Table Of Content

Computational Methods for Numerical Analysis with R James P. Howard, II The Johns Hopkins University Applied Physics Laboratory Laurel, Maryland, USA CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business Printed on acid-free paper Version Date: 20170516 International Standard Book Number-13: 978-1-4987-2363-3 (Hardback) Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents List of Figures xiii List of Tables xv List of R Functions xvii Preface xix 1 Introduction to Numerical Analysis 1 1.1 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 The Goals of Numerical Analysis . . . . . . . . . . . . 1 1.1.2 Numerical Analysis in R . . . . . . . . . . . . . . . . . 3 1.1.3 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Data Types in R . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Data Types . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 Data Structures . . . . . . . . . . . . . . . . . . . . . 11 1.3 Elementary Problems . . . . . . . . . . . . . . . . . . . . . . 16 1.3.1 Summation Algorithms . . . . . . . . . . . . . . . . . 16 1.3.2 Evaluating Polynomials . . . . . . . . . . . . . . . . . 20 1.3.3 The nth Root Algorithm . . . . . . . . . . . . . . . . 24 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 Error Analysis 29 2.1 True Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1.1 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.2 Precision . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Internal Data Storage . . . . . . . . . . . . . . . . . . . . . . 35 2.2.1 Binary Numbers . . . . . . . . . . . . . . . . . . . . . 35 2.2.2 Floating Point Numbers . . . . . . . . . . . . . . . . . 37 2.3 Numerical Error . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3.1 Round-Off Error and Machine (cid:15) . . . . . . . . . . . . 42 2.3.2 Loss of Significance. . . . . . . . . . . . . . . . . . . . 44 2.3.3 Overflow and Underflow . . . . . . . . . . . . . . . . . 48 2.3.4 Error Propagation and Stability . . . . . . . . . . . . 50 2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.4.1 Simple Division Algorithms . . . . . . . . . . . . . . . 53 2.4.2 Binary Long Division . . . . . . . . . . . . . . . . . . 55 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3 Linear Algebra 59 3.1 Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . . 59 3.1.1 Vector and Matrix Operations . . . . . . . . . . . . . 59 3.1.2 Elementary Row Operations. . . . . . . . . . . . . . . 64 3.2 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . 67 3.2.1 Row Echelon Form . . . . . . . . . . . . . . . . . . . . 67 3.2.2 Tridiagonal Matrices . . . . . . . . . . . . . . . . . . . 73 3.3 Matrix Decomposition . . . . . . . . . . . . . . . . . . . . . . 76 3.3.1 LU Decomposition . . . . . . . . . . . . . . . . . . . . 76 3.3.2 Cholesky Decomposition . . . . . . . . . . . . . . . . . 80 3.4 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . 82 3.4.1 Jacobi Iteration . . . . . . . . . . . . . . . . . . . . . . 83 3.4.2 Gauss–Seidel Iteration . . . . . . . . . . . . . . . . . . 86 3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.5.1 Least Squares . . . . . . . . . . . . . . . . . . . . . . . 89 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4 Interpolation and Extrapolation 95 4.1 Polynomial Interpolation . . . . . . . . . . . . . . . . . . . . 95 4.1.1 Linear Interpolation . . . . . . . . . . . . . . . . . . . 95 4.1.2 Higher-Order Polynomial Interpolation. . . . . . . . . 97 4.2 Piecewise Interpolation . . . . . . . . . . . . . . . . . . . . . 102 4.2.1 Piecewise Linear Interpolation . . . . . . . . . . . . . 103 4.2.2 Cubic Spline Interpolation . . . . . . . . . . . . . . . . 105 4.2.3 B´ezier Curves . . . . . . . . . . . . . . . . . . . . . . . 110 4.3 Multidimensional Interpolation . . . . . . . . . . . . . . . . . 115 4.3.1 Bilinear Interpolation . . . . . . . . . . . . . . . . . . 115 4.3.2 Nearest Neighbor Interpolation . . . . . . . . . . . . . 119 4.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.4.1 Time Series Interpolation . . . . . . . . . . . . . . . . 122 4.4.2 Computer Graphics . . . . . . . . . . . . . . . . . . . 125 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Contents xi 5 Differentiation and Integration 133 5.1 Numerical Differentiation . . . . . . . . . . . . . . . . . . . . 133 5.1.1 Finite Differences . . . . . . . . . . . . . . . . . . . . . 133 5.1.2 The Second Derivative . . . . . . . . . . . . . . . . . . 137 5.2 Newton–Cotes Integration . . . . . . . . . . . . . . . . . . . 138 5.2.1 Multipanel Interpolation Rules . . . . . . . . . . . . . 139 5.2.2 Newton–Cotes Errors . . . . . . . . . . . . . . . . . . 145 5.2.3 Newton–Cotes Forms, Generally . . . . . . . . . . . . 147 5.3 Gaussian Integration . . . . . . . . . . . . . . . . . . . . . . 149 5.3.1 The Gaussian Method . . . . . . . . . . . . . . . . . . 149 5.3.2 Implementation Details . . . . . . . . . . . . . . . . . 151 5.4 More Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.4.1 Adaptive Integrators . . . . . . . . . . . . . . . . . . . 153 5.4.2 Romberg’s Method . . . . . . . . . . . . . . . . . . . . 156 5.4.3 Monte Carlo Methods . . . . . . . . . . . . . . . . . . 159 5.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.5.1 Revolved Volumes . . . . . . . . . . . . . . . . . . . . 163 5.5.2 Gini Coefficients . . . . . . . . . . . . . . . . . . . . . 168 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6 Root Finding and Optimization 175 6.1 One-Dimensional Root Finding . . . . . . . . . . . . . . . . . 175 6.1.1 Bisection Method . . . . . . . . . . . . . . . . . . . . . 175 6.1.2 Newton–Raphson Method . . . . . . . . . . . . . . . . 179 6.1.3 Secant Method . . . . . . . . . . . . . . . . . . . . . . 183 6.2 Minimization and Maximization . . . . . . . . . . . . . . . . 185 6.2.1 Golden Section Search . . . . . . . . . . . . . . . . . . 185 6.2.2 Gradient Descent . . . . . . . . . . . . . . . . . . . . . 188 6.3 Multidimensional Optimization . . . . . . . . . . . . . . . . . 192 6.3.1 Multidimensional Gradient Descent. . . . . . . . . . . 192 6.3.2 Hill Climbing . . . . . . . . . . . . . . . . . . . . . . . 195 6.3.3 Simulated Annealing . . . . . . . . . . . . . . . . . . . 197 6.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.4.1 Least Squares . . . . . . . . . . . . . . . . . . . . . . . 200 6.4.2 The Traveling Salesperson . . . . . . . . . . . . . . . . 203 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 7 Differential Equations 213 7.1 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . 213 7.1.1 Euler Method . . . . . . . . . . . . . . . . . . . . . . . 213 7.1.2 Runge–Kutta Methods, Generally . . . . . . . . . . . 219 7.1.3 Linear Multistep Methods . . . . . . . . . . . . . . . . 224 7.2 Systems of Ordinary Differential Equations . . . . . . . . . . 227 7.2.1 Solution Systems and Initial Value Problems . . . . . 228 7.2.2 Boundary Value Problems . . . . . . . . . . . . . . . . 231 7.3 Partial Differential Equations . . . . . . . . . . . . . . . . . . 234 7.3.1 The Heat Equation . . . . . . . . . . . . . . . . . . . . 234 7.3.2 The Wave Equation . . . . . . . . . . . . . . . . . . . 238 7.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 7.4.1 Carbon Dating . . . . . . . . . . . . . . . . . . . . . . 242 7.4.2 Lotka–Volterra Equations . . . . . . . . . . . . . . . . 244 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Suggested Reading 249 Index 253 Figures 1.1 Assembly of partial sums in piecewise addition. . . . . . . . 18 2.1 Newark is near New York . . . . . . . . . . . . . . . . . . . 30 2.2 At least we got to New York, this time . . . . . . . . . . . . 32 2.3 A journey of a thousand miles starts with a good map . . . 33 2.4 Error in the measurement of length . . . . . . . . . . . . . . 51 2.5 Additive error in measurement of length . . . . . . . . . . . 51 2.6 Multiplicative error in measurement of length . . . . . . . . 52 4.1 Linear interpolation of two points . . . . . . . . . . . . . . . 96 4.2 Quadratic interpolation of three points . . . . . . . . . . . . 100 4.3 Cubic interpolation of y =tan((x−1/4))+1 . . . . . . . . 101 4.4 Piecewise linear interpolation of four points . . . . . . . . . 103 4.5 Cubic spline interpolation of four points . . . . . . . . . . . 109 4.6 Cubic spline interpolation of five points . . . . . . . . . . . . 110 4.7 Quadratic B´ezier interpolation of three points . . . . . . . . 113 4.8 Cubic B´ezier interpolation of four points . . . . . . . . . . . 115 4.9 Cubic B´ezier interpolation of four points, with crossover . . 116 4.10 Bilinear interpolation example . . . . . . . . . . . . . . . . . 118 4.11 Nearest neighbor interpolation in one dimension . . . . . . . 120 4.12 Two-dimensional nearest neighbor example. . . . . . . . . . 121 4.13 ckbhimageat400pixelsand200pixels,scaledusingnearest neighbor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.14 babhimageat400pixelsand200pixels,scaledusingbilinear interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.1 Derivative of sinx at x=π/4 for step sizes h={1,0.5,0.01} 135 5.2 The midpoint rule for f(x)=x2 with m=2 and m=4 . . 140 5.3 The trapezoid rule for f(x)=x2 with m=2 and m=4 . . 142 √ 5.4 The Monte Carlo method for f(x)= 1−x2 with m=20 . 160 5.5 Cylinder as a rotated solid around the x-axis and y-axis . . 165 5.6 Rotated volume around the x-axis. . . . . . . . . . . . . . . 166 5.7 Rotated volume around the y-axis . . . . . . . . . . . . . . . 167 5.8 Lorenz curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.1 The roots of f(x)=x2−1 . . . . . . . . . . . . . . . . . . . 176 6.2 The roots of f(x)=x3−x . . . . . . . . . . . . . . . . . . . 178 xiv Computational Methods for Numerical Analysis with R 6.3 Finding a real root of x3−2x2−x+2 . . . . . . . . . . . . 180 6.4 Golden section search on f(x)=−x2+4x+1 . . . . . . . . 186 6.5 Golden section search on f(x)=sinx−x2 . . . . . . . . . . 188 6.6 Gradient descent on 1x4+x3−x−1. . . . . . . . . . . . . 189 4 6.7 Plot of f(x ,x )=(x −1)2+(2x −2)2 . . . . . . . . . . 194 1 2 1 2 7.1 Vector field of f(cid:48)(x,y)= y . . . . . . . . . . . . . . . . . 214 2x+1 7.2 Euler method with h=0.5 and 0.2 . . . . . . . . . . . . . . 216 7.3 Integration of f(cid:48)(x,y)=−10y . . . . . . . . . . . . . . . . . 218 7.4 Euler and midpoint initial value problem example . . . . . . 220 7.5 Midpoint and fourth order Runge–Kutta methods with h = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.6 Adams–Bashforth with h=0.5 and 0.2 . . . . . . . . . . . . 227 7.7 Plot of bvpexample for x∈[−2,2] . . . . . . . . . . . . . . 233 7.8 Heat equation plot for ∆t=0.001 . . . . . . . . . . . . . . . 237 7.9 Heat equation plot for ∆t=0.0015 . . . . . . . . . . . . . . 238 7.10 Wave equation plot for ∆t=0.001 . . . . . . . . . . . . . . 241 7.11 Lotka–Volterra model for a hypothetical population dynamic 246 Tables 2.1 Hexadecimal, decimal, and binary numbers with equivalents 36 4.1 Time series interpolation examples . . . . . . . . . . . . . . 125 5.1 Gauss–Legendre points and weights for n=1,2,3,4 . . . . . 152 R Functions 1.1 The Fibonacci sequence . . . . . . . . . . . . . . . . . . . . 4 1.2 Test an integer for primality . . . . . . . . . . . . . . . . . . 7 1.3 The na¨ıve approach to summing a vector . . . . . . . . . . . 17 1.4 The pairwise summation algorithm . . . . . . . . . . . . . . 18 1.5 The Kahan summation algorithm . . . . . . . . . . . . . . . 19 1.6 The algebraic approach to evaluating polynomials . . . . . . 21 1.7 A better algebraic approach to polynomial evaluation . . . . 22 1.8 Horner’s rule for polynomial evaluation . . . . . . . . . . . . 23 1.9 The nth root algorithm . . . . . . . . . . . . . . . . . . . . . 25 2.1 The quadratic formula . . . . . . . . . . . . . . . . . . . . . 46 2.2 The quadratic formula rewritten . . . . . . . . . . . . . . . . 47 2.3 Na¨ıve division by repeated subtraction . . . . . . . . . . . . 54 2.4 Binary long division . . . . . . . . . . . . . . . . . . . . . . 56 3.1 The row scaling algorithm . . . . . . . . . . . . . . . . . . . 65 3.2 The row swapping algorithm . . . . . . . . . . . . . . . . . . 65 3.3 The row replacement algorithm . . . . . . . . . . . . . . . . 66 3.4 The row echelon matrix form . . . . . . . . . . . . . . . . . 68 3.5 The reduced row echelon matrix form . . . . . . . . . . . . . 70 3.6 Use reduced row echelon form to solve linear equations . . . 73 3.7 Tridiagonal matrix algorithm . . . . . . . . . . . . . . . . . 75 3.8 The matrix LU decomposition function . . . . . . . . . . . . 79 3.9 The Cholesky decomposition . . . . . . . . . . . . . . . . . . 82 3.10 Find the norm of a vector . . . . . . . . . . . . . . . . . . . 83 3.11 Jacobi iteration for diagonally dominant matrices . . . . . . 85 3.12 Gauss–Seidel iteration for diagonally dominant matrices . . 87 4.1 Linear interpolation . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 Vandermonde polynomial interpolation . . . . . . . . . . . . 99 4.3 Piecewise linear interpolation . . . . . . . . . . . . . . . . . 104 4.4 The natural cubic spline . . . . . . . . . . . . . . . . . . . . 108 4.5 Quadratic B´ezier curves . . . . . . . . . . . . . . . . . . . . 112 4.6 Cubic B´ezier curves . . . . . . . . . . . . . . . . . . . . . . . 114 4.7 Bilinear interpolation . . . . . . . . . . . . . . . . . . . . . . 117 4.8 Nearest neighbor interpolation . . . . . . . . . . . . . . . . . 121

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