18*(6*x^2*log(2)-6*x*log(2)+6*log(2)-
5/(x+1))/log(2)+1/18*(x+1)*(12*x*log(2)-6*log(2)+5/(x+1)^2)/log(2)
simplify(ans)
ans =
x^2
subs(u,x,0)
ans =
0
subs(u,x,1)
ans =
3.5594e-017
This last result is a bit of a surprise. However, every quantity in MATLAB is floating point by default, so
the "1" substituted into u is the floating point number 1, which causes the result to be computed in floating
point. Here is what we really want:
subs(u,x,sym(1))
ans =
0
Since we have the exact solution, let us compare it with our approximate solution:
plot(t,subs(u,x,t),t,subs(p,x,t))
87The computed solution is so close to the exact solution that the two graphs cannot be distinguished. Here is
the error:
plot(t,subs(u,x,t)-subs(p,x,t))
-6
x 10
6
4
2
0
-2
-4
-6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Computing the stiffness matrix and load vector in loops
The above calculation is easy to perform, but it is rather annoying to have to type in the formulas for K and
F in terms of all of the necessary inner products. It would be even worse were we to use an approximating
88
subspace with a higher dimension. Fortunately, we can compute K and F in loops and greatly reduce the
necessary typing. The key is to store the basis functions in a vector, so we can refer to them by index.
clear
syms x
p=[x*(1-x);x*(1/2-x)*(1-x);x*(1/3-x)*(2/3-x)*(1-x);x*(1/4-x)*(1/2-
x)*(3/4-x)*(1-x)];
pretty(p)
[ x (1 - x) ]
[ ]
[ x (1/2 - x) (1 - x) ]
[ ]
[ x (1/3 - x) (2/3 - x) (1 - x) ]
[ ]
[x (1/4 - x) (1/2 - x) (3/4 - x) (1 - x)]
k=1+x;
K=sym(zeros(4,4));
for ii=1:4
for jj=1:4
K(ii,jj)=eip(p(ii),p(jj),k);
end
end
K
K =
[ 1/2, -1/30, 1/90, -1/672]f=x^2;
F=sym(zeros(4,1));
for ii=1:4
F(ii)=l2ip(p(ii),f);
end
F
F =
[ 1/20]
[ -1/120]
[ 1/630]
[ -1/2688]
With this method, it is equally easy to compute K and F regardless of the number of basis functions.
Section 5.6: Piecewise polynomials and the finite element method
The finite element method is the Galerkin method with a piecewise linear basis. The computations are a bit
more complicated than in the previous section, since a basis function is not defined by a single formula, as
in the previous section.
Here is a typical basis function φi on a mesh with n elements:
clear
n=10;
89
x=linspace(0,1,n+1)';
phi=zeros(n+1,1);
ii=5;
phi(ii)=1;
plot(x,phi basis function φi is defined by
x xi 1
h , xi 1 x xi ,
xx
i ( x ) i
, xi x xi 1 ,
h
0, otherwise.
In these formulas, h=(b-a)/n, where the interval of interest is [a,b] and the number of elements is n, and
xi a ih, i 0,1,2,, n.
The easiest way to represent φi is to define two expressions representing the two nonzero pieces. I will
assume that a (the left endpoint of the interval) is zero:
clear
syms ii h x
phi1=(x-(ii-1)*h)/h
phi1 =
(x-(ii-1)*h)/h
phi2=-(x-(ii+1)*h)/h
phi2 =
(-x+(ii+1)*h)/h
90
Computing the load vector
Here is how I would use the above expressions to compute a load vector. Consider the following BVP:
d 2u
2 x 2 , 0 x 1,
dx
u (0) 0, u (1) 0.
I define the right-hand side:
f=x^2;
Now I choose n and define h:
n=10;
h=1/n;
Here is the computation:
F=zeros(n-1,1);
for ii=1:n-1
F(ii)=double(int(subs(phi1)*f,x,ii*h-
h,ii*h)+int(subs(phi2)*f,x,ii*h,ii*h+h));
end
(Notice that I use the double command to assign floating point values to the vector F.)
F
F =
0.0012
0.0042
0.0092
0.0162
0.0252
0.0362
0.0492
0.0642
0.0812
Notice the use of subs(phi1) and subs(phi2) in the above loop. When the subs command is called with a
single expression, it tells MATLAB to substitute, for variables in the expression, any value that exist in the
workspace. In the above example, those variables were h and ii:
phi1
phi1 =
(x-(ii-1)*h)/h
ii=10;
subs(phi1)
ans =
10*x-9
Computing the stiffness matrix
91
Computing the stiffness matrix is simple when the coefficient in the differential equation is a constant k,
because then we already know the entries in the stiffness matrix (they were derived in the text).
The diagonal entries in the stiffness matrix are all 2k/h, and the entries on the subdiagonal and
superdiagonal are all –k/h. All other entries are zero.
As I explained in the text, one of the main advantages of the finite element method is that the stiffness
matrix is sparse. One of the main advantages of MATLAB is that it is almost as easy to create and use
sparse matrices as it is to work with ordinary (dense) matrices! (There are some exceptions to this
statement. Some matrix functions cannot operate on sparse matrices. Also, a sparse matrix cannot hold
symbolic expressions, a limitation that we must take into account below.) Here is how I would create the
stiffness matrix. Notice that I first allocate an n-1 by n-1 sparse matrix and then write two loops, one to fill
the main diagonal, and the second to fill the subdiagonal and superdiagonal. (I take k=1 in this example.)
k=1;
K=sparse(n-1,n-1)
K =
All zero sparse: 9-by-9)=-k/h;
end
Notice that I wrote two loops because the subdiagonal and superdiagonal have only n-2 entries, while the
main diagonal has n-1 entries.
Here is the matrix K:
K
K =
(1,1) 20
(2,1) -10
(1,2) -10
(2,2) 20
(3,2) -10
(2,3) -10
(3,3) 20
(4,3) -10
(3,4) -10
(4,4) 20
(5,4) -10
(4,5) -10
(5,5) 20
(6,5) -10
(5,6) -10
(6,6) 20
(7,6) -10
(6,7) -10
(7,7) 20
(8,7) -10
(7,8) -10
(8,8) 20
(9,8) -10
(8,9) -10
(9,9) 20
92
Notice that MATLAB only stores the nonzeros, so the above output is rather difficult to read. I can, if I
wish, convert K to a full matrix to view it in the usual format:
full(K)
ans =
20 -10 0 0 0 0 0 0 0
-10 20 -10 0 0 0 0 0 0
0 -10 20 -10 0 0 0 0 0
0 0 -10 20 -10 0 0 0 0
0 0 0 -10 20 -10 0 0 0
0 0 0 0 -10 20 -10 0 0
0 0 0 0 0 -10 20 -10 0
0 0 0 0 0 0 -10 20 -10
0 0 0 0 0 0 0 -10 20
However, this is usually not a good idea, since we tend to use sparse matrices when the size of the problem
is large, in which case converting the sparse matrix to a dense matrix partially defeats the purpose.
The spy command is sometimes useful. It plots a schematic view of a matrix, showing where the nonzero
entries are:
spy(K)
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
nz = 25
I can now solve Ku=F to get the nodal values (I computed F earlier):
u=K\F
u =
0.0083
0.0165
0.0243
0.0312
0.0365
0.0392
93
0.0383
0.0325
0.0203
For graphical purposes, I often explicitly put in the nodal values (zero) at the endpoints:
u=[0;u;0]
u =
0
0.0083
0.0165
0.0243
0.0312
0.0365
0.0392
0.0383
0.0325
0.0203
0
If I now create the grid, I can easily plot the computed solution:
t=linspace(0,1,n+1)';
plot(t,u)
0.04
0.035
0.03As I mentioned earlier, MATLAB's plot command simply graphs the given data points and connects them
with straight line segments---that is, it automatically graphs the continuous piecewise linear function defined
by the data!
94
The nonconstant coefficient case
Now consider the BVPNow when I fill the stiffness matrix, I must actually compute the integrals, since their values are not known
a priori. I can, however, simplify matters because I know the derivatives of phi1 (1/h) and phi2 (-1/h).
Here is the computation of K:
k=x+1;
K=sparse(n-1,n-1);
for ii=1:n-1
K(ii,ii)=int(k/h^2,x,ii*h-h,ii*h)+int(k/h^2,x,ii*h,ii*h+h);
end
??? Conversion to double from sym is not possible.
Whoops! This is the limitation I mentioned above about sparse matrices: they cannot hold symbolic results.
Therefore, I must convert the results to floating point, using the double command, before assigning to K.
Here is the correct loop:
for ii=1:n-1
K(ii,ii)=double(int(k/h^2,x,ii*h-h,ii*h)+int(k/h^2,x,ii*h,ii*h+h));
end
Here is the loop that computes the subdiagonal and super diagonal. Recall that K is symmetric, which
means that I do not need to compute the subdiagonal once I have the superdiagonal:
for ii=1:n-2
K(ii,ii+1)=double(int(-k/h^2,x,ii*h,ii*h+h));
K(ii+1,ii)=K(ii,ii+1);
end
Now I can compute the nodal values and plot the result, as before:
u=K\F;
u=[0;u;0];
plot(t,u)
95Creating a piecewise linear function from the nodal values
Here is the exact solution to the previous BVP:
syms c1 c2
int(-x^2,x)+c1;
U=int(ans/k,x)+c2;
c=solve(subs(U,x,0),subs(U,x,1),c1,c2);
subs(U,c1,c.c1);
subs(ans,c2,c.c2);
U=simplify(ans);
pretty(U)
3 2
2 x log(2) - 3 x log(2) + 6 x log(2) - 5 log(x + 1)
- 1/18 -----------------------------------------------------
log(2)
Now I would like to compare this exact solution with the solution I computed above using the finite element
method. I can plot the error as follows:
plot(t,subs(U,x,t)-u)
96
-5
x 10
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
However, this is really a misleading graph, since I computed the errors at the nodes and assumed
(implicitly, by using the MATLAB plot command) that the error is linear in between the nodes. This is not
true.
To see the true error, I must create a piecewise linear function from the grid and the nodal values, and then
compare it to the true solution. MATLAB has some functions for working with piecewise polynomials,
including mkpp, which creates an array describing a piecewise polynomial according to MATLAB's own
data structure, and ppval, which evaluates a piecewise polynomial. Because mkpp is a little complicated, I
wrote a function, mkpl.m, specifically for creating a piecewise linear function. It takes as inputs the grid
and the nodal values:
help mkpl
pl = mkpl(x,v)
This function creates a continuous, piecewise linear function
interpolating the data (x(i),v(i)), i=1,...,length(x). The
vectors x and v must be of the same length. Also, it is
assumed that the components of x are increasing.
Here is the piecewise linear function we computed using the finite element method:
pu=mkpl(t,u);
I now create a fine grid:
tt=linspace(0,1,10*n+1)';
I can now use ppval to evaluate pu on the grid (and subs to evaluate U, as usual):
plot(tt,subs(U,x,tt)-ppval(pu,tt))
97
-4
x 10
7
6
5
4
3
2
1
0
-1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
You should notice that the error is very small at the nodes---graphing only the nodal error gives a
misleading impression of the size of the error.
More about sparse matrices
If a matrix B is stored in ordinary (dense) format, then the command S =sparse(A) creates a copy of the
matrix stored in sparse format. For example:
clear
A = [0 0 1;1 0 2;0 -3 0]
A =
0 0 1
1 0 2
0 -3 0
S = sparse(A)
S =
(2,1) 1
(3,2) -3
(1,3) 1
(2,3) 2
whos
Name Size Bytes Class
A 3x3 72 double array
S 3x3 64 double array (sparse)
Grand total is 13 elements using 136 bytes
Unfortunately, this form of the sparse command is not particularly useful, since if A is large, it can be very
time-consuming to first create it in dense format. The command S = sparse(m,n) (as we have already seen)
creates an m by n zero matrix in sparse format. Entries can then be added one-by-one:
98
A = sparse(3,2)
A =
All zero sparse: 3-by-2
A(1,2)=1;
A(3,1)=4;
A(3,2)=-1;
A
A =
(3,1) 4
(1,2) 1
(3,2) -1
(Of course, for this to be truly useful, the nonzeros would be added in a loop.)
Another version of the sparse command is S = sparse(I,J,S,m,n,maxnz). This creates an m by n sparse
matrix with entry (I(k),J(k)) equal to S(k). The optional argument maxnz causes MATLAB to pre-allocate
storage for maxnz nonzero entries, which can increase efficiency in the case when more nonzeros will be
later added to S.
There are still more versions of the sparse command. See "help sparse" for details.
The most common type of sparse matrix is a banded matrix, that is, a matrix with a few nonzero diagonals.
Such a matrix can be created with the spdiags command. Consider the following matrix:
A =[
64 -16 0 -16 0 0 0 0 0
-16 64 -16 0 -16 0 0 0 0
0 -16 64 0 0 -16 0 0 0
-16 0 0 64 -16 0 -16 0 0
0 -16 0 -16 64 -16 0 -16 0
0 0 -16 0 -16 64 0 0 -16
0 0 0 -16 0 0 64 -16 0
0 0 0 0 -16 0 -16 64 -16
0 0 0 0 0 -16 0 -16 64];
(notice the technique for entering the rows of a large matrix on several lines). This is a 9 by 9 matrix with 5
nonzero diagonals. In MATLAB's indexing scheme, the nonzero diagonals of A are numbers -3, -1, 0, 1,
and 3 (the main diagonal is number 0, the first subdiagonal is number -1, the first superdiagonal is number
1, and so forth). To create the same matrix in sparse format, it is first necessary to create a 9 by 5 matrix
containing the nonzero diagonals of A. Of course, the diagonals, regarded as column vectors, have different
lengths; only the main diagonal has length 9. In order to gather the various diagonals in a single matrix, the
shorter diagonals must be padded with zeros. The rule is that the extra zeros go at the bottom for
subdiagonals and at the top for superdiagonals. Thus we create the following matrix:
B = [
-16 -16 64 0 0
-16 -16 64 -16 0
-16 0 64 -16 0
-16 -16 64 0 -16
-16 -16 64 -16 -16
-16 0 64 -16 -16
0 -16 64 0 -16
0 -16 64 -16 -16
0 0 64 -16 -16];
99
The spdiags command also needs the indices of the diagonals:
d = [-3,-1,0,1,3];
The matrix is then created as follows:
S = spdiags(B,d,9,9);
The last two arguments give the size of S.
Perhaps the most common sparse matrix is the identity. Recall that an identity matrix can be created, in
dense format, using the command eye. To create the n by n identity matrix in sparse format, use
I = speye(n). For example:
I=speye(3)
I =
(1,1) 1
(2,2) 1
(3,3) 1
Recall that the spy command is very useful for visualizing a sparse matrix:
spy(A)
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
nz = 33
100
Chapter 6: Heat flow and diffusion
Section 6.1: Fourier series methods for the heat equation
Example 6.2: An inhomogeneous example
Consider the IBVP
u 2u
A 2 10 7 ,The constant A has value 0.208 cm2/s.
clear
A=0.208
A =
0.2080
The solution can be written as
n x
u ( x, t ) a n (t ) sin
n 1 100
where the coefficient an(t) satisfies the IVP
dan An2 2
a n cn , t 0,
dt 1002
a n (0) 0.
The values c1, c2, c3, ... are the Fourier sine coefficients of the constant function 10 -7:
syms n x pi
2/100*int(10^(-7)*sin(n*pi*x/100),x,0,100)
ans =
-1/5000000*(cos(n*pi)-1)/n/pi
mysubs(ans,n)
ans =
-1/5000000*((-1)^n-1)/n/pi
c=ans
c =
-1/5000000*((-1)^n-1)/n/pi
I can compute an(t) by the following formula:
101
syms t s
int(exp(-A*n^2*pi^2*(t-s)/100^2)*c,s,0,t)
ansa=simplify(ans)
aNext I define the (partial) Fourier series of the solution:
S=symsum(a*sin(n*pi*x/100),n,1,10);
I can now look at some "snapshots" of the solution. For example, I will show the concentration distribution
after 10 minutes (600 seconds). Some trial and error may be necessary to determine how many terms in the
Fourier series are required for a qualitatively correct solution. (As discussed in the text, this number
decreases as t increases.)
S600=subs(S,t,600);
xx=linspace(0,100,201);
plot(xx,subs(S600,x,xx))
-5
x 10
7
6
5
4
3
2
1
0
-1
0 10 20 30 40 50 60 70 80 90 100
The wiggles in the graph suggest that 10 terms is not enough for a correct graph (after 10 minutes, the
concentration distribution ought to be quite smooth). Therefore, I will try again with 20 terms:
S=symsum(a*sin(n*pi*x/100),n,1,20);
S600=subs(S,t,600);
plot(xx,subs(S600,x,xx))
102
-5
x 10
6
5
4
3
2
1
0
-1
0 10 20 30 40 50 60 70 80 90 100
Now with 40 terms:
S=symsum(a*sin(n*pi*x/100),n,1,40);
S600=subs(S,t,600);
plot(xx,subs(S600,x,xx))
-5
x 10
6
5
4
3
2
1
0
-1
0 10 20 30 40 50 60 70 80 90 100
It appears that 20 terms is enough for a qualitatively correct graph at t=600.
103
Section 6.4: Finite element methods for the heat equation
Now I will show how to use the backward Euler method with finite element discretization to
(approximately) solve the heat equation. Since the backward Euler method is implicit, it is necessary to
solve an equation at each step. This makes it difficult to write a general-purpose program implementing
backward Euler, and I will not attempt to do so. Instead, the function beuler.m applies the backward Euler
method to the system of ODEs
da
M Ka f (t ), t 0,
dt
a ( 0) a 0 ,
which is the result of applying the finite element to the heat equation.
type beuler
function [a,t]=beuler(M,K,f,a0,N,dt)
%[a,t]=beuler(M,K,f,a0,N,dt)
%
% This function applies the backward Euler method to solve the
% IVP
%
% Ma'+Ka=f(t),
% a(0)=a0,
%
% where M and K are mxm matrices (with M invertible) and f is
% a vector-valued function taking two arguments, t and m+1 (m+1 is
% the number of elements in the underlying finite element mesh).
%
% N steps are taken, each of length dt.
m=length(a0);
a=zeros(m,N+1);
a(:,1)=a0;
t=linspace(0,N*dt,N+1)';
L=M+dt*K;
for ii=1:N
a(:,ii+1)=L\(M*a(:,ii)+dt*feval(f,t(ii+1),m+1));
end
To solve a specific problem, I must compute the mass matrix M, the stiffness matrix K, the load vector f(t),
and the initial data a0. The techniques should by now be familiar.
Example 6.9
A 100 cm iron bar, with ρ=7.88, c=0.437, and κ=0.836, is chilled to an initial temperature of 0 degrees, and
then heated internally with both ends maintained at 0 degrees. The heat source is described by the function
F ( x, t ) 10 8 tx(100 x) 2 .
The temperature distribution is the solution of
104
u 2u
c 2 F ( x, t ),I will use standard piecewise linear basis functions and the techniques introduced in Section 5.6 of this
tutorial to compute the mass and stiffness matrices:
n=100
n =
100
h=100/n
h =
1
k=0.836
k =
0.8360
p=7.88
p =
7.8800
c=0.437
c =
0.4370
K=sparse(n-1,n-1);)=K(ii,ii+1);
end
M=sparse(n-1,n-1);
for ii=1:n-1
M(ii,ii)=2*p*c*h/3;
end
for ii=1:n-2
M(ii,ii+1)=p*c*h/6;
M(ii+1,ii)=M(ii,ii+1);
end
(Note that, for this constant coefficient problem, we do not need to perform any integrations, as we already
know the entries in the mass and stiffness matrices.)
Now I compute the load vector. Here is the typical entry:
clear ii
syms x t ii
phi1=(x-(ii-1)*h)/h
phi1 =
x-ii+1
phi2=-(x-(ii+1)*h)/h
phi2 =
105
-x+ii+1
F=10^(-8)*t*x*(100-x)^2
F =
1/100000000*t*x*(100-x)^2
int(F*subs(phi1),x,ii*h-h,ii*h)+int(F*subs(phi2),x,ii*h,ii*h+h)
ans =
1/500000000*t*(ii^5-(ii-1)^5)+1/4*(-1/500000*t+1/100000000*t*(-
ii+1))*(ii^4-(ii-1)^4)+1/3*(1/10000*t-1/500000*t*(-ii+1))*(ii^3-(ii-
1)^3)+1/20000*t*(-ii+1)*(ii^2-(ii-1)^2)-1/500000000*t*((ii+1)^5-
ii^5)+1/4*(1/500000*t+1/100000000*t*(ii+1))*((ii+1)^4-ii^4)+1/3*(-
1/10000*t-1/500000*t*(ii+1))*((ii+1)^3-ii^3)+1/20000*t*(ii+1)*((ii+1)^2-
ii^2)
simplify(ans)
ans =
-1/3000000*t+1/100000000*t*ii^3-1/500000*t*ii^2+20001/200000000*t*ii
Now I need to turn this formula into a vector-valued function that I can pass to beuler. I write an M-file
function f6.m:
type f6
function y=f(t,n)
ii=(1:n-1)';
y=-1/3000000*t+1/100000000*t*ii.^3-1/500000*t*ii.^2+...
20001/200000000*t*ii;
(Note the clever MATLAB programming in f6: I made ii a vector with components equal to 1,2 ,...,n-1.
Then I can compute the entire vector in one command rather than filling it one component at a time in a
loop.)
Next I create the initial vector a0. Since the initial value in the IBVP is zero, a0 is the zero vector:
a0=zeros(n-1,1);
Now I choose the time step and invoke beuler:
dt=2;
N=180/dt;
[a,t]=beuler(M,K,'f6',a0,N,dt);
The last column of a gives the temperature distribution at time t=180 (seconds). I will put in the zeros at the
beginning and end that represent the Dirichlet conditions:
T=[0;a(:,N+1);0];
xx=linspace(0,100,n+1)';
Here is a plot of the temperature after 3 minutes:
plot(xx,T)
106
7
6
5
4
3
2
1
0
0 10 20 30 40 50 60 70 80 90 100
Chapter 8: Problems in multiple spatial dimensions
Section 8.2: Fourier series on a rectangular domain
Fourier series calculations on a rectangular domain proceed in almost the same fashion as in one-
dimensional problems. The key difference is that we must compute double integrals and double sums in
place of single integrals and single sums. Fortunately, this is not difficult, since a double integral over a
rectangle is just an iterated integral.
As an example, I will compute the Fourier double sine series of the following function f on the unit square:
clear
syms x y
f=x*y*(1-x)*(1-y)
f =
x*y*(1-x)*(1-y)
The Fourier series has the form
a
m 1 n 1
mn sin(mx) sin(ny ),
where amn is computed as follows:
syms m n pi
4*int(int(f*sin(m*pi*x)*sin(n*pi*y),y,0,1),x,0,1)
ans =
107
4*(n*pi^2*sin(n*pi)*sin(m*pi)*m+2*n*sin(n*pi)*cos(m*pi)*pi+2*cos(n*pi)*s
in(m*pi)*m*pi+4*cos(n*pi)*cos(m*pi)-4*cos(m*pi)-2*m*pi*sin(m*pi)-
2*n*pi*sin(n*pi)-4*cos(n*pi)+4)/m^3/pi^6/n^3
Recall the command mysubs for simplifying such an expression when n and n are known to be integers:
a=mysubs(ans,m,n);
pretty(a)
(n + m) (1 + m) (1 + n)
1 + (-1) + (-1) + (-1)
16 -------------------------------------------
3 6 3
m pi n
Here is the partial Fourier series with a total of 100 terms:
S=symsum(symsum(a*sin(m*pi*x)*sin(n*pi*y),n,1,10),m,1,10);
Now I would like to graph f and the approximation S. This is the first time we have needed to graph a
function of two variables.
Two-dimensional graphics in MATLAB
Recall that to plot a function of one variable, we create a grid using the linspace command and then
evaluate the desired function on the grid. We can then call the plot command. For a function of two
variables, the procedure is similar. However, we need to create a grid on a rectangle rather than on an
interval, which is a bit more complicated.
The meshgrid command takes two one-dimensional grids, on the intervals a<x<b and c<y<d, and creates
the necessary grid on the rectangle {(x,y) : a<x<b and c<y<d}. This grid is represented as two matrices X
and Y; the points in the grid are then (Xij,Yij). Evaluating f(x,y) on the grid means producing a matrix Z such
that
Z ij f ( X ij , Yij ) for all i, j.
This last step is easy, since MATLAB supports vectorized operations.
Here, then, is the procedure:
First, I create the two one-dimensional grids:
xx=linspace(0,1,21)';
yy=linspace(0,1,21)';
Next, I invoke meshgrid to create the two-dimensional grid:
[X,Y]=meshgrid(xx,yy);
Finally, I compute the function f, from the previous example, on the grid:
Z=subs(f,{x,y},{X,Y});
108
(Notice how I can substitute for two variables at one time by listing the variables and the values between
curly brackets.)
The command for plotting the surface z=f(x,y) is called surf:
surf(X,Y,Z)
An alternative to surf is mesh, which does not shade or color the patches on the surface:
I can also evaluate the Fourier series on the two-dimensional grid. Notice that this may take some time if
the grid is very fine or there are many terms in the series, or both.
Z1=subs(S,{x,y},{X,Y});
surf(X,Y,Z1)
109
Here is the approximation error:
surf(X,Y,Z-Z1)
Looking at the vertical scale on the last two plots, we see that the error of approximation is only about one-
half of one percent.
110
Section 8.3: Fourier series on a disk
The Bessel functions Jn(s) are built-in functions in MATLAB, just as are the more common elementary
function such as sine and cosine, and can be used just as conveniently. For example, here is the graph of J0:
clear
t=linspace(0,10,101);
plot(t,besselj(0,t));grid
1
0.5
0
-0.5
0 1 2 3 4 5 6 7 8 9 10
(Notice that the first argument to besselj is the index n.)
As an example, I will compute the smallest root s01, s02, s02 of J0. The above graph shows that these roots
are approximately 2.5, 5.5, and 8.5. Recall that the command fzero finds a (floating point estimate of) a
root, near a given estimate, of a function. We have a slight difficulty in applying fzero to besselj, however,
since besselj takes two arguments, and the first is the parameter n. Like most MATLAB functions that
operate on functions, fzero will allow us to pass a parameter through to the user-defined function, but the
parameter must come after the variable in the calling sequence. I will get around this problem by defining
an inline function that represents J0:
J0=inline('besselj(0,x)','x');
Now I can invoke fzero:
fzero(J0,2.5)
ans =
2.4048
fzero(J0,5.5)
ans =
5.5201
fzero(J0,8.5)
ans =
8.6537
111
Graphics on the disk
Functions on a disk are naturally described by cylindrical coordinates, that is, as z=f(r,θ), where (r,θ) are
polar coordinates. We can use surf to graph such functions; however, it requires a little extra work to set up
the grid. Here is how it works:
First, set up one-dimensional grids in r and θ (A is the radius of the disk):
clear
A=1;
r=linspace(0,A,21);
th=linspace(0,2*pi,21);
Next, use meshgrid to set up a rectangular grid in r and θ:
[R,Th]=meshgrid(r,th);
Now compute the function z=f(r,θ). I will use f(r,θ)=rcos(θ) for this example:
Z=R.*cos(Th);
Next, create matrices X and Y containing the (x,y) coordinates:
X=R.*cos(Th);
Y=R.*sin(Th);
Now I can plot the surface:
surf(X,Y,Z)
Here is a more interesting example. I will graph the eigenfunction φ(r,θ)=J1(s11r)cos(θ) on the unit disk.
First I must compute the root s11 of J1:
112
clear
J1=inline('besselj(1,x)','x');
t=linspace(0,10,101)';
plot(t,J1(t));grid
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0 1 2 3 4 5 6 7 8 9 10
s11=fzero(J1,4)
s11 =
3.8317
Now I define the various grids:
r=linspace(0,1,21);
th=linspace(0,2*pi,21);
[R,Th]=meshgrid(r,th);
X=R.*cos(Th);
Y=R.*sin(Th);
Next I define the function z=φ(r,θ):
Z=besselj(1,s11*R).*cos(Th);
Finally, I can graph the surface:
surf(X,Y,Z)
113
Chapter 9: More about Fourier series
Section 9.1: The complex Fourier series
It is no more difficult to compute complex Fourier series than the real Fourier series discussed earlier. You
should recall that the imaginary unit sqrt(-1) is denoted by i (or j, but I will use i) in MATLAB. As an
example, I will compute the complex Fourier series of f(x)=x2 on the interval [-1,1].
clear
f=inline('x.^2','x');
syms x pi i n
1/2*int(f(x)*exp(-i*pi*n*x),x,-1,1)
ans =
-1/2*(-i*exp(-i*pi*n)*pi^2*n^2-2*exp(-i*pi*n)*n*pi+2*i*exp(-
i*pi*n)+i*exp(i*pi*n)*pi^2*n^2-2*exp(i*pi*n)*n*pi-
2*i*exp(i*pi*n))/pi^3/n^3
We now have a simplification problem similar to what we first encountered in Chapter 5, and for which I
wrote the function mysubs. In this case, since n is an integer, exp(-iπn) equals (-1)n, as does exp(iπn). For
this reason, I wrote another version of mysubs, namely mysubs1, to handle this case:
type mysubs1
function expr=mysubs1(expr,varargin)
%expr=mysubs1(expr,m,n,...)
%
% This function substitutes (-1)^m for exp(-i*pi*m) and
% for exp(i*pi*m), and similarly for exp(-i*pi*n) and
% exp(i*pi*n), and any other symbols given as inputs.
114
syms pi i
k=length(varargin);
for jj=1:k
expr=subs(expr,exp(-i*(pi*varargin{jj})),(-1)^varargin{jj});
expr=subs(expr,exp(i*pi*varargin{jj}),(-1)^varargin{jj});
end
expr=simplify(expr);
Now I apply mysubs1:
a=mysubs1(ans,n);
pretty(a)
n
(-1)
2 ------
2 2
n pi
This formula obviously does not hold for n=0, and so I need to compute the coefficient a0 separately:
a0=1/2*int(f(x),x,-1,1)
a0 =
1/3
Now I can define the partial Fourier series. Recall from Section 9.1 of the text that, since f is real-valued,
the following partial Fourier series is also real-valued (I choose to use 21 terms in the series):
S=a0+symsum(a*exp(i*n*pi*x),n,-10,-1)+symsum(a*exp(i*n*pi*x),n,1,10)
S =
1/3+1/50/pi^2*exp(-10*i*pi*x)-2/81/pi^2*exp(-9*i*pi*x)+1/32/pi^2*exp(-
8*i*pi*x)-2/49/pi^2*exp(-7*i*pi*x)+1/18/pi^2*exp(-6*i*pi*x)-
2/25/pi^2*exp(-5*i*pi*x)+1/8/pi^2*exp(-4*i*pi*x)-2/9/pi^2*exp(-
3*i*pi*x)+1/2/pi^2*exp(-2*i*pi*x)-2/pi^2*exp(-i*pi*x)-
2/pi^2*exp(i*pi*x)+1/2/pi^2*exp(2*i*pi*x)-
2/9/pi^2*exp(3*i*pi*x)+1/8/pi^2*exp(4*i*pi*x)-
2/25/pi^2*exp(5*i*pi*x)+1/18/pi^2*exp(6*i*pi*x)-
2/49/pi^2*exp(7*i*pi*x)+1/32/pi^2*exp(8*i*pi*x)-
2/81/pi^2*exp(9*i*pi*x)+1/50/pi^2*exp(10*i*pi*x)
I will check the computation with a graph:
t=linspace(-1,1,101);
plot(t,subs(S,x,t))
Warning: Imaginary parts of complex X and/or Y arguments ignored.
115
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
The approximation is not bad, as the following graph of the error shows:
plot(t,f(t)-subs(S,x,t))
Warning: Imaginary parts of complex X and/or Y arguments ignored.
0.04
0.03
0.02
0.01
0
-0.01
-0.02
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
You should notice the above warning messages complaining that the quantity I am trying to graph is
complex. This is not important, because I know from the results in the text that the partial Fourier series
must evaluate to a real number (since the underlying function f is real-valued). Therefore, any imaginary
part must be due to roundoff error, and it is right to ignore it. Here is an example:
116
subs(S,x,0.5)
ans =
0.2484 + 0.0000i
imag(ans)
ans =
1.3384e-017
(The imag command extracts the imaginary part of a complex number.)
Section 9.2: Fourier series and the FFT
MATLAB implements the fast Fourier transform in the command fft. As mentioned in Section 9.2.3 in the
text, there is more than one way to define the DFT (although all are essentially equivalent), and MATLAB's
definition differs slightly from that used in the text: Instead of associating the factor of 1/M with the DFT,
MATLAB associates it with the inverse DFT (cf. the formulas for the DFT and inverse DFT in Section
9.2.2 of the text). Otherwise, the definition used by MATLAB is the same as that in the text.
The inverse FFT is implemented in the command ifft.
As I explained in Section 9.2.2 of the text, a common operation when using the FFT is to swap the first and
second halves of a finite sequence. Since this operation is so common, MATLAB implements it in the
command fftshift:
help fftshift
FFTSHIFT Shift zero-frequency component to center of spectrum.
For vectors, FFTSHIFT(X) swaps the left and right halves of
X. For matrices, FFTSHIFT(X) swaps the first and third
quadrants and the second and fourth quadrants. For N-D
arrays, FFTSHIFT(X) swaps "half-spaces" of X along each
dimension.
FFTSHIFT(X,DIM) applies the FFTSHIFT operation along the
dimension DIM.
FFTSHIFT is useful for visualizing the Fourier transform with
the zero-frequency component in the middle of the spectrum.
See also IFFTSHIFT, FFT, FFT2, FFTN, CIRCSHIFT.
I can now reproduce Example 9.3 from the text to illustrate the use of these commands.
First, I define the function and the initial sequence:
clear
ff=inline('x.^3','x')
ff =
Inline function:
ff(x) = x.^3
x=linspace(-2/3,2/3,5);
f=[0,ff(x)]
f =
Columns 1 through 4
0 -0.2963 -0.0370 0
Columns 5 through 6
0.0370 0.2963
117
Next, I shift the sequence using fftshift, and then apply fft. Recall that, to reproduce the results in the text,
I must use the same definition for the DFT, which means that I must divide the output of fft by 6, the length
of the sequence:
f1=fftshift(f)
f1 =
Columns 1 through 4
0 0.0370 0.2963 0
Columns 5 through 6
-0.2963 -0.0370
F1=fft(f1)/6
F1 =
Columns 1 through 2
0 0 - 0.0962i
Columns 3 through 4
0 + 0.0748i 0
Columns 5 through 6
0 - 0.0748i 0 + 0.0962i
Now I apply fftshift to F1 to obtain the desired sequence F:
F=fftshift(F1The result is the same as in the text, up to round-off error.
I can perform the entire computation above in one line as follows:
F=fftshift(fft(fftshift(f))/6Chapter 10: More about finite element methods
Section 10.1 Implementation of finite element methods
In this section, I do more than just explain MATLAB commands, as has been my policy up to this point. I
define a collection of MATLAB M-file functions that implement piecewise linear finite elements on two-
dimensional polygonal domains. The interested reader can experiment with and extend these functions to
see how the finite element method works in practice. The implementation of the finite element method
follows closely the explanation I give in Section 10.1 of the text.
118
The main commands are Stiffness and LoadV, which assemble the stiffness matrix and load vectors for the
BVP
a( x, y )u f ( x, y ) in ,
u 0 on 1 ,
du
0 on 2 ,
dn
where Γ1 and Γ2 partition the boundary of Ω. However, before any computation can be done, the mesh must
be described. I provide two routines for creating meshes, RectangleMeshD and RectangleMeshM.
Creating a mesh
The data structure is described in the M-file FemMesh.m (this M-file contains only comments, which are
displayed when you type "help FemMesh"):
help FemMesh
The Fem package uses a structure to describe a triangular mesh.
This structure contains four arrays:
NodeList: Mx3 matrix, where M is the number of nodes in
the mesh (including boundary nodes). Each row
corresponds to a node and contains the
coordinates of the node
FreeNodePtrs: Nx1 vector, where N is the number of free
nodes. NodePtrs(i) is the index into
NodeList of the ith free node. So, for
example, NodeList(NodePtrs(i),1:2)
are the coordinates of the ith free node.
NodePtrs: Mx1 vector, where M is the number of nodes in
the mesh (including boundary nodes). The ith
entry is zero if the ith node in NodeList is
constrained, and equals the index of that
node in FreeNodePtrs if that node is free.
ElList: Lx3 matrix, where L is the number of triangular
elements in the mesh. Each row corresponds to
one element and contains pointers to the nodes
of the triangle in NodeList.
For more information, see Section 10.1.1 of "Partial Differential
Equations: Analytical and Numerical Methods" by Mark S. Gockenbach
(SIAM, 2002).
The command RectangleMeshD creates a regular triangulation of a rectangle
[0, l x ] [0, l y ] ,
assuming Dirichlet conditions on the boundary.
help RectangleMeshD
T=RectangleMeshD(nx,ny,lx,ly)
This function creates a regular, nx by ny finite element
mesh for a Dirichlet problem on the rectangle
[0,lx]x[0,ly].
119
The last three arguments can be omitted; their default
values are ny=nx, lx=1, ly=1. Thus, the command
"T=RectangleMeshD(m)" creates a regular mesh with
2m^2 triangles on the unit square.
For a description of the data structure describing T,
see "help FemMesh".
The command RectangleMeshM creates a regular triangulation of the same rectangle, assuming Dirichlet
conditions on the bottom and left boundaries, Neumann conditions on the top and right
(the "M" stands for "mixed").
help RectangleMeshM
T=RectangleMeshM(nx,ny,lx,ly)
This function creates a regular, nx by ny finite element
mesh for the rectangle [0,lx]x[0,ly]. Dirichlet conditions
are assumed on the bottom and left of the rectangle, and
Neumann conditions on the top and right.
The last three arguments can be omitted; their default
values are ny=nx, lx=1, ly=1. Thus, the command
"T=RectangleMeshM(m)" creates a regular mesh with
2m^2 triangles on the unit square.
For a description of the data structure describing T,
see "help FemMesh".
Thus I only provide the means to deal with a single domain shape (a rectangle), and only under two
combinations of boundary conditions. To use my code to solve BVPs on other domains, you will have to
write code to generate the mesh yourself. Note, however, that the rest of the code is completely general: it
handles any polygonal domain, with any combination of (homogeneous) Dirichlet and Neumann conditions.
Here I create a mesh:
clear
T=RectangleMeshD(4)
T =
NodeList: [25x2 double]
NodePtrs: [25x1 double]
FreeNodePtrs: [9x1 double]
ElList: [32x3 double]
The mesh I just created is shown in Figure 10.1 in the text. I have provided a means of viewing a mesh:
help ShowMesh
ShowMesh(T,flag)
This function displays a triangular mesh T. Free nodes
are indicated by a 'o', and constrained nodes by an 'x'.
For a description of the data structure T, see
"help FemMesh".
The optional argument flag has the following effect:
flag=1: the triangles are labeled by their indices
120
flag=2: the nodes are labeled by their indices
flag=3: both the nodes and triangles are labeled
flag=4: the free nodes are labeled by their indices.(Notice that it is also possible to label the triangles and/or nodes by their indices.)
Computing the stiffness matrix and the load vector
Here are the main commands:
help Stiffness
K=Stiffness(T,fna)
Assembles the stiffness matrix for the PDE
-div(a(x,y)*grad u)=f(x,y) in Omega,
u=0 on Gamma,
du/dn=0 on Bndy(Omega)-Gamma.
The coefficient a(x,y) must be implemented in the function fna,
while T describes the triangulation of Omega. For a description of
the data structure T, see "help FemMesh".
help LoadV
F=LoadV(T,fnf)
Assembles the load vector for the BVP
-div(a(x,y)*grad u)=f(x,y) in Omega,
u=0 on Gamma,
du/dn=0 on Bndy(Omega)-Gamma.
The right hand side function f(x,y) must be implemented in the
function fnf.
121
Thus, to apply the finite element method to the BVP given above, I need to define the coefficient a(x,y) and
the forcing function f(x,y). As an example, I will reproduce the computations from Example 8.10 from the
text, in which case a(x,y)=1 and f(x,y)=x.
a=inline('1','x','y');
f=inline('x','x','y');
Now I compute the stiffness matrix K and the load vector F:
K=Stiffness(T,a);
F=LoadV(T,f);
Finally, I solve the system Ku=F to get the nodal values:
u=K\F;
Given the vector of nodal values (and the mesh), you can graph the computed solution using the
ShowPWLinFcn command:
help ShowPWLinFcn
ShowPWLinFcn(T,U)
This function draws a surface plot of a piecewise
linear function defined on a triangular mesh. The
inputs are T, the mesh (see "help FemMesh" for details
about this data structure) and the vector U, giving
the nodal values of the function (typically U would
be obtained by solving a finite element equation).
ShowPWLinFcn(T,u)122
The above solution does not look very good (not smooth, for instance); this is because the mesh is rather
coarse. I will now repeat the calculation on a finer mesh.
T=RectangleMeshD(16);K=Stiffness(T,a);
F=LoadV(T,f);
u=K\F;
ShowPWLinFcn(T,u)
123For the sake of illustrating mixed boundary conditions, I will solve the same PDE with mixed boundary
conditions:
T1=RectangleMeshM(16);
K1=Stiffness(T1,a);
F1=LoadV(T1,f);
u1=K1\F1;
ShowPWLinFcn(T1,u1)
0.25124
Testing the code
To see how well the code is working, I can solve a problem with a known solution, and compare the
computed solution with the exact solution. I can easily create a problem with a known solution; I just
choose a(x,y) and any u(x,y) satisfying the boundary conditions, and then compute
f ( x, y) a( x, y)u
to get the right-hand side f. For example, suppose I take
clear
a=inline('1+x.^2','x','y');
u=inline('x.*(1-x).*sin(pi*y)','x','y');
(Notice that u satisfies homogeneous Dirichlet conditions on the unit square.) Then I can compute f:
syms x y
-diff(a(x,y)*diff(u(x,y),x),x)-diff(a(x,y)*diff(u(x,y),y),y)
ans =
-2*x*((1-x)*sin(pi*y)-x*sin(pi*y))+2*(1+x^2)*sin(pi*y)+(1+x^2)*x*(1-
x)*sin(pi*y)*pi^2
f=inline(char(ans),'x','y');
Now I will create a coarse mesh and compute the finite element approximation:
T=RectangleMeshD(2);
K=Stiffness(T,a);
F=LoadV(T,f);
U=K\F;
Here is the approximate solution:
ShowPWLinFcn(T,U)
125(The mesh is so coarse that there is only one free node!)
For comparison purposes, let me compute the nodal values of the exact solution on the same mesh. I have
provided a command to do this:
help NodalValues
v=NodalValues(T,u)
This function sets v equal to the vector of
values of u(x,y) at the nodes of the mesh T.
The function implementing u must be vectorized.
See "help FemMesh" for a description of the
data structure for T.
V=NodalValues(T,u);
Now I will plot the difference between the computed solution and the piecewise linear interpolant of the
exact solution (notice the scale on the graph):
ShowPWLinFcn(T,V-U)
126
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1
0.8 1
0.6 0.8
0.4 0.6
0.4
0.2
0.2
0 0
I now repeat with a sequence of finer meshes:
T=RectangleMeshD(4127
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
1
0.8 1
0.6 0.8
0.4 0.6
0.4
0.2
0.2
0 0
T=RectangleMeshD(8128
-3
x 10
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
1
0.8 1
0.6 0.8
0.4 0.6
0.4
0.2
0.2
0 0
T=RectangleMeshD(16-3
x 10
1.2
1
0.8
0.6
0.4
0.2
0
1
0.8 1
0.6 0.8
0.4 0.6
0.4
0.2
0.2
0 0
129
Using the code
My purpose for providing this code is so that you can see how finite element methods are implemented in
practice. To really benefit from the code, you should extend its capabilities. By writing some code
yourself, you will learn how such programs are written. Here are some projects you might undertake, more
or less in order of difficulty:
1. Write a command called Mass that computes the mass matrix. The calling sequence should be
simply M=Mass(T), where T is the triangulation.
2. Choose some other geometric shapes and/or combinations of boundary conditions, and write some
mesh generation routines analogous to RectangleMeshD.
3. Extend the code to handle inhomogeneous Dirichlet conditions. Recall that such boundary
conditions change the load vector, so the routine LoadV must be modified.
4. Extend the code to handle inhomogeneous Neumann conditions. Like inhomogeneous Dirichlet
conditions, the load vector is affected. However, it is not enough to simply modify LoadV,
because the data structure describing T does not contain all of the necessary information. The data
structure must indicate which edges (of triangles) lie on the boundary and are free. You will have
to devise a way to store this information in the data structure. Here is a key point: a boundary edge
is free if one or both of its endpoints is a free node. To incorporate inhomogeneous Neumann
conditions, you must integrate over each free boundary edge.
5. (Hard) Write a routine to refine a given mesh, according to the standard method suggested in
Exercise 10.1.4 of the text. You will need to extend the data structure in order to do this, since
during the refinement process, you need to know, for each edge, whether it lies on the boundary or
not. This information is not found in the data structure I defined, but it can be easily added.
130 |
It should come as no surprise that a textbook has been written that
attempts to teach foundational math and introduce programming. Indeed,
several undergraduate Computer Science departments schedule both
courses side by side. This book, unlike those programmes, attempts to
teach the subjects as complementary, using one to learn about the
other. Requiring only a secondary education in math, this textbook's
goal is to teach the reader Haskell programming and theorem proving.
The first chapter serves as a short crash-course in Haskell and the
hugs interpreter. Subsequent chapters are each about a mathematical
topic and use both text and Haskell code to illustrate these
concepts. Chapters two through seven analyze foundational math
constructs including sets, relations, functions, and induction. The
final four chapters touch on more difficult concepts such as
corecursion and infinite sets.
Throughout the book, programs are used to illustrate the text. This is
one area where the book really excels. As well as illustrating the
text, the programs incite further thinking, ensuring a deeper
understanding of the mathematical concepts. For example, when teaching
recursive proofs over the natural numbers, they solidify understanding
by defining, in Haskell, the natural numbers in terms of successors of
zero. This style of integrating programs within the text gave the math
an applicable feel, absent from most pure math books.
The coverage of proofs, sets, relations, functions, and induction is
gentle and effective. New notation is always accompanied by easily
understood explanations. Those explanations are supplemented with
common mistakes and a variety of examples. This methodology allows one
to learn the math with very little background and external support.
In some instances, because the book focuses primarily on mathematical
concepts, the reader is deprived of necessary and basic programming
knowledge. An example of this is chapter one, where Haskell is
introduced. Although someone familiar with programming could read
through the chapter with ease, a beginner would likely stumble on some
of the undefined words, like stack-overflow and floating point
numbers. Furthermore, additional material about how Haskell relates to
programming languages in general could have better set the stage.
The Haskell programs in the book are very concise and directly
connected to the math they demonstrate. The high-level math constructs
which are applied enhance the reader's ability to write more compact
and conceptually elegant programs. Moreover, the reader is forced to
think about programming from a declarative point of view, which
encourages using higher-level constructs for problem solving. By
contrast, most introductory programming books concentrate mostly on
successfully solving the problem.
Instead of locating exercises at the end of the each chapter, the
authors mixed them within the text and examples. Designed to flow with
the text, most of them are quick and have an easy to medium
difficulty. The integration of exercises and text gives the book a
enjoyable, hands-on feel. Some difficult proof problems, marked with a
'*', are scattered throughout the text. Although few in number, they
provide excellent preparation for upper level math classes.
On the whole, I think this book would be very successful for use in an
introductory college level math course or self learning. The
integration of programming provides a unique and enjoyable way to
learn math.
The most hard core programming language ever - Haskell - presented in
a really different and thought provoking way. For those interested in
these topics - functional languages, logics and mathematical
foundations - i think it's must have.
I'm enjoying the book very much. I wish I had had a book like
this when I first started learning programming.
Charles Turner, December 10, 2010:
I would like to express my gratitude, I've been struggling
with the discrete maths strand of my CS course, but the book really
has helped. It's given me confidence tackling proof problems in
particular.
Shyamal Prasad, December 14, 2008:
Thanks very much for a wonderful little book - it's really one of the
best texts I've read in computing in a long time!
B. Edds (Greenville, SC USA): We need more like it!, September
18, 2012:
Let me start by saying I am not a Haskell programmer. I
program mostly with C# and F#, and study Haskell as a educational
exercise. However, Haskell's ability to express formulae makes it the
perfect 'pseudo-code' for presenting abstract mathematical concepts in
a concrete way to modern programmers.
Us workaday programmers have a hard time picking up maths or
remembering the maths we learned in school. This book gives us a great
alternative to grinding through old textbooks to learn the math we
need in our projects. Really, the concreteness of seeing mathematical
concepts implemented in working code can make all the difference.
I love this book so much, I can only wish there were more in the same
vein. What about a `Haskell Road to Linear Algebra', or a `Haskell
Road to Calculus'? Wouldn't such a series be magnificent?!
If I could ask for anything more from the authors, I would ask for
more books like this! I can't praise the work enough!
Andrey Smorodin (USA): This is a great book, May 31, 2012:
I am really happy to have this item which is great book for
learning math with real applications in Haskell. Author demonstrate
elegant of math theory and pure functional language. This pair
together can be used for solving and modeling math concepts.
I really recommend this book to math hobbits and/or Haskell
newcomers. And thank you again Kees Doets and Jan van Eijck for
several happy weeks and good time with reading and solving
problems. That was really fun.
Want to learn discrete math as in the non-calculus version of Math
101? Willing to work hard to learn it? Then this is the book for you!
It is written in a user friendly style. The book has a chatty tone
when explaining serious topics. The chat is the talk of experts,
however, so it should never be underestimated. The book will teach you
the functional programming language Haskell. This language is the most
modern version of the Lisp family to have received any significant
attention. It features advanced type features and lazy evaluation. The
book covers all the "classic" topics of a discrete math course, to a
considerable depth. Best of all, they are all implemented in
Haskell. All except for the predicate calculus, that is. It would be a
major piece of work indeed to have any implementation of this topic in
a work at this level! The book teaches logic by example. One is taught
the meaning, the rules and the insider's "recipes" of mathematical
proofs. The book then covers the standard topics of sets, relations,
functions, and induction / recursion with impressive computer
implementations. The book goes on to a quick overview of the
construction of the number systems of mathematics. It proceeds to a
substantial treatment of combinatorics in a short space which includes
implementation of polynomial solutions to recursion relations. Then it
is on to corecursion, power series, and a hint of domain theory. The
book ends with a modern chapter on the elements of set theory. All in
all an excellent book! IMHO it is the best book out there on the
topics in a discrete math course. It is amazing that all of this
(except the serious parts of logic) could be implemented in Haskell so
successfully. The one caveat about the book is that if you want to
benefit fully from it, you are going to have to do some hard work.
Code Monkey, The math book all programmers should read,
December 2, 2009:
The 'Haskell Road to Logic' is a wonderful introduction to the
mathematics that lie behind functional programming and computer
science. Readers should however be aware that this book is not, and
does not pretend to be, a book about programming in Haskell. It is
really a text book about topics in mathematics that are of particular
interest to computer scientists. What distinguishes the book from many
others is its use of Haskell to implement mathematical structures that
are usually taught as abstract concepts. This approach makes the
mathematics far more approachable for computer programmers than many
other text books. Presumably it should also make for an excellent
introduction to computer programming for the mathematically inclined.
While the book is easy to read and has a friendly writing style, it
not particularly well suited to casual reading. To really understand
the subject being discussed the reader will probably need to solve
most of the exercises in the text. The good news is this requires
minimal prior mathematical training (the authors expect familiarity
with "secondary school mathematics"). Solving the exercises will also
train the reader in writing, and proving the correctness of, short
functional programs.
The book has a minor few faults. One is a relatively large number of
minor errors (many of which are noted in the errata available on the
book's website). Another is that some major topics are introduced in
exercises without much discussion, particularly in the later
chapters. But these are but quibbles in a review of a fantastic math
book for programmers.
When I was a Math undergrad back in the 70's, we had a 5 hr course
called Foundations of Mathematics. This was an intro to symbolic
logic, propositional calculus, and methods of proof. Deadly boring,
dry material that we either knew, grasped by intuition, or ignored.
This textbook covers this material in a constructive fashion by using
the Haskell programming language. Haskell is a modern form of lisp,
one of the original programming languages, from the '50's, the
language used for most Artificial Intelligence work. The breadth of
Haskell allows it to be used for logic and proof, as well as the usual
numerical and string processing. Pattern matching and list processing
is built into the basic structure of the Haskell language.
This text's exercises are mostly Haskell programming
assignments. Turning the abstract ideas of the math into the concrete
statements of Haskell (if statements in a program can be considered
concrete) will make the ideas familiar and real. Free, useful versions
of Haskell (Hugs) are available for readers or students to use, even
on Windows systems. Any familiarity with any programming system and a
text editor should be enough to get started.
What is the difference between proving a theorem and debugging a
program? The way I do it, not much. That has much to do with me, I am
a programmer first, and a Math second. This statement is the
Curry-Howard correspondence, connecting computability and proof or
truth. This text is a step on that road to truth.
This book is breathtaking in its clarity and depth. I'm into the
chapter on using Haskell to prove logical theorems.
One of my all-time favourite subjects in undergraduate study was
Logic. That you could translate English language arguments into
logical symbolism and test their validity and soundness by following a
set of rules was, to me, a revelation. By following a number of
logical inference rules, one could build an argument that proves the
validity of a conclusion. The use of truth tables was intriguing and
so simple in concept.
Blaise Pascal (French philosopher, and whose name was given to the
Pascal programming language) speculated about the possibility of a
logic calculating machine. Well, this book shows how you can do it
with Haskell.
The Haskell Road To Logic, Maths and Programming provides Haskell
source code that you can run to demonstrate the validity of all the
traditional logical inference rules that I learned in Logic.
It's a pleasure to read.
Alexey Romanov (Moscow, Russia),
A perfect fit, August 20, 2009:
As a mathematician, I find that of all current programming languages
Haskell is the best for describing mathematical notions in thanks to
such features as newtypes, type classes, list comprehensions, purity,
etc. It is also much better suited to reasoning about programs than
the imperative languages are.
Therefore, it makes perfect sense to write a book which covers the
basic notions of discrete math and implements them in Haskell, which
is just what this book does. Along the way the reader will see many
examples of strict and rigorous logical proofs, and learn how to
approach them.
Luby Liao (University of San Diego, San Diego, CA),
I recommend every university library to have this book,
March 16, 2006:
In March 2006, there are only three or four books that boast Haskell
in their titles. They are all excellent books. But the world can
benefit from more Haskell books. In fact, we are in need of such
books, especially CS1 texts. This book is a pleasure to read. I
suspect that even math haters will not find it hostile. Anyone
learning Haskell will find the book and its companion web site a
valuable resouce. On the web site, you will find the source code from
the book, such as Powerseries.hs. You can quickly play and experiment
with it; read and learn from it. I wish more books are as pleasant
and affordable ($25).
Qual Highety (Austin, Texas), Enlightening, September 17, 2010
As a professional programmer who dropped out of college too early to
get a good grounding in mathematics, this book has been a wonder to
me. I've been trying to learn more advanced mathematics in the last
year or so, and struggled because I didn't know how to do proofs.
Haskell is prominent in the title, but is secondary in the book. The
vast majority of my time has been spent reading and performing proofs
with with pencil and paper. The mathematics portions are all fairly
rigorous. Once proofs are fully introduced, nearly all the exercises
are proving theorems given in the book, or providing counter
examples. There isn't much in the way of computational type exercises
I've seen in some other discrete math texts, stuff like "what's the
transitive closure of this relation". Instead, a theorem will be given
and the author will ask you to prove it.
The Haskell exercises typically follow pure math introductions and
exercises, and are used to help develop a stronger, more intuitive
understating of the subjects. I can now write enough Haskell to take
on the exercises, but to this point, the Haskell has all be very
compact - 10 lines of code to answer a question at the max, usually
more like 3. There's no way I could go and do something 'practical'
with Haskell at this point, but I can grok some of it and am starting
to appreciate functional programming.
This is easily the most challenging self study book I've taken on, but
also the most rewarding. Not only have I gained proof writing
knowledge, and a solid understanding of the fundamentals of modern
mathematics, but it's helped me develop a measure of discipline in
thought. I've noticed improvements in the clarity of my day to day
software development - in design, development, and verification. The
methodical approach to breaking down a proof into cases and sub-cases
has been most helpful in that regard.
I recommend this to anyone, especially developers who haven't had the
benefit of a thorough mathematical education.
J. Burton, First class introduction to Haskell,
4 April 2007:
This is a great book which I enjoyed reading and found more useful
than other introductory Haskell texts with the exception of Bird,
although it may not be the best choice if you are most interested in
the practical aspects of the Haskell. The emphasis is on foundations
and language features (which are all Haskell 98) are introduced as
they occur naturally in the discussion of a number of Number Theory
and Logic problems.
This book is well named -- it aims to teach the three disciplines of
its title in equal measure and learning Haskell is presented as the
channel for that, rather than being a goal in itself.
Le Serf, Lovely, 23 April 2009:
The Haskell Road is a truly enjoyable little book.
What it is not: HR is not one of those vast towers of paper that
introductory computer science books seem to have become. The kind of
book that's called "Discrete Mathematics" and essentially contains all
the stuff that nobody wants to teach but everyone wants you to
know. Those books and the courses they support are often a student's
first introduction to thinking about computing, and it's
shameful. They are a patchwork of misaligned topics - and the
students' thinking begins to resemble them, unsurprisingly.
Rather, the Haskell road is elegant - clean, concise, yet informal and
approachable. Like the title says, it is an introduction to Logic,
Maths and Programming. The book takes the enlightened viewpoint that
these are unified concepts. The book begins by introducing basic
Haskell syntax, and all of a sudden, we are implementing a prime
number test. Simple; yes, but we also learn how to _prove_ that a
procedure is a prime number test.
This approach continues throughout the book. The ideas of formal logic
and deductive reasoning are made approachable by the fact that we
implement the rules in Haskell. Sometimes, the exercise is in Haskell,
and the answer is in logic. The point is that the reader is made from
the first instant to see the equivalence, the shared foundations
between these different means of expressing thought.
This is also one of the few books that teaches, explicitly, the means
of proof. It does not do so abstractly, but quite straightforwardly,
using the tools of formal logic. A few somewhat difficult chapters are
the result (2-4); but they are greatly enlightened by enjoyable
exercises. This treatment of proof was a first for me - though I am
currently a graduate student, it made clear much that had been opaque
to me. I read the chapters and did the exercises in a sitting; the
following day (literally) I was finding my quantum computing proofs
easier than I had the day before. Few books are able to have such a
direct, jolting impact - indeed, that experience compelled me to write
this review.
The exercises are not too difficult nor too easy; they are not all
gathered at the end but rather placed in exactly the right place. Five
minutes attempting an exercise is usually enough to see the trick of
it. Some of them take seconds, however, and some take quite a long
time. Those are marked as such. The exercises are the glue that sticks
the book's ideas into the reader's mind, and it works.
The basic ideas of programming, like lists and functions are
brilliantly intertwined with the equivalent ideas in mathematics,
namely set theory. Haskell's lazy evaluation enables us to start
puzzling about infinite sets early on. Throughout, one learns a
reasoned, careful, elegant approach to programming. Too many students
learn to program by throwing Java API calls at the problem until most
of the output is correct. A more thorough, more disciplined mind can
go much further, and the Haskell Road seeks to develop this.
I can't say enough good things about this book, so I will stop now. If
you have been doing computer science for 30 years, or if it's your
first day, or (especially!) if you're a programmer that wants to learn
to do "real math" - this is the book for you.
For some reason, I keep picking up "The Haskell Road to Logic, Maths and Programming"
All my life I have avoided proofs and mathematics, but perhaps now is
the moment to meet my Maker. I keep going back to this book.
For some reason, it is infectious. A beginner introduction to how
thought is structured. A beginner introduction to what math is all
about.
I had been wondering what I was going to do with Haskell once I
finished learning it. I am a Perl professional by trade and even
though some interplay has been going on, I never really knew why I was
learning Haskell. I just liked the conciseness and elegance of the
language and the mathematical purity.
The most important thing is to have passion behind what you do. If I
want to go deeper into math for math's sake, then so be it... no need
to artificially create projects for myself. Just enjoy haskell and
math. It is a very very good fit.
Metaperl again (Sat, 11/05/2005):
I am grateful to be studying Haskell. It has
done wonders for my corporate-level Perl development. The book to read
is "The Haskell Road to Logic, Maths and Computer Programming." This
book will force you to think clearly. It does not cover monads, but
monads can wait until you can specify and solve problems. But my
improved thought processes are one thing. Delivering projects on a
deadline is another and Perl continues to deliver through a
cooperative community based around CPAN. The Haskell community is
based around the reason of masterminds doing the intelligent
things. The Perl community is based around the needs of people doing
practical things in the best way that they can.
Maiz, Jan 4, 2010:
The most hard core programming language ever - Haskell -
presented in a really different and thought provoking way. For those
interested in this topics - functional languages, logics and
mathematical foundations - i think it's must have.
GoodReads Review
I'm getting better with producing proofs because I found the Haskell
Road to Logic Maths and Programming. It's a good remedial course for
those of us familiar with programming but weak in abstract math chops.
I finished my 2 year degree with distance courses 10 years
ago. I remember the feeling of isolation both from the professor and
the other students. Were there any other students? I don't even
know. I did finish my courses though.
I can add two observations from own experience.
1. I desperately wanted to socialize with the other students. Even
though the work had some challenge to it, I was bored. Half the fun of
the class is doing it together and the together was completely missing
for me.
However...
2. When I'm interested in a subject on my own, and select a text on my
own, I feel completely different about it. I email the author of the
book to get answer keys, I do problem sets. I work alone at reading
and understanding the text. If I get stuck on something I email the
author for answers.
Does anyone else spot the irony? The situation is exactly the same,
worse even, yet I'm having fun. I don't necessarily "finish" because I
usually have a particular goal which usually doesn't involve a
comprehensive understanding of my text.
For instance, I wanted to understand a little linear algebra and
realized that what little knowledge I ever had about constructing
proofs was completely rusted away.
I got the Haskell Road to Logic Maths and Programming. I did the first
3 chapters and part of chapter 4, just to learn how to do a proof the
right way. I did every stinking exercise up to the point I felt I had
reached my goal.
The professor/author was responsive to my occasional questions and
provided me an answer key. (And a little admonition not to peek. Still
makes me smile to this day.)
My wonderful experience was exactly the same as my dreadful experience
in many ways.
The Haskell Road To Logic, Maths And Programming has a very seductive
intro and table of contents... I am starting to realize that I am
going nowhere in Haskell unless I learn to think and know functions
well... humbling after 5 years as a well-paid Perl programmer.
Herbert Carl Meyer wrote (June 1, 2009):
I find the book fascinating and instructive. I am afraid I am a
programmer first and mathematician second. It is hard for me to read the
book and think abstractly, and easy to fire up the laptop and write and
run haskell code.
I had a similar problem when I was an undergrad, my instructor in a
Linear Algebra course used an APL (Iverson) system as part of the
course. I think at least part of it was to automate the drudgery of
marking homework. I learned more about APL than linear algebra.
But I will continue. I plan on picking up the other texts in the series.
Thank you for writing the book.
Chris Kimm wrote (November 4, 2008):
I'm about 2/3 though the book and I've found it to be
excellent in all ways! Though I have no formal math or computer
science training, I've found the book to be accessible and perfectly
paced. Also, though I was initially doubtful that Haskell could be a
learn-as-you-go language, (ala Scheme in SICP), there haven't been any
sections so far where I've needed to consult external sources to get
through your material.
My initial hope was to use the material in this book to prepare me for
"Concrete Mathematics" by Graham, Knuth, and Patashnik. Even if I
need another stepping stone before approaching that book, "The Haskell
Road" has offered an enormous amount of clarity for me in math
reasoning and also in functional programming.
Iain Barnett wrote (October 3, 2008):
I'm finding the book challenging but very interesting. I like
the pace, the simple presentation, and that the only challenge is the
subject matter -- not anything to do with the writing (unlike the other
Haskell book I've got -- or a lot of computing books).
Bernardo Szpilman wrote (May 24, 2008):
I'm learning Haskell and extinguishing my Linear Algebra hate
thanks to your book. I love programming and have always done, but
mine was a bad relationship with math, especially proofs. Thanks so
much for bringing the two together, thus bringing me much closer to
the math I'll need throughout my whole Computer Engineering career
(I'm in college). It's a splendid idea, and for that your work is
completely unique.
I must say that I was really pleased to find your book, because I
have been trying to find a way of studying logic using a computer for
some time and this was the first time it all seems to come together.
I am currently going through your excellent book, "The Haskell Road to
Logic, Maths and Programming". Many thanks for taking the time to
write such a fine book, utilizing such a fine language as Haskell, and
teaching both programming, logic, and the fundamentals of math.
I'd like to thank you for writing such an informative book on
Haskell. I'm a senior in high school from New Jersey. I've been
trying to wrap my mind around functional programming and formal
mathematics to get a headstart on my college studies and your text has
proved to be a formidable companion thus far.
I'm enjoying your book a lot, this combination of Haskell plus
mathematics and logic is just awsome :-)
John O'Keeffe wrote (August 8, 2007):
I'd just like to say what a fabulous book it is. It exactly
targets the maths I'm working through as well as opening up the
delightful world of Haskell programming. So thank you very much
indeed!
One of the nicest things about writing Haskell code is that it's the
closest one can probably get to writing pure maths while
programming. Thus, learning Haskell is an excellent way to learn more
about maths and logic. If that sounds to you like doubling the fun
(and you're not yet a math wizard), The Haskell Road to Logic, Maths
and Programming by Jan van Eijck and Kees Doets is definitely the book
for you. Its purpose is to teach logic and mathematical reasoning in
practice, and to connect logical reasoning with Haskell
programming. And, in my opinion, it does a pretty good job at
that. Although it begins with the very basics, it includes chapters on
far from trivial (i.e., fun!) stuff like, for instace, corecursion or
Cantor sets. And i found amazing how natural it was to express logical
and mathematical ideas in Haskell.
I'd recommend this one: The Haskell Road to Logic, Maths and
Programming -- would fit best if you'd also have an interest in
Haskell/FP, but it's also a good introduction to formal proofs --
might be a bit too basic if you're interested in higher maths, tho,
but a good starting point for math education. |
Precalculus: Mathematics for Calculus
9780840068071
ISBN:
0840068077
Edition: 6 Pub Date: 2011 Publisher: Brooks Cole
Summary: Designed to give students a background in mathematics theory and introduce them to mathematics concepts this textbook is comprehensive without being daunting. Students are introduced to modelling and problem solving and they are given a rigorous workout on what they have learned as they work through the book. It has many graphs that chart mathematical ideas that students can assimilate with ease. It is written in a c...lear and readable style that will aid comprehension and enjoyment. This is just one of the many cheap math textbooks we have available for students to acquire in great condition.
Stewart, James is the author of Precalculus: Mathematics for Calculus, published 2011 under ISBN 9780840068071 and 0840068077. One thousand two hundred twenty eight Precalculus: Mathematics for Calculus textbooks are available for sale on ValoreBooks.com, four hundred eighty one used from the cheapest price of $100.45, or buy new starting at $101.25.[read moreThis book did a very good job of explaining concepts step by step. Compared to other math textbooks, this one was actually easy to follow, the problems and examples were similar and flowed in a logical way.
I learned about vectors, conic equations,trigonometry and various functions from this book. |
A systematic, research-based introduction to the principles and practice of teaching mathematics at primary school level. This second edition includes new material on middle years, and on numeracy, early numbers and fractions.Praise for the Second Edition "This book is a systematic, well-written, well-organized text on multivariate analysis packed with intuition and insight . . . There is much practical wisdom in this book that is hard to find elsewhere." —IIE Transactions Filled with new and timely content, Methods of Multivariate Analysis,...In the vein of A Beautiful Mind, The Man Who Loved Only Numbers, and Rosalind Franklin: The Dark Lady of DNA, this volume tells the poignant story of the brilliant, colorful, controversial mathematician named Dorothy Wrinch. Drawing on her own personal and professional relationship with Wrinch and archives in the United States, Canada, and England,... more... |
About:
Techniques of Estimation: Estimation by Rounding
Metadata
Name:
Techniques of Estimation: Estimation by Rounding
ID:
m35011
Language:
English
(en)
Summary:
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to estimate by rounding. By the end of the module students should understand the reason for estimation and be able to estimate the result of an addition, multiplication, subtraction, or division using the rounding technique. |
Mathematica is a computer system that integrates symbolic and numerical mathematics with powerful computer graphics. These are supported by a concise and flexible programming language. These six documents provide an intensive study of Mathematica Graphics from a basic introduction to advanced graphics programming. Reprint from the Mathematica Conference, June 1992, Boston. 92 pages. |
Screencasts explaining curriculum changes
The Group 5 academic team has developed some useful screencasts to help explain the changes in the new mathematics curriculum (first teaching September 2012). The first screencast details general changes in the mathematics courses (mathematical studies SL, mathematics SL, mathematics HL and further mathematics HL). The second discusses changes to internal assessment in mathematics SL and HL.
Group 5 mathematics overview of new IA
Excellent new publications for mathematics
Supporting the existing course
Supporting the new course
Three new Course Companions supporting the new course curriculum are available from OUP (Oxford University Press), published in collaboration with the IB.
These new Course Companions contain comprehensive syllabus coverage and lots of practice exercises. They are developed with the IB and written by experienced authors, teachers and examiners. They take an inquiry-based approach, provide mathematical theory framed in a meaningful context, thoroughly integrate international-mindedness and TOK and ensure that study is fully aligned with the Learner Profile.
4 Comments
I have a question regarding the new Math curriculum. My daughter is in 1st year of IBDP taking Math SL. Will the change in curriculum and textbook affect her in her 2nd and final year of IBDP starting Aug 2012 (USA)? If so, what are the differences between the current and new curricula – in terms of rigor, course details, format etc?
Hi,
I wonder where can I buy a textbook covering the curriculum of the new further mathematics HL course?
I could only find the Cambridge one published in 2008 which does not include the linear algebra topic.
Hi Crystal,
Thanks for your post. We will be publishing a new resource for further mathematics HL in a couple of months – an online Questionbank dedicated to this level. We do not have any other publications for this level I'm afraid.
Regards
Kate |
Jen1Document Transcript
Chapter 1
THE PROBLEM AND ITS BACKGROUND
Introduction
We live in a mathematical world. Whenever we decide on a purchase, choose an
insurance or health plan, or use a spreadsheet, we rely on mathematical understanding.
The World Wide Web, CD-ROMs, and other media disseminate vast quantities of
quantitative information. The level of mathematical thinking and problem solving needed
in the workplace has increased dramatically. In such a world, those who understand and
can do mathematics will have opportunities that others do not. Mathematical competence
opens doors to productive futures. A lack of mathematical competence closes those
doors. Students have different abilities, needs, and interests. Yet everyone needs to be
able to use mathematics in his or her personal life, in the workplace, and in further study.
All students deserve an opportunity to understand the power and beauty of mathematics.
Students need to learn a new set of mathematics basics that enable them to compute
fluently and to solve problems creatively and resourcefully.
It has been long time to discover the importance of Mathematics in our world.
And these discoveries lead us to more technological or what so called Industrial era,
wherein the different usage of technological devices occur. In this era, application of
Mathematics helps to develop and invent such technological devices. Through these
applications our life became easier. Now a day, Mathematics is the key to all Sciences.
Despite explaining more about mathematics and the proof that it's really
important, the students today don't like this subject. They think that the Mathematics is a
boring subject, and it's hard to understand formulas, they always say "Why should we
study Mathematics, only four major operations are enough and the rest no longer needed.
We do use graphs and formulas in our daily living." Only if they understand the logic
behind this subject and the principles applied in different problems, if they get what
Mathematics really meant to be, they will find that it is not a boring subject, that
mathematics is an interesting one. Mathematics becomes part of our life, not only in our
academic subjects, but in all part of our integral life. We don't see that even in simple
conversation mathematics take place. In our transportation it also occurs, and in our daily
living it definitely applied.
Background of the Study
According to Schereiber (2000) those who have positive attitudes toward
mathematics have a better performance in this subject.
Mathematics achievement has shown that the students from each major level of
Education in Asia seemed to outperform their counterparts. Many studies have examined
students' thinking about school and their attitude toward Mathematics. Mathematics
performance involves a complex interaction of factors on school outcome. Although the
relationship between mathematics performance and students factor has been studied
widely, it is important to explore the factors that contribute students' mathematics
performance.
Wendy Hansen (2008) stated that boys are more likely than girls to be math
geniuses. Girls scored in the top 5% almost as often as boys, the data showed Comparing
the average scores of girls and boys in California and nine other states, the researchers
found that neither gender consistently outpaced the other in any state or at any grade
level. Even on test questions from the National Assessment of Education Progress that
were designed to measure complex reasoning skills, the gender differences were
minuscule, according to the study.
Student engagement in mathematics refers to students' motivation to learn
mathematics, their confidence in their ability to succeed in mathematics and their
emotional feelings about mathematics. Student engagement in mathematics plays a key
role in the acquisition of math skills and knowledge – students who are engaged in the
learning process will tend to learn more and be more receptive to further learning.
Student engagement also has an impact upon course selection, educational pathways and
later career choices.
Math performance has improved, again, through expecting students to achieve,
providing instruction based on individual student needs and using a variety of methods to
reach all learners. One factor...has been aligning the math curriculum to ensure that the
delivery of instruction is consistent with the assessment frequency.
This particular study attempts to determine the factors affecting mathematics
performance of Laboratory High School at Laguna State Polytechnic University
Academic Year 2009-2010.
Theoretical Framework
Dweck, C. S. (1999) stated that students believe that their ability is fixed,
probably at birth, and there is very little if anything they can do to improve it is called
fixed IQ theorists. They believe ability comes from talent rather than from the slow
development of skills through learning. "It's all in the genes". Either you can do it with
little effort, or you will never be able to do it, so you might as well give up in the face of
difficulty. E.g. " I can't do math". And Untapped Potential theorists, students believe that
ability and success are due to learning, and learning requires time and effort. In the case
of difficulty one must try harder, try another approach, or seek help etc.
Inzlicht (2003) stated that entity and incremental theories of ability were assessed
separately so that their separate influences could be examined; math performance was
examined by controlling for prior math performance. Entity theory was expected to be a
negative predictor of performance, whereas incremental theory was expected to be a
positive predictor.
Guohua Peng (2002) stated that simple traditional methods gradually make the
students feel that mathematics is pointless and has little value to them in real life. It
becomes a subject they are forced to study, but one that is useless to them in real life. In
the traditional classroom setting discussed above, both students and teacher are often
frustrated because the students' individual needs are not met. Another disadvantage of the
conventional teaching method is that it provides no way for students to practice generic
skills such as communication and teamwork, which are very important for every
university student.
Dan Hull (1999) stated that growing numbers of teachers today—especially those
frustrated by repeated lack of student success in demonstrating basic proficiency on
standard tests are discovering that most students' interest and achievement in math,
science, and language improve dramatically when they are helped to make connections
between new information (knowledge) and experiences they have had, or with other
knowledge they have already mastered. Students' involvement in their schoolwork
increases significantly when they are taught why they are learning the concepts and how
those concepts can be used outside the classroom. And most students learn much more
efficiently when they are allowed to work cooperatively with other students in groups or
teams.
Conceptual Framework
The major concept of this study is focused on factors affecting Mathematics
Performance of Laboratory High School at Laguna State Polytechnic University
Academic Year 2009-2010.
Figure 1; shows the relationship of input variables which contain the extent of the
student-related factors and the extent of the teacher-related factors. While in the process
Statement of the Problem
The study attempts to determine the factors affecting mathematics performance of
Laboratory High School at Laguna State Polytechnic University Academic Year
2009-2010.
Specifically, it sought to answer the following questions:
1. What is the extent of the student-related factors in terms of:
1.1 Interest
1.2 Study Habits
2. What is the extent of teacher-related factors as evaluated by the students in terms
of:
2.1 Personality Traits
2.2 Teaching Skills
2.3 Instructional Materials
3. What is the level of students' mathematics performance?
4. Is there significant relationship between students' mathematics performance and
students-related factors?
5. Is there significant relationship between students' mathematics performance and
teacher-related factors?
Hypothesis
The following are the null hypothesis of this research:
There is no significant relationship between students' mathematics performance
and students-related factors.
There is no significant relationship between students' mathematics performance
and teacher-related factors.
Significance of the Study
The result of the study will merit the following:
School Administrator. The result of this study could serve as a baseline data to improve
programs for school advancement.
Curriculum Planner. The result of this study will help them appraise the existing
programs in terms of the student's needs and abilities and make changes as required.
Guidance Councilor. This study will help develop the guidance program in line with
individual needs and abilities of the students.
Facilitators. The results of this study may serve as an eye opener to create and innovate
instructional materials, and to use varied and appropriate teaching strategies.
Students. This study will help the students to develop their interest toward Mathematics
and appreciate the importance of Mathematics in their daily lives.
Parents. Who are directly concerned with the education of their children considering
school performance in different discipline.
Future Researcher. The result of this study can serve as basis for further study on
teaching learning activities and student mathematical performance.
Scope and Limitation
This study is limited only to Laboratory High School Students of Laguna State
Polytechnic University during the Academic Year 2009-2010.
Determining the factors affecting Mathematics Performance of Laboratory High
School Students was the focus of this research. The information needed will be gathered
using the checklist style research-made questionnaire. All information and conclusions
drawn from this study were obtained only to this particular group of students.
Definition of Terms
For better clarification and understanding of the terms related to this study, the
following terms are defined conceptually and operationally.
Mathematics Performance. This refers to the degree or capacity of students'
knowledge in Mathematics.
Instructional Materials. This refers to motivating techniques that teaching materials
or equipment used. It can high technology or simple materials that can use in learning
preference.
Interest. This refers to the amount of the students' dislike or like of particular things.
Study Habits. This refers to usual form or action of a person in studying.
Teaching Skills. This refers to the skills of teachers in mathematics in terms of
teaching her/his lesson.
Personality Traits. This refers to the good relationship of the mathematics teachers
with the students.
List of Tables
Table
1 Level of Interest in Mathematics as Perceived by the students
2 Level of Study Habits as Perceived by the Students
3 Student-related factors in terms of Interest and Study Habits
4 Personality Traits of Mathematics Teachers as Perceived by the students
5 Teaching Skills in Mathematics as Perceived by the Students
6 Instructional Materials used by the teachers in Mathematics as Perceived by the students
7 Teacher-related factors in terms of Personality Traits, Teaching Skills, and Instructional
Materials
8 Significant relationship between students' mathematics performance and students-related
factors.
9 Significant relationship between students' mathematics performance and teacher-related
factors. |
students find the leap between school and university level mathematics to be significantly greater than they expected. Success with Mathematics has been devised and written especially in order to help students bridge that gap. It offers clear, practical guidance from experienced teachers of mathematics in higher education on such key issues as:
After reading this book, students will find themselves better prepared for the change in pace, rigour and abstraction they encounter in degree level mathematics. They will also find themselves able to broaden their learning strategies and improve their self-directed study skills.
This book is essential reading for anyone following, or about to undertake, a degree in mathematics, or other degree courses with mathematical content. |
Eighth Grade Math:Algebra 1 course includes the study of rational number properties, variables, polynomials, and factoring. Students learn to write, solve, and graph linear and quadratic equations and to solve systems of equations. They also learn to model real-world applications, including statistics and probability investigations.
6th grde Religionprogram is The Sadlier "We Believe" Program. It is drawn from the wisdom of the community. It was developed by nationally recognized experts in catechesis, curriculum, and child development. |
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This is the 2nd edition with a publication date of 5Basic College Mathematics with Early Integers
Basic College Mathematics with Early Integers, 3/E
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Basic Mathematics with Early Integers
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Summary
Worksheets for Classroom or Lab Practice offer extra practice exercises for every section of the text, with ample space for students to show their work. These lab- and classroom-friendly workbooks also list the learning objectives and key vocabulary terms for every text section, along with vocabulary practice problems. |
Summary:Elementary Algebra is a work text coefficient familiar with algebraic expressions, understand the difference between a term and a factor, be familiar with the concept of common factors, know the function of a coefficient polynomial equations familar with polynomials, be able classify polynomials and polynomial equations containing parentheses concept of like terms, be able to combine like terms, be able to simplify expressions containing parentheses polynomial multiply a polynomial by a monomial, be able to simplify +(a + b) and -(a - b), be able to multiply a polynomial by a polynomial numerical evaluations meaning of an equation, be able to perform numerical evaluations provides an exercise supplement objectives of Special proficiency exam - b).[ expand (a + b)^2, (a - b)^2, and (a + b)(a - b).[Collapse Summary] |
In this powerful and dramatic biography Sylvia Nasar vividly re-creates the life of a mathematical genius whose career was cut short by schizophrenia and who, after three decades of devastating mental illness,...
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Word problems are the most difficult part of any math course –- and the most important to both the SATs and other standardized tests. This book teaches proven methods for analyzing and solving any type of... |
This packet contains a copy of the original problem used to create the activity, rationale and explanation behind the "Change the Representation" focal activity, and some thoughts on why this activity... More: lessons, discussions, ratings, reviews,...
This TI Interactive file is best suited for teacher demonstration. This document allows the user to define a function f(x) and guess the derivative of this function. The slider bar's values range f... More: lessons, discussions, ratings, reviews,...
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The SimCalc Project aims to democratize access to the Mathematics of Change for mainstream students by combining advanced simulation technology with innovative curriculum that begins in the early g... More: lessons, discussions, ratings, reviews,...
Simplesim is suited for modelling of non-analytic relations in systems which are causal in the sense that different courses of events interact in a way that is difficult to see and understand. Exam... More: lessons, discussions, ratings, reviews,...
We present two versions of a 3D function grapher--one on a white background, one on a black background. The user enters a formula for f(x,y) in terms of x and y and the applet draws its graph in 3D. T |
201539799 / ISBN-13: 9780201539790
Mathematics for Elementary School Teachers
Future elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be ...Show synopsisFuture elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be called upon to teach. This text uniquely balances "what" they will teach (concepts and content) with "how" to teach (processes and communication). As a result, students using "Mathematics for Elementary School Teachers" leave the course knowing more than basic math skills; they develop a deep understanding of concepts that enables them to effectively teach others. This Fourth Edition features an increased focus on the 'big ideas' of mathematics, as well as the individual skills upon which those ideas are built.Hide synopsis |
Developmental Mathematics-Text - 8th edition
Summary: TheBittinger serieschanged the face of developmental education with the introduction of objective-based worktexts that presented math one concept at a time. This approach allowed readers to understand the rationale behind each concept before practicing the associated skills and then moving on to the next topic. With this revision, Marv Bittinger continues to focus on building success through conceptual understanding, while also supporting readers with quality applications, exercises,...show more and new review and study materials to help students apply and retain their68.23 +$3.99 s/h
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How does reading a good book help you in real life?
How does playing a game of golf help you in real life?
How does listening to Mozart help you in real life ?
How does doing a cross-word puzzle help you in real life ?
How does listening to your mp3 players help you in real life ?
How does solving a tough math problem help you in real life ?
etc.
Actually, it doesn't necessarily depend what your so-called real life is like.
Real life is real life, believe it or not. Oh my god. So confusing. Not.
Sorry to be so blunt.
So back to the question. It could help you solve problems that could actually occur to you, like complex interest equations can help you save money. It's like having an inside look at your life. You can know exactly what to do in certain situations.
Algebra - Here is the problem: A two-digit number is eight times the sum of its ...
Algebra - Here is the problem: A two-digit number is eight times the sum of its ...
Algebra II - We are learning how to solve systems using elimination. 2y - 1 = 3x...
Algebra - 2 (3a - 4) - 2 (4a + 3) simplify 2 (a + 4) = 4 (a - 3)solve the ...
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myTestBook.com Home Page mytestbook.com Math, Science, Reading and Grammar online tests ; Math tests based on "Critical Thinking" and "Problem Solving" Scheduled weekly tests for regular practiceLogin Please · Games · 4th Grade Math Worksheet · 3rd Grade Math WorksheetBooks in the Mathematical Sciences site is intended as a resource for university students in the mathematical sciences. Books are recommended on the basis of readability and other pedagogical value. |
This course explores major modern mathematical developments and helps students to understand and appreciate the unique approach to knowledge employed by mathematics. The course is organized around three major mathematical ideas that have emerged since the time of Sir Isaac Newton. These three ideas may be drawn from: game theory, graph theory, knot theory, logic, mathematics of finance, modern algebra, non-Euclidean geometry, number theory, probability, set theory and the different sizes of infinity, and topology. Students will learn how to solve basic problems in the three areas covered in the course and how to present their solutions concisely, coherently, and rigorously. |
erence Equations: An Introduction with Applications
Difference Equations, An Introduction with Applications, Second Edition, uses elementary analysis and linear algebra to investigate solutions of ...Show synopsisDifference Equations, An Introduction with Applications, Second Edition, uses elementary analysis and linear algebra to investigate solutions of difference equations. Some of the techniques discussed are summation methods, generating functions, z-transforms, theory of linear equations, matrix methods, stability, chaos, asymptotic methods, Green's functions, finite Fourier analysis, variational methods, fixed point theorems, and connections with differential equations. Applications of difference equations to combinatorics, geometry, epidemiology, special functions, economics, population biology, numerical analysis, circuit analysis, differential equations, along with other areas of study. Many examples of the theory are given, and there are a large number of exercises, with difficulty ranging from elementary calculation to investigation of new ideas or 012403330X Brand New International Edition. Cover and ISBN...New. 012403330X Brand New International Edition. Cover and ISBN of the books might be different in some cases. Fast Shipping, Super Quality & Awesome Customer Support is Guaranteed! |
Online Algebra 1 Tutoring and Math Help
Algebra I is viewed as the introductory course to high school mathematics. Instruction focuses on using modern concepts to develop traditional basic principles. Topics include: real number system, algebraic functions and relations and their graphs, solutions of linear and quadratic equations, inequalities, and data analysis.
Free Algebra 1 Videos
Algebra 1 Knowledge Center Articles
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Solving multi-step equations: Increased Complexity
Understanding key Algebraic Terminology - Equations vs Expressions
Understanding Solving Equations
Finding slope given two points
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How to use Algebraic Properties in Basic Proofs - Utilizing Algebraic Properties in Basic Proofs |
Book summary
Linear algebra and matrix theory have long been fundamental tools in mathematical disciplines as well as fertile fields for research. In this book the authors present classical and recent results of matrix analysis that have proved to be important to applied mathematics. Facts about matrices, beyond those found in an elementary linear algebra course, are needed to understand virtually any area of mathematical science, but the necessary material has appeared only sporadically in the literature and in university curricula. As interest in applied mathematics has grown, the need for a text and reference offering a broad selection of topics in matrix theory has become apparent, and this book meets that need. This volume reflects two concurrent views of matrix analysis. First, it encompasses topics in linear algebra that have arisen out of the needs of mathematical analysis. Second, it is an approach to real and complex linear algebraic problems that does not hesitate to use notions from analysis. Both views are reflected in its choice and treatment of topics. [via] |
Conroe Algebra 2Studying smart requires methodical approach to learning. For example in the study of a course like biology, the student learns how to approach the material, highlight key points, and check in the margin items that are not clear. He/she works on the items that require clarification and erases the pencil checks as he/she gains understanding |
So You Really Want to Learn Maths Book 2
A Textbook for Key Stage 3 and Common Entrance
Pupil's Book
Suitable for those beginning their Key Stage 3 studies as well as those working towards entrance examinations at 13+. This book offers explanations of various mathematical methods and plenty of exercises for consolidation.
Designed for use by those in Years 6/7, "So You Really Want to Learn Maths 2" is the perfect resource for those beginning their Key Stage 3 studies as well as those working towards entrance examinations at 13+. The So you really want to learn Maths course is a rigorous, thorough mathematical course for those who really want to learn. Clear explanations are followed by an impressive amount of practice exercises which will ensure that even the fastest mathematicians will never run out of exercises!There are also extension questions which are ideal for gifted and talented pupils as well as those working towards scholarship examinations. Originally designed for those working towards papers 1 and 3 of the current Common Entrance examination, this textbook is ideal for any teacher looking to stretch certain pupils as well as those preparing pupils for Common Entrance at 13+. This course is also ideal for parents and home schoolers looking for a textbook which takes a rigorous approach to mathematics whilst also providing clear explanations of current mathematical methods and plenty of exercises for consolidation. An accompanying Answer Book is also available to purchase separately.
Chapter 1: Working with numbers The four rules Adding and subtracting Harder calculations Addition Subtraction Adding and multiplying Multiplication and division Harder calculations Estimating Multiplication Long multiplication Division Long division Mixed operations Babylonian numbers End of chapter 1 activity: Dice games Chapter 2: Back to Babylon Factors and multiples Prime numbers and prime factors Squares and their roots Cubes and their roots Triangle numbers More clues Inspired guesswork Pythagoras' perfect numbers Using factors to multiply Using factors to find a square roots End of chapter activity: The number game Chapter 3: Fractions Equivalent fractions Equivalent fractions, decimals and percentages Mixed numbers and improper fractions Adding fractions Addition with mixed numbers Subtraction Subtracting with mixed numbers A fraction of an amount Multiplying fractions Dividing with fractions Problems with fractions End of chapter activity: 'Equivalent fraction' dominoes Chapter 4: Probability The probability scale The probability of an event Games of chance Finding the probability of an event NOT happening Sometimes one action can change the situation Probability with two events End of chapter 4 activity: Designing a board game Chapter 5: Handling data Mean and range Looking at data: The median The mode Displaying data: Frequency tables and frequency diagrams Displaying data: Pie charts Interpreting pie charts End of chapter 5 activity: A traffic survey Chapter 6: Working with decimals Decimal arithmetic: Adding and subtracting Multiplying decimals Multiplying decimals by a decimal Dividing decimals Division by decimals Measurement and the metric system Metric units Calculating with quantities End of chapter 6 activity: Imperial units Chapter 7: Algebra 1 - Expressions and formulae The rules of algebra Multiples of x and powers of x More about algebra and index numbers Combining multiplication and division Which number is indexed? More about algebra Negative numbers Multiplying with negative numbers Using negative numbers with a calculator Dividing with negative numbers Substitution Formulae Substituting into formulae End of chapter 7 activity: Maths from stars 1 - Symmetry Chapter 8: Angles and polygons Parallel lines Naming angles Polygons Sum of interior angles of a regular polygon Finding the interior angle of a regular polygon Finding angles in polygons End of chapter 8 activity: Nets of prisms Chapter 9: Percentages Fractions, decimals and percentages Rules of conversion A percentage of an amount Calculating the percentage Profit and loss End of chapter 9 activity: Tormenting tessellations Chapter 10: Ratio and enlargement Ratio Simplifying the ratio Finding the ratio Ration as a fraction Ratio as parts of a whole Solving problems with ratio Scale drawings Using scale Proportional division: The unitary method Ratio and enlargement Enlargement on a grid Drawing the enlargement End of chapter 10 activity: Christmas lunch investigation Chapter 11: Algebra 2 - Equations and brackets Brackets and minus signs Making bundles or factorising Equations Squares and square roots Solving equations with x squared Equations with brackets Fractions and equations Writing story puzzles with brackets End of chapter 11 activity: Dungeons and dragons Chapter 12: Scale drawings and bearings How to use a compass Using bearings Calculating bearings Measuring bearings End of chapter 12 activity: Black-eyed Jack's treasure Chapter 13: Area The parallelogram The triangle Finding the height Some other quadrilaterals The trapezium Area of a kite, a rhombus and other quadrilaterals End of chapter 14 activity: Real life graphs Chapter 15: More about numbers Decimal places: Zooming in on the number line Significant figures Rounding Large and small numbers Decimals as powers of ten Standard index form Standard index form and the scientific calculator More estimating End of chapter 15 activity: Calculator games Chapter 16: Circles
"Those who appreciate Book 1 will find in Book 2 an equally challenging and enjoyable experience. There is more than enough here to meet the needs of the most able pupils." David E. Hanson, Leader of the ISEB 11+ Mathematics setting team, member of the ISEB 13+ Mathematics setting team, Member of the ISEB Editorial Board "These books provide a thorough grounding in maths. Detailed explanations are given through graded exercises, practical work, investigations and puzzles. The material is neatly laid-out, well organised with clear instructions and excellent diagrams, and very informative. The author even brings in fascinating snippets of history throughout the books: how the Egyptians wrote fractions, the history of the penny and of percentages, the origin of imperial units and much more. For instance, did you know that it was the Greeks who established the 'foot' as their basic unit of length and that, according to legend, this Greek unit was based on the measurement of Hercules' foot? I am very impressed by the quality of material, which is very substantial both in content and in weight (these books are heavy!) and would definately recommend them as a complete course." Home Education Advisory Service Bulletin |
Product Description
Students will realize that the power of algebra can help them discover fascinating things about places and people through real-life examples. This series concentrates on the essentials of algebra plus provides a thorough breakdown of difficult concepts using step-by-step explanations and visual examples.Topics Covered: What is Algebra, anyway? Kinds of numbers Order of Operations - PEMDAS Rules of exponents Properties of operations Grade Level: 8 - 12. 26 |
Merchandising Mathematics for Retailing
Merchandising Mathematics for Retailing
Merchandising Mathematics for Retailing
Summary
Written by experienced retailers, MECHANDISING MATH FOR RETAILING, 5/e introduces students to the essential principles and techniques of merchandising mathematics, and explains how to apply them in solving everyday retail merchandising problems. Instructor- and student-friendly, it features clear and concise explanations of key concepts, followed by problems, case studies, spreadsheets, and summary problems using realistic industry figures. Most chapters lend themselves to spreadsheet use, and skeletal spreadsheets are provided to instructors. This edition is extensively updated to reflect current trends, and to discuss careers from the viewpoint of working professionals. It adds 20+ new case studies that encourage students to use analytic skills, and link content to realistic retail challenges. This edition also contains a focused discussion of profitability measures, and an extended discussion of assortment planning. |
Grid Algebra
Read a Review
Grid Algebra is a new visual and kinaesthetic way to learn number concepts and pre-algebra at KS2 and to learn about number and algebra for KS3 and KS4. What is more, it supports more developed algebra ideas as well.
Grid Algebra is a software package developed by Dave Hewitt to support learners as they develop their understanding of early algebra. Learners can be observed as they deal confidently with notation that is far from simplistic. The software enables a powerful visual, and dynamic representation of the ideas that underpin this aspect of mathematics.
• lesson ideas with prepared grids and worksheets
• computer generated tasks, with a wide range of difficulty levels
• an interactive grid to explore and design grids for whole class use or to engage with individual learners.
Screenshots, Worksheets, Ideas
Technical Support
Installing Grid Algebra
To install Grid Algebra successfully at least Windows 2000 Service Pack 4 should be installed. This was issued by Microsoft in 2003 so what follows should only concern users whose machine is over 4 years old. In particular the file 'GDIPlus.dll' is needed. It is used to process many types of pictures with great speed and quality. Note that this file should be on your system already if you are using Windows XP.
If, when you run Grid Algebra you get a message saying that there is no 'GDIPlus.dll' file you will need to download the file.
Grid Algebra may be available as an instant download in the future, but is only available on CD at the moment.
Student Extension Licences
Universities and Colleges who own a Licence for Grid Algebra will be able to purchase extension licences for their students. The extension licences will enable the university to load the software onto students own laptop computers allowing them to use it in schools as part of their training and teaching in the classroom.
Extension licence per student (where University is an ATM member): £15.00
Extension licence per student (where University is NOT an ATM member): £20.00
This will allow the user to use the software on students' own laptops and retain the software once the course has been completed for use in their teaching practice. An individual licence will be issued for each student. |
For equations involving multiple operations, such as 3x+4–2x=8, they erroneously generalized their method and simply undid each operation as they came to it. For example, they would take 8, divide it by 2, add 4, and then subtract 3. (They had to ignore the last operation of multiplication because they had run out of operands.) A preference for the undoing method of equation solving seemed to work against the students when they were later taught the procedure of performing the same operation on both sides of an equation. The students who preferred the undoing method were, in general, unable to make sense of "performing the same operation on both sides." The instruction seemed to have its greatest impact on those students who had an initial preference for the informal method of substitution and who viewed the equation as a balance between left and right sides. This observation suggests that learning to operate on the structure of a linear equation by performing the same operation on both sides may be easier for students who already view equations as entities with symmetric balance and not as statements about a calculation on the left side and the answer on the right.
Despite the considerable body of research on creating meaning for the transformational activities of algebra, few researchers have been able to shed light on the long-term acquisition and retention of transformational fluency. In one study, students were able to produce a meaningful justification for |
Programs and Services
Algebra I
The standards below outline the content for a one-year course in Algebra I. All students are expected to achieve the Algebra I standards. When planning for instruction, consideration will be given to the sequential development of concepts and skills by using concrete materials to assist students in making the transition from the arithmetic to the symbolic. Students should be helped to make connections and build relationships between algebra and arithmetic, geometry, and probability and statistics. Connections also should be made to other subject areas through practical applications. This approach to teaching algebra should help students attach meaning to the abstract concepts of algebra.
These standards require students to use algebra as a tool for representing and solving a variety of practical problems. Tables and graphs will be used to interpret algebraic expressions, equations, and inequalities and to analyze behaviors of functions.
Graphing calculators, computers, and other appropriate technology tools will be used to assist in teaching and learning. Graphing utilities enhance the understanding of functions; they provide a powerful tool for solving and verifying solutions to equations and inequalities.
Throughout the course, students should be encouraged to engage in discourse about mathematics with teachers and other students, use the language and symbols of mathematics in representations and communication, discuss problems and problem solving, and develop confidence in themselves as mathematics students. |
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Elements of Scientific Computing
Publisher:
Springer
Number of Pages:
459
Price:
74.95
ISBN:
9783642112980
This gentle introduction to scientific computing aims to convey the basic ideas, principles and techniques of computational science to undergraduates in mathematics, science and engineering. The required background is a first course in calculus, the basics of linear algebra, and familiarity with elementary programming. The real strength of the text is its adroit mix of analytical and numerical, theoretical and practical. It is imbued throughout with uncommon good sense.
What the book is not: a text in programming, a guide to modeling, a how-to manual for using canned software. What it is: an invitation to students to apply their mathematical skills to computing problems in a variety of contexts. Programming, modeling and the use of commercial software are all addressed, but they don't intrude on the overall theme.
The authors begin with a relatively simple problem: a model that requires computing an integral numerically, using a combination of analytical techniques and variations of the trapezoidal rule. This establishes a pattern we see throughout: set up the model with care; understand what needs to be computed, what technique is appropriate, and what the relative error is likely to be; carry out the computation, determine if the answer makes sense, and interpret the solution in scientific terms.
This introductory piece is followed by two chapters on numerical solutions of ordinary differential equations that emphasize numerical stability and accuracy. Both chapters are based on case studies (rabbit cultivation and predator-prey). They begin with model building, and then go on to selection of numerical methods, computation of solutions and alternative methods. Following this are two chapters that explore methods for solving nonlinear algebraic equations and computing least squares solutions.
The next three chapters are quite distinctive. The first is about making choices about commercially available scientific software (including C, C++, Java, Python, Matlab and others). The authors talk about the characteristics of each language or program, the advantages and disadvantages of each, and then provide and compare code for two different algorithms in several different languages.
Two long chapters on diffusion processes follow. The first one includes: developing models for diffusion, deriving the diffusion equations, describing relevant numerical methods, and talking about how one can verify a computer implementation. The second looks at the diffusion equation again from slightly different perspectives. What are the important characteristics of the solution? Does the solution meet sense physically? How might one solve the diffusion equation analytically?
This is a top-notch book on scientific computing written with clarity, rigor where it matters, and a good sense of what students need to learn. It is among the best books in this area that I have seen.
Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics. |
Discovery Math was designed to meet 21 century learning and meet common core standards. This version was specifically designed for Kindergarten students, but with some curriculum modification it can... More > be applied to 1st grade and special educationThis book probes the universal system and how it works in your life. By simple calculations, logical and rational conclusions can be made concerning your purpose for living and why you experience any... More > and all things from birth to death and beyond. The equations presented here will give a person a sense of security and significance preventing failures and disappointments in life before they happen.
Here in the second of his LifeCode series, Swami Ram Charran answers more of your questions about your own life and how to know ahead of time what you need to do to make your life prosperous and successful. Knowing yourself, your karmas, your past, your present and your future will make you live your life happier and more prosperous<Living on the Palos Verdes Peninsula has afforded us the opportunity to research and study a habitat that truly is within our own backyard. To write and publish this work, the students blended... More > academic classroom curriculum with the use of technology. The children incorporated their computer skills, research abilities, writing, social studies and math skills to produce this dazzling final product. This project is a prime example of differentiated instruction at its best. Not only did this project focus on the scientific side of the curriculum, but it spanned across all subject matters giving the students real world experience and built upon 21st-century skills.< Less
A miracle is defined as a highly improbable or extraordinary accomplishment. The story of the Algebra program at JEB Stuart High School in Fairfax, Virginia, qualifies for such a designation. Over a... More > period of fifteen years, a series of ambitious, no-cost innovations which challenged the prevailing status quo in math education led to a set of academic accomplishments that were indeed improbable and extraordinary. This miracle was achieved by a high-poverty, ethnically diverse student body that was unique at the time but is now representative of schools found throughout the U.S.
For everyone touched by education from parents and students to teachers and administrators, "The Algebra Miracle" will provide insights into the complexity of finding a low-cost formula for academic success in the tight budgetary times of the 21st century. This story serves as a model of what can be accomplished when a dedicated school staff commits its time, energy and creativity to the needs of their students.< Less |
Precalculus
9780136015789
ISBN:
0136015786
Edition: 5 Pub Date: 2008 Publisher: Prentice Hall
Summary: The authors understand what it takes to be successful in mathematics, the skills the students bring to the course, and the way that technology can be used to enhance learning without sacrificing maths skills. They have created a textbook with an overall learning system involving preparation, practice and review.
Sullivan, Michael is the author of Precalculus, published 2008 under ISBN 9780136015789 and 01360...15786. Thirty four Precalculus textbooks are available for sale on ValoreBooks.com, twenty used from the cheapest price of $13.32, or buy new starting at $96 |
Precalculus - Text and helpful features. The result is an easy-to-use, comprehensive text that is the best edition yet321528841 |
Precalculus focuses only on topics that students actually need to succeed in calculus. Because of this, Precalculus is a very manageable size even though it includes a student solutions manual. The book is geared towards courses with intermediate algebra prerequisites and it does not assume that students remember any trigonometry. It covers topics such as inverse functions, logarithms, half-life and exponential growth, area, e, the exponential function, the natural logarithm and trigonometry. The Student Solutions Manual is integrated at the end of every section. The proximity of the solutions encourages students to go back and read the main text as they are working through the problems and exercises. The inclusion of the manual also saves students money. |
Re: Learning Graph Theory
I learned graph theory from "Graph Theory and Its Applications" by Yellen and Gross. There is some coding info in it and I found the book very readable. Perhaps your local university library has this one or similar.
Last edited by pellerinb (2012-12-27 04:51:36)
Prime numbers have got to be the neatest things; they are like atoms. Composites are two or more primes held together by multiplication. In biology, we use math like we know what we are talking about. Sad isn't it.
Re: Learning Graph Theory
pellerinb wrote:
It's not really the kind of thing I was looking for, though if it's readable and targets CS students it might be worth a look.
bobbym wrote:
let me know how it is...
This is exactly the kind of thing I'm looking for, though I don't think the page has been updated in 4 odd years - which is a pity.
I've had someone recommend Reinhard Diestel's "Graph Theory." The first chapter seems to cover most of what I'm interested in. I could probably read some of the later chapters to get an overview of the other stuff. |
UltimaCalc 2.1.255
Description of UltimaCalc 2.1.255: UltimaCalc is a scientific and mathematical calculator designed to occupy minimum screen area, making it immediately available for use. UltimaCalc can stay on top of other windows. Type a calculation as plain text, evaluate it, maybe edit it and re-calculate. Has a comprehensive context-sensitive help system.
CalculatesFunctions available include logarithms to base 2, exp, two-argument inverse tangent; cube root (even of negative numbers); factorials, combinations, permutations, powers, modulus, GCD. Also floor and ceiling functions, absolute value, min, max, extract the fractional part of a number. Calculate the slope of a line given its end points. Even calculate definite integrals.
Define your own functions and constants, saved as a plain text file.
Find the date of Easter. Calculate future or past dates. Julian day numbers.
Calculate the mean, median and standard deviation of a sample and its population.
Create bar, line, pie charts. Add title, subtitle, labels. Adjust the layout, choose colours and hatching, save as an image.
Regression: Plot a scatter chart and regression line. Fit a polynomial to data. Analyse the effects of multiple variables on data. Absolute deviation fit minimises the effects of outlying values.
Plot functions: Specify starting and ending conditions and how variables change. Choose axis locations. Combine multiple plots. Save as an image.
Solve triangles: Given one side and two other facts, calculate the unknowns. View the result graphically.
Solve Simultaneous Linear Equations, and do Navigational calculations.
Log calculations to a text file, or copy and paste into other programs. Save specialised calculations with notes in data files.
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MOS-OXP Practice Exam Test Questions
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Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). |
'This listing is divided into two key groups – tools intended specifically for educators, and general applications intended...
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'Over 100 math formulas at high school level. The covered areas include algebra, geometry, calculus, trigonometry,...
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'Over 100 math formulas at high school level. The covered areas include algebra, geometry, calculus, trigonometry, probability and statistics. Most of the formulas come with examples for better understanding. Use the powerful search function to find what you are looking for and mark your favorites for easier access. A convenient tool for students and teachers and a handy reference for anyone interested in math!'This app costs $0.99
This is a collection of resources by a librarian. Therea re over 50 links to resources, many of which have further links. ...
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This is a collection of resources by a librarian. Therea re over 50 links to resources, many of which have further links. They are all designed to be useful to "middle and high school teachers, students, and parents.״.'This is a free app. Welcome to the Louvre!'This is a free app |
vanessa v comment: whats the easiest way to learn calculus when you're 10 years old?
Krista K comment: make sure first that you're really good with algebra. from there, google "calculus and limits".... and let me know if you have trouble! :)
Allan P comment: Thanks for the refresher. I think this course will be very useful. One Love!
joni d comment: It's been 40 years since I took calculus. This is on my bucket list!
bailey m comment: when I was in the sixth grade i didn't turn my homework in so they stuck me ion regular math for another year so in the eighth grade i took pre-algebra now I'm getting ready to go into the ninth grade and i really want to get a good grade in algebra. How would I go about doing that? do you have any suggestions?
Christopher T comment:
"jacinta k commented:
If i invest K50 a year for 40 years toward my POSF savings, and earn 8% a year on my investments, how much will i have when i retire?"
Answer: 1,086,226.075
Students in this lesson (483)
About the Teacher
Whether you're starting your first semester of calculus, or cramming for your final exam, integralCALC will help you quickly build skills with simple, step-by-step lessons.
Krista King (founder of integralCALC) worked as a calculus tutor while she was a student at the University of Notre Dame. After graduation, she started integralCALC.com to serve as a resource for calculus students. Since the beginning of 2010, Krista's calculus lessons have helped students around the world and she continues to publish lessons regularly.
17 Lessons
59
190
15K
1.2K
Table of Contents
1.
Lesson Intro
0:23
2.
Functions vs. Equations
0:23
3.
What are Functions?
1:06
4.
Domain and Range
1:32
5.
What Functions are Not
1:02
6.
Combinations and Compositions
0:54
Lesson Description
Lost in the world of calculus? Start at the beginning with integralCALC's online tutorials. The first in a series, this lesson introduces the idea of functions - number 'machines' that describe a relationship between two or more variables (like 'x' and 'y'). Learn how functions differ from equations, what it means to find a function's domain and range, and how to use the vertical line test to check a function's validity. |
of Change in Temperature
Huttner of Burlington County College designed this activity to help students learn and practice their calculus skills. Students will collect data on the monthly temperature of a city. They will then graph and analyze that data using their knowledge of differentiation of trigonometric functions and critical thinking skills. The procedure for this activity is clearly outlined. Detailed directions for students are located in the "Content Materials" section. A short list of weather websites that provide temperature information is included in the "Supplementary Resources" section. This activity is a great opportunity to incorporate real data and concrete examples into math education.Mon, 16 Nov 2009 17:03:23 -0600Graph 4.3
is "an open source application used to draw mathematical graphs in a coordinate system." Anyone who wants to draw graphs of functions will find this program useful. The program makes it very easy to visualize a function and paste it into another program. It is also possible to do some mathematical calculations on the functions. Visitors will find all sorts of helpful information about the software, its uses, history, and features following the links on the left side of the page.Tue, 16 Jun 2009 12:46:19 -0500Geometric Construction with the Compass Alone
and exercises about geometric constructions: almost everything you can do with a ruler and a compass you can do with the compass alone.Fri, 28 Dec 2007 03:00:01 -0600The Four Colour Theorem
essay describing work on the theorem from its posing in 1852 through its solution in 1976, with two other web sites and 9 references (books/articles).Thu, 20 Dec 2007 03:00:02 |
...
More About
This Book
measurement, data collection and storage, computation, and communication of information." Black Hole Math is designed to be used as a supplement for teaching mathematical topics. The problems can be used to enhance understanding of the mathematical concept, or as a good assessment of student mastery. This collection of activities is based on a weekly series of space science problems distributed to thousands of teachers during the 2004-2008 school years. They were intended as supplementary problems for students looking for additional challenges in the math and physical science curriculum in grades 10 through 12. The problems are designed to be 'one-pagers' consisting of a Student Page, and Teacher's Answer Key. This compact form was deemed very popular by participating teachers. The topic for this collection is Black Holes, which is a very popular, and mysterious subject among students hearing about astronomy. Students have endless questions about these exciting and exotic objects as many of you may realize! Amazingly enough, many aspects of black holes can be understood by using simple algebra and pre-algebra mathematical skills. This booklet fills the gap by presenting black hole concepts in their simplest mathematical form. General Approach: The activities are organized according to progressive difficulty in mathematics. Students need to be familiar with scientific notation, and it is assumed that they can perform simple algebraic computations involving exponentiation, square-roots, and have some facility with calculators. The assumed level is that of Grade 10-12 Algebra II, although some problems can be worked by Algebra I students. Some of the issues of energy, force, space and time may be appropriate for students taking high school Physics |
Contents
GeoGebra is dynamic geometry software. Constructions can be made with points, vectors, segments, lines, polygons, conic sections, inequalities, implicit polynomials and functions. All of them can be changed dynamically afterwards. Elements can be entered and modified directly on screen, or through the Input Bar. GeoGebra has the ability to use variables for numbers, vectors and points, find derivatives and integrals of functions and has a full complement of commands like Root or Extremum. Teachers and students can use GeoGebra to make conjectures and to understand how to prove geometric theorems.
GeoGebra can also create code that can be used inside LaTeX files in order to create that same images that GeoGebra generates, through the PSTricks package, the PGF/TikZ packages, or Asymptote code.
GeoGebraTube is the official repository website of GeoGebra products and GeoGebra related free resources.[7] It started working in June 2011, and it contains about 60 000 (in January 2014) materials, like interactive presentation, games, and lessons made by GeoGebra.[8]
GeoGebra depends on software licensed under the GNU General Public License (GPL), the LGPL, the Apache license and others.[9] The software is licensed under the "GeoGebra Non-Commerical License Agreement",[10] which asserts that while the "source code is licensed […] under the terms of the GNU General Public License", the translation files, installers, and web services are licensed under non-GPL-compatible terms. Commercial use is prohibited without the purchase of a separate license.[10] It has been noted by the maintainer of the software in Debian GNU/Linux that these terms, a change from earlier releases, prevent the resulting combined work from being considered free software and may be nondistributable.[11]
The GeoGebra Institutes (IGI) are more than 120 (in 2013 March) non-profit organizations around the world. GeoGebra Institutes join teachers, students, software developers and researchers to support, develop, translate and organise the Geogebra related tasks and projects. Local GeoGebra Institutes are groups at schools and universities who support students and teachers in their region. As part of the International GeoGebra Institute network they share free educational materials, organize workshops, and work on projects related to GeoGebra. GeoGebra Institute may certify local GeoGebra users, experts, and trainers according Guidelines. |
Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in one real variable. Organized into the topics of sets and relations, infinity and induction, sequences of numbers,...
A discussion of the history of conic sections, one of the oldest math subjects studied systematically and thoroughly, with a description, formulas, properties, a proof, Mathematica notebooks, the ellipse seen as a...
What is the difference between the arithmetic 3+5 = 5+3 and the algebraic a+b = b+c? One is a specific fact, another is a pattern valid in a multitude of situations. While arithmetic may hint at some regularities,...
Specify a type of combinatorial object, together with specific parameter values, and COS will return to you a list of 200 such objects. Generation pages: permutations, combinations, various types of trees, unlabelled... |
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