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

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$Fractional and operational calculus with generalized fractional ...$

iv CONTENTS 3.4.6 Gauss Seidel Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.7 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.8 A Free Lunch? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4 Finding Roots 47 4.1 Bisection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1.1 Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2.4 Using Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Secant Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5 Interpolation 61 5.1 Polynomial Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1.1 Lagranges Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.1.3 Newton’s Nested Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.1.4 Divided Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2 Errors in Polynomial Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2.1 Interpolation Error Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2.2 Interpolation Error for Equally Spaced Nodes . . . . . . . . . . . . . . . . . . 71 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6 Spline Interpolation 77 6.1 First and Second Degree Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.1.1 First Degree Spline Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.1.2 Second Degree Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.1.3 Computing Second Degree Splines . . . . . . . . . . . . . . . . . . . . . . . . 80 6.2 (Natural) Cubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.2.1 Why Natural Cubic Splines? . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.2.2 Computing Cubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.3 B Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7 Approximating Derivatives 87 7.1 Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.1.1 Approximating the Second Derivative . . . . . . . . . . . . . . . . . . . . . . 89 7.2 Richardson Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.2.1 Abstracting Richardson’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.2.2 Using Richardson Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . 91 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
CONTENTS v 8 Integrals and Quadrature 95 8.1 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.1.1 Upper and Lower Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.1.2 Approximating the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.1.3 Simple and Composite Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.2 Trapezoidal Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.2.1 How Good is the Composite Trapezoidal Rule? . . . . . . . . . . . . . . . . . 99 8.2.2 Using the Error Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.3 Romberg Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.3.1 Recursive Trapezoidal Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 8.4 Gaussian Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.4.1 Determining Weights (Lagrange Polynomial Method) . . . . . . . . . . . . . . 105 8.4.2 Determining Weights (Method of Undetermined Coefficients) . . . . . . . . . 106 8.4.3 Gaussian Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.4.4 Determining Gaussian Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 8.4.5 Reinventing the Wheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 9 Least Squares 115 9.1 Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9.1.1 The Definition of Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9.1.2 Linear Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 9.1.3 Least Squares from Basis Functions . . . . . . . . . . . . . . . . . . . . . . . 117 9.2 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 9.2.1 Alternatives to Normal Equations . . . . . . . . . . . . . . . . . . . . . . . . 120 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 10 Ordinary Differential Equations 123 10.1 Elementary Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 10.1.1 Integration and ‘Stepping’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 10.1.2 Taylor’s Series Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 10.1.3 Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 10.1.4 Higher Order Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 10.1.5 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 10.1.6 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 10.1.7 Backwards Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 10.2 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 10.2.1 Taylor’s Series Redux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 10.2.2 Deriving the Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . 132 10.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 10.3 Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 10.3.1 Larger Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.3.2 Recasting Single ODE Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.3.3 It’s Only Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 10.3.4 It’s Only Autonomous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Page 1: Numerical Methods Course Notes Vers
Page 5: Contents Acknowledgments i 1 Introd
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 and 18: 1.3. EIGENVALUES 9 for all vectors
Page 19 and 20: Chapter 2 A “Crash” Course in o
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
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Page 55 and 56: 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
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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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Chapter 6 Spline Interpolation Spli
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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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