Scientific Computing: For Scientists and Engineers

Description

Scientific Computing for Scientists and Engineers is designed to teach undergraduate students relevant numerical methods and required fundamentals in scientific computing. Most problems in science and engineering require the solution of mathematical problems, most of which can only be done on a computer. Accurately approximating those problems requires solving differential equations and linear systems with millions of unknowns, and smart algorithms can be used on computers to reduce calculation times from years to minutes or even seconds. This book explains: How can we approximate these important mathematical processes? How accurate are our approximations? How efficient are our approximations? Scientific Computing for Scientists and Engineers covers: An introduction to a wide range of numerical methods for linear systems, eigenvalue problems, differential equations, numerical integration, and nonlinear problems; Scientific computing fundamentals like floating point representation of numbers and convergence; Analysis of accuracy and efficiency; Simple programming examples in MATLAB to illustrate the algorithms and to solve real life problems; Exercises to reinforce all topics. Acknowledgments 􀱩 IX 1 Introduction 􀱩 1 1.1 Why study numerical methods? 􀱩 1 1.2 Terminology 􀱩 2 1.3 Convergence terminology 􀱩 3 1.4 Exercises 􀱩 5 2 Computer representation of numbers and roundoff error 􀱩 7 2.1 Examples of the effects of roundoff error 􀱩 7 2.2 Binary numbers 􀱩 10 2.3 64-bit floating-point numbers 􀱩 12 2.4 Floating-point arithmetic 􀱩 14 2.4.1 Avoid adding large and small numbers 􀱩 14 2.4.2 Subtracting two nearly equal numbers is bad 􀱩 14 2.5 Visualizing a floating-point number system 􀱩 16 2.6 Exercises 􀱩 17 3 Solving linear systems of equations 􀱩 20 3.1 Linear systems of equations and solvability 􀱩 20 3.2 Solving triangular systems 􀱩 22 3.3 Gaussian elimination 􀱩 24 3.4 The backslash operator 􀱩 28 3.5 LU decomposition 􀱩 28 3.6 Exercises 􀱩 30 4 Finite difference methods 􀱩 33 4.1 Approximating the first derivative 􀱩 34 4.1.1 Forward and backward differences 􀱩 34 4.1.2 Centered difference 􀱩 37 4.1.3 Three-point difference formulas 􀱩 40 4.1.4 Further notes 􀱩 41 4.2 Approximating the second derivative 􀱩 41 4.3 Application: initial value ODE’s using the forward Euler method 􀱩 42 4.4 Application: boundary value ODE’s 􀱩 45 4.5 Exercises 􀱩 50 5 Solving nonlinear equations 􀱩 53 5.1 The bisection method 􀱩 54 5.2 Newton’s method 􀱩 58 5.3 Secant method 􀱩 60 5.4 Comparing bisection, Newton, and secant methods 􀱩 61 5.5 Combining methods, inverse interpolation, and the fzero command 􀱩 62 5.6 Newton’s method in higher dimensions 􀱩 64 5.7 Fixed point theory and algorithms 􀱩 67 5.7.1 Nonlinear Helmholtz 􀱩 70 5.7.2 Navier–Stokes 􀱩 71 5.7.3 Anderson acceleration 􀱩 74 5.8 Exercises 􀱩 77 6 Accuracy in solving linear systems 􀱩 80 6.1 Gauss–Jordan elimination and finding matrix inverses 􀱩 80 6.2 Matrix and vector norms and condition number 􀱩 83 6.3 Sensitivity in linear system solving 􀱩 85 6.4 Exercises 􀱩 87 7 Eigenvalues and eigenvectors 􀱩 90 7.1 Mathematical definition 􀱩 90 7.2 Power method 􀱩 92 7.3 Application: population dynamics 􀱩 95 7.4 Exercises 􀱩 96 8 Fitting curves to data 􀱩 98 8.1 Interpolation 􀱩 98 8.1.1 Interpolation by a single polynomial 􀱩 99 8.1.2 Piecewise polynomial interpolation 􀱩 101 8.2 Curve fitting 􀱩 104 8.2.1 Line of best fit 􀱩 104 8.2.2 Curve of best fit 􀱩 107 8.3 Exercises 􀱩 110 9 Numerical integration 􀱩 113 9.1 Newton–Cotes methods 􀱩 113 9.2 Composite rules 􀱩 117 9.3 MATLAB’s integral function 􀱩 122 9.4 Gauss quadrature 􀱩 122 9.5 Exercises 􀱩 125 10 Initial value ODEs 􀱩 128 10.1 Reduction of higher-order ODEs to first-order ODEs 􀱩 128 10.2 Common methods and derivation from integration rules 􀱩 130 10.2.1 Backward Euler 􀱩 131 10.2.2 Crank–Nicolson 􀱩 132 10.2.3 Runge–Kutta 4 􀱩 133 10.3 Comparison of speed of implicit versus explicit solvers 􀱩 134 10.4 Stability of ODE solvers 􀱩 135 10.4.1 Stability of forward Euler 􀱩 136 10.4.2 Stability of backward Euler 􀱩 137 10.4.3 Stability of Crank–Nicolson 􀱩 138 10.4.4 Stability of Runge–Kutta 4 􀱩 139 10.5 Accuracy of ODE solvers 􀱩 139 10.5.1 Forward Euler 􀱩 140 10.5.2 Backward Euler 􀱩 140 10.5.3 Crank–Nicolson 􀱩 141 10.5.4 Runge–Kutta 4 􀱩 142 10.6 Summary, general strategy, and MATLAB ODE solvers 􀱩 143 10.7 The 1D heat equation 􀱩 145 10.8 Exercises 􀱩 152 A Getting started with Octave and MATLAB 􀱩 155 A.1 Basic operations 􀱩 155 A.2 Arrays 􀱩 158 A.3 Operating on arrays 􀱩 160 A.4 Script files 􀱩 161 A.5 Function files 􀱩 162 A.5.1 Inline functions 􀱩 162 A.5.2 Passing functions to other functions 􀱩 163 A.6 Outputting information 􀱩 163 A.7 Programming in MATLAB 􀱩 164 A.8 Plotting 􀱩 165 A.9 Exercises 􀱩 166 Index 􀱩 169