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

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

Index Backwards Euler’s Method, 129 big-O, 3 bisection method, 48 centered difference, 89 Chebyshev Polynomials, 67, 120 Chebyshev Quadrature, 112 composite quadrature rule, 98 Dirichlet function, 97 divided differences, 65 eigenvalue, 5 eigenvector, 5 elementary row operations, 23 error roundoff, 87, 126 truncation, 87, 126 Euler’s Method, 124, 125 Gaussian Elimination, Naïve, 23 Gaussian Quadrature, 105 Heaviside function, 96 Hilbert Matrix, 46 Intermediate Value Theorem, 47 knot, 77 Lagrange Polynomials, 61 least squares, 117 definition, 115 linear, 117 LU Factorization, 31 normal equations, 116 ODE errors, 126 higher order methods, 125 stability, 126 stepping, 124 partition, 77 phase curve, 138 Richardson Extrapolation, 90, 102 Riemann Integrable, 96 Riemann integral, 95 Romberg Algorithm, 102, 103 Runge Function, 67 Runge-Kutta Method, 131 secant method, 56 splines, 77 basis splines, 82 linear, 77 quadratic, 79 subtractive cancellation, 4 Taylor’s Series Method, 124 Taylor’s Theorem, 1, 3 trapezoidal rule, 98 error, 100 recursive, 104 method of undetermined coefficients, 106 multiplier, 26 natural cubic spline, 81 Newton’s Method, 50 norm, 6 subordinate, 6 160
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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
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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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Page 123 and 124: Chapter 9 Least Squares 9.1 Least S
Page 125 and 126: 9.1. LEAST SQUARES 117 The answer i
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Page 131 and 132: Chapter 10 Ordinary Differential Eq
Page 133 and 134: 10.1. ELEMENTARY METHODS 125 value
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Page 137 and 138: 10.1. ELEMENTARY METHODS 129 It tur
Page 139 and 140: 10.2. RUNGE-KUTTA METHODS 131 Examp
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Page 143 and 144: 10.3. SYSTEMS OF ODES 135 Example 1
Page 145 and 146: 10.3. SYSTEMS OF ODES 137 Euler’s
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Page 149 and 150: 10.3. SYSTEMS OF ODES 141 4.5 4 3.5
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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
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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: Bibliography [1] Guillaume Bal. Lec
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