Visual Mathematics, lllustrated by the Tl-92 and the Tl-89

Transcription

1 Visual Mathematics, lllustrated by the Tl-92 and the Tl-89

2 Springer-V erlag France S.A.R.L

3 George C. DORNER, Jean Michel FERRARD, Henri LEMBERG Visual Mathematics, Illustrated by the TI-92 and the TI-89 Springer

4 Pr. George C. DORNER William Rainey Rarper College Palatine, Illinois, USA Dr. Jean Michel FERRARD Professem en classe PC Lycee Jean Perrin Lyon, France Dr. Henri LEMBERG Professem en classe PC College Stanislas Paris, France Springer-Verlag France 2000 ISBN ISBN (ebook) DOI / Apart from any fair dealing for the purposes of the research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or, in the case of reprographic reproduction, in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiry conceming reproduction outside those terms should be sent to the publishers The use of registered names, trademark etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and are, therefore, free for general use. Product liability : the publisher can give no guarantee for information about drug dosage and application thereof contained in this book. In every individual case the respective user must check its accuracy by consulting other pharmaceuticalliterature. SPIN:

5 Preface This book is unlike other mathematics books. Same basic old topics and themes and some quite advanced, new concepts and themes of mathematics are covered. There is a constant concern for rigor and precision in the presentation of the material. A mathematics student will see here the integration of many themes which run throughout the history of mathematics and which are still the subject of intensive research. The emphasis is on concrete results which may see application by applied mathematicians, computer scientists, engineers, and other scientists. The topics covered are in classical analysis ( dynamic systems, Fourier series, differential equations, function interpolation, etc.) and in linear algebra ( orthogonality, eigenvalues and eigenvectors ). The breadth of topics is difficult to describe simply. Perhaps "introductory applied functional analysis" would come close. An unstated subtheme is how to evaluate or Iook at a useful function. Classical topics of solution of equations, representation, approximation, and polynomial, rational function, and trigonometric interpolation are treated. These subjects are revisited in several settings. Bezier curves, splines, wavelets and other topics of interest in computer science applications are included in this comprehensive introduction. This is classical, "post-calculus" mathematics which is unified and integrated with the tools and ideas of linear algebra. Because of the breadth of material covered and the diversity of the intended audience it may be incumbent on the reader to dig out an idea or a term which is not detailed here. For example, we did not cite some of the farnaus theorems of calculus, such as Rolle's Theorem or the Intermediate Value Theorem, even though they are frequently used. Similarly, a property of compact sets may be called upon or the idea of a basis may be used even though the definitions are not given in this book. On occasion there is a forward-reference in the book which may usually be skipped on first reading. Certainly, the history of this mathematics is not "linear" and probably the same may be said about the learning. We attempted to present not a treatise but a useful introduction to many topics, illustrated by concrete examples. So far, this preface reads like many others. The mathematics is classical, rigorous, and comprehensive. Then, why is it not like the others? Most of the concepts which are developed are developed and illustrated by formal calculations using the TI-89 and TI-92 graphic and symbolic calculators from Texas Instruments. These calculators readily provide graphic displays and formal computations of a computer algebra system (CAS) which belie their portability and low cost. Little is sacrificed and much is gained by use of these machin es.

6 2 Visual Mathematics Numerous programs, all very brief, are used to give examples, to illustrate significant points, and even to point toward extensions of the theory. This is what makes the breadth of material covered more accessible and what renders it more concrete. The reader may "see" and almost "touch" the mathematical themes studied which, without this clarification, may remain abstract theoretical constructions. While no Iist of problems or exercises accompany the text, the reader with calculator in band may explore, experiment, and play with new ideas until they are comprehended. We are in fact persuaded that from now on it is no Ionger possible to learn and to understand mathematics as one did not so long ago, before the daily and intensive use of computers. Today's formal calculators are true "pocket computers" dedicated to mathematics which permit both the professor and the student to renew their approach to understanding this science. For the former, it renders illustrations and demonstrations more accessible. For the latter, it provides immediate visual experiences and Ieads to better understanding. We want you to take part in our experiences. We are convinced of the benefits, and we definitely believe that using the calculator in this way yields a "less is more" result. This work is thus really a book "unlike the others": it is a book of concrete mathematics. Finally, the book differs not only in its conception and execution but also in its production. The book first appeared in France as: Mathematiques concrete, illustrees par Ia TI-92 et Ia TI-89 J.M. Ferrard - H. Lernberg Springer-Verlag France 1998 ISBN X This English version appears as the result of a trans-oceanic collaboration via the internet between mathematics professors with similar interests and almost identical philosophies about the use of technology in teaching and learning of mathematics. The authors became collaborators and friends through this technology as a result of their common interests in mathematics, technology, and pedagogy. We accept responsibility for any errors which may appear in this work, but, as another innovation, we attribute them to lost bits and bytes which Iinger somewhere over the Atlantic.

7 Contents 1. Discrete Dynamical Systems 1. Dynamical systems in IR The logistic model of Verhulst The case of convergence Cycles The bifurcation diagram The Feigenbaum constant Study of cycles 2. Newton's method in IR 3. Equivalence between different systems 4. Dynamical systems of the plane Julia sets The Mandelbrot set Revisiting Newton's method The Henon attractor 2. Differential Equations 1. Definition of the problern 2. Linear equations of first order Non normalized equations 3. Non linear first order equations Examples of the Cauchy-Lipschitz theorem 4. Systems of differential equations of first order Linear differential systems of first order Differential systems with constant coefficients 5. Linear differential equations of order n Linear equations of order 2 with constant coefficients Linear equations of order n with constant coefficients 6. Autonomaus systems of the plane Linear systems Non linear systems 7. Numeric solutions Euler's method Runge-Kutta method 8. The Laplace transformation

8 3. Fourier analysis 1. Fourier series Convergence of Fourier series The Gibbs phenomenon Cesara summability 2. Aceeieration of convergence of Fourier series 3. Hilbert analysis 4. Discrete Fourier Transform 5. Fast Fourier Transform Principle of the FFT Programming the FFT Applications of the FFT 6. An introduction to wavelets Gabor windows The Marlet wavelets The multi-resolution analysis Interpolation and approximation 1. Interpolating polynomials The Lagrange form of the interpolating polynomial The Vandermonde form Newton's interpolating polynomial Neville's algorithm 2. The Runge phenomenon 3. Interpolating by equally spaces x-values Calculation of the interpolation error 4. Hermite interpolation From Lagrange to Hermite Chebyshev to the rescue of Hermite Return to divided differences Polynomial interpolation "a Ia carte" 5. Bezier curves The Casteljau algorithm 6. Spline functions A first example Definitions and forst properties for splines Interpolation using spline functions Convergence Algorithm for calculation of cubic splines

9 7. Interpolating by rational fractions 204 A "Vandermonde style" method 205 Reciprocal divided differences 209 Use of reciprocal divided differences Trigonometrie interpolation 213 Return to polynomial interpolation 214 Using undertermined coefficients 216 equidistants abscissas Orthogonality PreHilbert or Inner Product Spaces 221 Inner product 221 Classical examples 222 Orthonormal families 224 Orthogonal complement of a subspace 226 Gram-Schmidt orthogonalization Problems of least squares 230 Distance to a subspace 231 "Continuous" or "discrete" least squares: a comparison 233 "Discrete" least squares. A generalization Orthogonal polynomials 244 First properties 244 Chebyshev polynomials 248 Chebyshev polynomials and discrete least squares 254 Legendre polynomials 259 Laguerre and Hermite polynomials Gaussian quadrature 267 Introduction to the method 267 The use of orthogonal polynomials 268 Precision of the method 269 The classic cases of gaussian quadrature Orthogonal operators 274 General information about orthogonal operators 275 Isometries of the plane 278 Isometries of space 279 Isometries in dimension n 286 Unitary matrices QR factorization 288 Use of the Gram-Schmidt algorithm 290 Toward other methods 291 Givens' method 292 Householder's method 295 Questions of precision 300

10 6. Eigenvalues and eigenvectors 1. Review of theory First definitions Polynomials of operators Polynomials of matrices Polynomials annihilators The characteristic polynomial 2. Reduction of operators Eigenvalues and multiplicities Finding eigenspaces Diagonalization Triangularization Jordan matrices Jordan reduction Characteristic subspaces The Dunford decomposition and applications 3. Localization of eigenvalues The instructions eigvl and eigvc Matrix norms and the spectral radius Gershgorin disks 4. Power methods of finding eigenvalues Direct iterations (favorable case) Direct iterations ( unfavorable case) Iterated inverse powers Iterations to an arbitrary eigenvalue Improvement of the method 5. Other iterative methods Use of the LU decomposition Use of the QR decomposition The Jacobi method for real symetric matrices 6. Symetric operators Review of theory Symetric or Hermitian matrices Diagonalization in an orthorrormal basis Orthogonal polynomials: a return visit Tridiagonalization of symetric matrices 7. Positive symetric matrices Preliminary theory Square root of a positive symetric matrix Polar decomposition The Cholevsky decomposition

11 Calculator guide 421 Expansion, factorization 421 Expansion 421 Factorization 423 Equations and systems 424 The equation f(x) = The numerical solver 426 Systems of equations 427 Differential equations 428 Differentiation and integration 430 Symbolic differentiation 430 N umeric differentiation 431 Taylor formula 431 Integration 432 Matrices and lists 433 Matrices 433 Lists 437 Sums, products. Iimits 439 Expressions, sub-expressions 440 Bibliography 441 List of the programs 443 Symbols used in the book 447 Index 449