Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

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10 CHAPTER 1. INTRODUCTION
Chapter 2 A “Crash” <strong>Course</strong> in octave/Matlab 2.1 Getting Started Matlab is a software package that allows you to program the mathematics of an algorithm without getting too bogged down in the details of data structures, pointers, and reinventing the wheel. It also includes graphing capabilities, and there are numerous packages available for all kinds of functions, enabling relatively high-level programming. Unfortunately, it also costs quite a bit of money, which is why I recommend the free Matlab clone, octave, available under the GPL 1 , freely downloadable from http://www.octave.org. In a lame attempt to retain market share, <strong>Math</strong>works continues to tinker with Matlab to make it noncompatible with octave; this has the side effect of obsoletizing old Matlab code. I will try to focus on the intersection of the two systems, except where explicitly noted otherwise. What follows, then, is an introduction to octave; Matlab users will have to make some changes. You can find a number of octave/Matlab tutorials for free on the web; all of them are certainly better than this one. A brief web search reveals the following excellent tutorials: • http://www.math.mtu.edu/~msgocken/intro/intro.html • http://www.cyclismo.org/tutorial/matlab/vector.html • http://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/usna.html Matlab has some demo programs covering a number of topics– from the most basic functionality to the more arcane toolboxes. In Matlab, simply type demo. What follows is a lame demo for octave. Start up octave. You should get something like: GNU Octave, version 2.1.44 (i686-pc-linux-gnu). Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003 John W. Eaton. This is free software; see the source code for copying conditions. There is ABSOLUTELY NO WARRANTY; not even for MERCHANTIBILITY or FITNESS FOR A PARTICULAR PURPOSE. For details, type ‘warranty’. Please contribute if you find this software useful. For more information, visit http://www.octave.org/help-wanted.html Report bugs to . octave:1> 1 Gnu Public License. See http://www.gnu.org/copyleft/gpl.html. 11
Page 1: Numerical Methods Course Notes Vers
Page 5 and 6: Contents Acknowledgments i 1 Introd
Page 7 and 8: CONTENTS v 8 Integrals and Quadratu
Page 9 and 10: Chapter 1 Introduction 1.1 Taylor
Page 11 and 12: 1.1. TAYLOR’S THEOREM 3 with the
Page 13 and 14: 1.3. EIGENVALUES 5 To correct this
Page 15 and 16: 1.3. EIGENVALUES 7 Example Problem
Page 17: 1.3. EIGENVALUES 9 for all vectors
Page 21 and 22: 2.1. GETTING STARTED 13 77 22 333 o
Page 23 and 24: 2.2. USEFUL COMMANDS 15 octave:22>
Page 25 and 26: 2.3. PROGRAMMING AND CONTROL 17 x1
Page 27 and 28: 2.4. PLOTTING 19 %just plot Y plot(
Page 29 and 30: 2.4. PLOTTING 21 Exercises (2.1) Wh
Page 31 and 32: Chapter 3 Solving Linear Systems A
Page 33 and 34: 3.1. GAUSSIAN ELIMINATION WITH NAÏ
Page 35 and 36: 3.2. PIVOTING STRATEGIES FOR GAUSSI
Page 37 and 38: 3.2. PIVOTING STRATEGIES FOR GAUSSI
Page 39 and 40: 3.3. LU FACTORIZATION 31 appropriat
Page 41 and 42: 3.3. LU FACTORIZATION 33 We pivot o
Page 43 and 44: 3.4. ITERATIVE SOLUTIONS 35 3.3.3 S
Page 45 and 46: 3.4. ITERATIVE SOLUTIONS 37 3.4.2 D
Page 47 and 48: 3.4. ITERATIVE SOLUTIONS 39 We star
Page 49 and 50: 3.4. ITERATIVE SOLUTIONS 41 Now not
Page 51 and 52: 3.4. ITERATIVE SOLUTIONS 43 Remembe
Page 53 and 54: 3.4. ITERATIVE SOLUTIONS 45 is the
Page 55 and 56: Chapter 4 Finding Roots 4.1 Bisecti
Page 57 and 58: 4.2. NEWTON’S METHOD 49 Algorithm
Page 59 and 60: 4.2. NEWTON’S METHOD 51 4.2.2 Pro
Page 61 and 62: 4.3. SECANT METHOD 53 The following
Page 63 and 64: 4.3. SECANT METHOD 55 Since the roo
Page 65 and 66: 4.3. SECANT METHOD 57 relation betw
Page 67 and 68: 4.3. SECANT METHOD 59 (b) f(x) = x
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Chapter 5 Interpolation 5.1 Polynom
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5.1. POLYNOMIAL INTERPOLATION 63 2
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5.1. POLYNOMIAL INTERPOLATION 65 Su
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5.2. ERRORS IN POLYNOMIAL INTERPOLA
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frag replacements 5.2. ERRORS IN PO
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Chapter 6 Spline Interpolation Spli
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frag replacements 6.1. FIRST AND SE
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6.2. (NATURAL) CUBIC SPLINES 81 pro
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6.3. B SPLINES 83 The zero degree B
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6.3. B SPLINES 85 Exercises (6.1) I
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Chapter 7 Approximating Derivatives
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7.2. RICHARDSON EXTRAPOLATION 89 7.
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7.2. RICHARDSON EXTRAPOLATION 91 7.
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7.2. RICHARDSON EXTRAPOLATION 93 Ex
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Chapter 8 Integrals and Quadrature
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8.1. THE DEFINITE INTEGRAL 97 Examp
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8.2. TRAPEZOIDAL RULE 99 The compos
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frag replacements 8.2. TRAPEZOIDAL
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8.3. ROMBERG ALGORITHM 103 then def
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8.4. GAUSSIAN QUADRATURE 105 8.4 Ga
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8.4. GAUSSIAN QUADRATURE 107 Exampl
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8.4. GAUSSIAN QUADRATURE 109 Simple
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8.4. GAUSSIAN QUADRATURE 111 Exerci
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8.4. GAUSSIAN QUADRATURE 113 functi
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Chapter 9 Least Squares 9.1 Least S
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9.1. LEAST SQUARES 117 The answer i
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9.2. ORTHONORMAL BASES 119 g 0 (x 0
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frag replacements 9.2. ORTHONORMAL
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Chapter 10 Ordinary Differential Eq
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10.1. ELEMENTARY METHODS 125 value
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frag replacements 10.1. ELEMENTARY
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10.1. ELEMENTARY METHODS 129 It tur
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10.2. RUNGE-KUTTA METHODS 131 Examp
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10.2. RUNGE-KUTTA METHODS 133 We no
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10.3. SYSTEMS OF ODES 135 Example 1
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10.3. SYSTEMS OF ODES 137 Euler’s
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10.3. SYSTEMS OF ODES 139 have to b
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10.3. SYSTEMS OF ODES 141 4.5 4 3.5
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10.3. SYSTEMS OF ODES 143 (10.9) Im
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Appendix A Old Exams A.1 First Midt
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A.3. FINAL EXAM, FALL 2003 147 •
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A.4. FIRST MIDTERM, FALL 2004 149
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A.5. SECOND MIDTERM, FALL 2004 151
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A.6. FINAL EXAM, FALL 2004 153 P5 (
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Appendix B GNU Free Documentation L
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157 F. Include, immediately after t
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Bibliography [1] Guillaume Bal. Lec
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