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

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114 CHAPTER 8. INTEGRALS AND QUADRATURE
Chapter 9 Least Squares 9.1 Least Squares Least squares is a method of fitting observed data to a theoretical model function. In the general setting we are given a set of data x x 0 x 1 . . . x n y y 0 y 1 . . . y n and some class of functions, F. The goal then is to find the “best” f ∈ F to fit to the data. Usually the class of functions F will be determined by some small number of parameters; the number of parameters will be smaller (usually much smaller) than the number of data points. The theory here will be concerned with defining “best,” and examining methods for finding the “best” function to fit the data. 9.1.1 The Definition of Least Squares First we consider the data to be two vectors of length n + 1. That is we let x = [x 0 x 1 . . . x n ] ⊤ , and y = [y 0 y 1 . . . y n ] ⊤ . The error, or residual, of a given function with respect to this data is the vector r = y − f(x). That is r = [r 0 r 1 . . . r n ] ⊤ , where r i = y i − f(x i ). Our goal is to find f ∈ F such that r is reasonably small. We measure the size of a vector by the use of norms, which were explored in Subsection 1.3.1. The most useful norms are the l p norms. For a given p with 0 < p < ∞, the l p norm of r is defined as ( n∑ ) 1/p ‖r‖ p = r p i i=0 For the purposes of approximation, we the easiest norm to use is the l 2 norm: Definition 9.1.1 (Least Squares Best Approximant). The least-squares best approximant to a set of data, x, y from a class of functions, F, is the function f ∗ ∈ F that minimizes the l 2 norm of the error. That is, if f ∗ is the least squares best approximant, then ‖y − f ∗ (x)‖ 2 = min f∈F ‖y − f(x)‖ 2 We will generally assume uniqueness of the minimum. 115
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Numerical Methods Course Notes Vers
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Contents Acknowledgments i 1 Introd
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CONTENTS v 8 Integrals and Quadratu
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Chapter 1 Introduction 1.1 Taylor
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1.1. TAYLOR’S THEOREM 3 with the
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1.3. EIGENVALUES 5 To correct this
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1.3. EIGENVALUES 7 Example Problem
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1.3. EIGENVALUES 9 for all vectors
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Chapter 2 A “Crash” Course in o
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2.1. GETTING STARTED 13 77 22 333 o
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2.2. USEFUL COMMANDS 15 octave:22>
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2.3. PROGRAMMING AND CONTROL 17 x1
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2.4. PLOTTING 19 %just plot Y plot(
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2.4. PLOTTING 21 Exercises (2.1) Wh
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Chapter 3 Solving Linear Systems A
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3.1. GAUSSIAN ELIMINATION WITH NAÏ
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3.2. PIVOTING STRATEGIES FOR GAUSSI
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3.2. PIVOTING STRATEGIES FOR GAUSSI
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3.3. LU FACTORIZATION 31 appropriat
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3.3. LU FACTORIZATION 33 We pivot o
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3.4. ITERATIVE SOLUTIONS 35 3.3.3 S
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3.4. ITERATIVE SOLUTIONS 37 3.4.2 D
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3.4. ITERATIVE SOLUTIONS 39 We star
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3.4. ITERATIVE SOLUTIONS 41 Now not
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3.4. ITERATIVE SOLUTIONS 43 Remembe
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3.4. ITERATIVE SOLUTIONS 45 is the
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Chapter 4 Finding Roots 4.1 Bisecti
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4.2. NEWTON’S METHOD 49 Algorithm
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4.2. NEWTON’S METHOD 51 4.2.2 Pro
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4.3. SECANT METHOD 53 The following
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4.3. SECANT METHOD 55 Since the roo
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4.3. SECANT METHOD 57 relation betw
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4.3. SECANT METHOD 59 (b) f(x) = x
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Chapter 5 Interpolation 5.1 Polynom
Page 71 and 72: 5.1. POLYNOMIAL INTERPOLATION 63 2
Page 73 and 74: 5.1. POLYNOMIAL INTERPOLATION 65 Su
Page 75 and 76: 5.2. ERRORS IN POLYNOMIAL INTERPOLA
Page 77 and 78: frag replacements 5.2. ERRORS IN PO
Page 85 and 86: Chapter 6 Spline Interpolation Spli
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Page 89 and 90: 6.2. (NATURAL) CUBIC SPLINES 81 pro
Page 91 and 92: 6.3. B SPLINES 83 The zero degree B
Page 93 and 94: 6.3. B SPLINES 85 Exercises (6.1) I
Page 95 and 96: Chapter 7 Approximating Derivatives
Page 97 and 98: 7.2. RICHARDSON EXTRAPOLATION 89 7.
Page 99 and 100: 7.2. RICHARDSON EXTRAPOLATION 91 7.
Page 101 and 102: 7.2. RICHARDSON EXTRAPOLATION 93 Ex
Page 103 and 104: Chapter 8 Integrals and Quadrature
Page 105 and 106: 8.1. THE DEFINITE INTEGRAL 97 Examp
Page 107 and 108: 8.2. TRAPEZOIDAL RULE 99 The compos
Page 109 and 110: frag replacements 8.2. TRAPEZOIDAL
Page 111 and 112: 8.3. ROMBERG ALGORITHM 103 then def
Page 113 and 114: 8.4. GAUSSIAN QUADRATURE 105 8.4 Ga
Page 115 and 116: 8.4. GAUSSIAN QUADRATURE 107 Exampl
Page 117 and 118: 8.4. GAUSSIAN QUADRATURE 109 Simple
Page 119 and 120: 8.4. GAUSSIAN QUADRATURE 111 Exerci
Page 121: 8.4. GAUSSIAN QUADRATURE 113 functi
Page 125 and 126: 9.1. LEAST SQUARES 117 The answer i
Page 127 and 128: 9.2. ORTHONORMAL BASES 119 g 0 (x 0
Page 129 and 130: frag replacements 9.2. ORTHONORMAL
Page 131 and 132: Chapter 10 Ordinary Differential Eq
Page 133 and 134: 10.1. ELEMENTARY METHODS 125 value
Page 135 and 136: frag replacements 10.1. ELEMENTARY
Page 137 and 138: 10.1. ELEMENTARY METHODS 129 It tur
Page 139 and 140: 10.2. RUNGE-KUTTA METHODS 131 Examp
Page 141 and 142: 10.2. RUNGE-KUTTA METHODS 133 We no
Page 143 and 144: 10.3. SYSTEMS OF ODES 135 Example 1
Page 145 and 146: 10.3. SYSTEMS OF ODES 137 Euler’s
Page 147 and 148: 10.3. SYSTEMS OF ODES 139 have to b
Page 149 and 150: 10.3. SYSTEMS OF ODES 141 4.5 4 3.5
Page 151 and 152: 10.3. SYSTEMS OF ODES 143 (10.9) Im
Page 153 and 154: Appendix A Old Exams A.1 First Midt
Page 155 and 156: A.3. FINAL EXAM, FALL 2003 147 •
Page 157 and 158: A.4. FIRST MIDTERM, FALL 2004 149
Page 159 and 160: A.5. SECOND MIDTERM, FALL 2004 151
Page 161 and 162: A.6. FINAL EXAM, FALL 2004 153 P5 (
Page 163 and 164: Appendix B GNU Free Documentation L
Page 165 and 166: 157 F. Include, immediately after t
Page 167 and 168: Bibliography [1] Guillaume Bal. Lec
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