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# Lectures In Basic Computational Numerical Analysis

Added On :  03.06.2016 01:57 pm
Author :  J. M. McDonough

LECTURES IN BASIC COMPUTATIONAL NUMERICAL ANALYSIS - By J. M. McDonough ( Departments of Mechanical Engineering and Mathematics, University of Kentucky).

Contents
1 Numerical Linear Algebra 1
1.1 Some Basic Facts from Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Solution of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Numerical solution of linear systems: direct elimination . . . . . . . . . . . . 5
1.2.2 Numerical solution of linear systems: iterative methods . . . . . . . . . . . . 16
1.2.3 Summary of methods for solving linear systems . . . . . . . . . . . . . . . . . 24
1.3 The Algebraic Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.1 The power method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.2 Inverse iteration with Rayleigh quotient shifts . . . . . . . . . . . . . . . . . . 29
1.3.3 The QR algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.4 Summary of methods for the algebraic eigenvalue problem . . . . . . . . . . . 31
1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Solution of Nonlinear Equations 33
2.1 Fixed-Point Methods for Single Nonlinear Equations . . . . . . . . . . . . . . . . . . 33
2.1.1 Basic fixed-point iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.2 Newton iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Modifications to Newton?s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.1 The secant method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.2 The method of false position . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Newton?s Method for Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.1 Derivation of Newton?s Method for Systems . . . . . . . . . . . . . . . . . . . 41
2.3.2 Pseudo-language algorithm for Newton?s method for systems . . . . . . . . . 43
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Approximation Theory 45
3.1 Approximation of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.1 The method of least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.2 Lagrange interpolation polynomials . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.3 Cubic spline interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.4 Extraplotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2 Numerical Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.1 Basic Newton?Cotes quadrature formulas . . . . . . . . . . . . . . . . . . . . 60
3.2.2 Gauss?Legendre quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.3 Evaluation of multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 Finite-Difference Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.2 Use of Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.3 Partial derivatives and derivatives of higher order . . . . . . . . . . . . . . . . 70
3.3.4 Differentiation of interpolation polynomials . . . . . . . . . . . . . . . . . . . 71
3.4 Richardson Extrapolation Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5 Computational Test for Grid Function Convergence . . . . . . . . . . . . . . . . . . . 73
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Numerical Solution of ODEs 77
4.1 Initial-Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1.1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1.2 Basic Single-Step Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.1.3 Runge?Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.1.4 Multi-Step and Predictor-Corrector, Methods . . . . . . . . . . . . . . . . . . 94
4.1.5 Solution of Stiff Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Boundary-Value Problems for ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.2 Shooting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.3 Finite-Difference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3 Singular and Nonlinear Boundary-Value Problems . . . . . . . . . . . . . . . . . . . 108
4.3.1 Coordinate Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3.2 Iterative Methods for Nonlinear BVPs . . . . . . . . . . . . . . . . . . . . . . 110
4.3.3 The Galerkin Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5 Numerical Solution of PDEs 119
5.1 Mathematical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.1.1 Classification of Linear PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.1.2 Basic Concept of Well Posedness . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Overview of Discretization Methods for PDEs . . . . . . . . . . . . . . . . . . . . . . 122
5.3 Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.3.1 Explicit Euler Method for the Heat Equation . . . . . . . . . . . . . . . . . . 124
5.3.2 Backward-Euler Method for the Heat Equation . . . . . . . . . . . . . . . . . 128
5.3.3 Second-Order Approximations to the Heat Equation . . . . . . . . . . . . . . 130
5.3.4 Peaceman?Rachford Alternating-Direction-Implicit Scheme . . . . . . . . . . 136
5.4 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.4.1 Successive Overrelaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.4.2 The Alternating-Direction-Implicit Scheme . . . . . . . . . . . . . . . . . . . 148
5.5 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.5.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.5.2 First-Order Hyperbolic Equations and Systems . . . . . . . . . . . . . . . . . 155
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
References 160

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