Preface     vii
Introduction     1
Background     1
Representation of Numbers on a Computer     4
Errors in Numerical Solutions     10
Round-Off Errors     10
Truncation Errors     13
Total Error     14
Computers and Programming     15
Problems     18
Mathematical Background     21
Background     21
Concepts from Pre-Calculus and Calculus     22
Vectors     26
Operations with Vectors     28
Matrices and Linear Algebra     30
Operations with Matrices     31
Special Matrices     33
Inverse of a Matrix     34
Properties of Matrices     35
Determinant of a Matrix     35
Cramer's Rule and Solution of a System of Simultaneous Linear Equations     36
Norms     38
Ordinary Differential Equations (ODE)     39
Functions of Two or More Independent Variables     42
Definition of the Partial Derivative     42
Chain Rules     43
The Jacobian     44
Taylor Series Expansion of Functions     45
Taylor Seriesfor a Function of One Variable     45
Taylor Series for a Function of Two Variables     47
Problems     48
Solving Nonlinear Equations     53
Background     53
Estimation of Errors in Numerical Solutions     55
Bisection Method     57
Regula Falsi Method     60
Newton's Method     62
Secant Method     67
Fixed-Point Iteration Method     70
Use of MATLAB Built-In Functions for Solving Nonlinear Equations     73
The fzero Command     74
The roots Command     75
Equations with Multiple Solutions     75
Systems of Nonlinear Equations     77
Newton's Method for Solving a System of Nonlinear Equations     78
Fixed-Point Iteration Method for Solving a System of Nonlinear Equations     82
Problems     84
Solving a System of Linear Equations     93
Background     93
Overview of Numerical Methods for Solving a System of Linear Algebraic Equations     94
Gauss Elimination Method     96
Potential Difficulties When Applying the Gauss Elimination Method     104
Gauss Elimination with Pivoting     106
Gauss-Jordan Elimination Method     109
LU Decomposition Method     112
LU Decomposition Using the Gauss Elimination Procedure     114
LU Decomposition Using Crout's Method     115
LU Decomposition with Pivoting     122
Inverse of a Matrix     122
Calculating the Inverse with the LU Decomposition Method     123
Calculating the Inverse Using the Gauss-Jordan Method     125
Iterative Methods     126
Jacobi Iterative Method     127
Gauss-Seidel Iterative Method     127
Use of MATLAB Built-In Functions for Solving a System of Linear Equations     130
Solving a System of Equations Using MATLAB's Left and Right Division     130
Solving a System of Equations Using MATLAB's Inverse Operation     131
MATLAB's Built-In Function for LU Decomposition     132
Additional MATLAB Built-In Functions     133
Tridiagonal Systems of Equations     135
Error, Residual, Norms, and Condition Number     140
Error and Residual     140
Norms and Condition Number     142
Ill-Conditioned Systems     147
Eigenvalues and Eigenvectors     149
The Basic Power Method     152
The Inverse Power Method      156
The Shifted Power Method     157
The QR Factorization and Iteration Method     157
Use of MATLAB Built-In Functions for Determining Eigenvalues and Eigenvectors     167
Problems     169
Curve Fitting and Interpolation     179
Background     179
Curve Fitting with a Linear Equation     181
Measuring How Good Is a Fit     181
Linear Least-Squares Regression     183
Curve Fitting with Nonlinear Equation by Writing the Equation in a Linear Form     187
Curve Fitting with Quadratic and Higher-Order Polynomials     191
Interpolation Using a Single Polynomial     196
Lagrange Interpolating Polynomials     198
Newton's Interpolating Polynomials     202
Piecewise (Spline) Interpolation     209
Linear Splines     209
Quadratic Splines     211
Cubic Splines     215
Use of MATLAB Built-In Functions for Curve Fitting and Interpolation     222
Curve Fitting with a Linear Combination of Nonlinear Functions     224
Problems     227
Numerical Differentiation     233
Background     233
Finite Difference Approximation of the Derivative      235
Finite Difference Formulas Using Taylor Series Expansion     240
Finite Difference Formulas of First Derivative     240
Finite Difference Formulas for the Second Derivative     245
Summary of Finite Difference Formulas for Numerical Differentiation     247
Differentiation Formulas Using Lagrange Polynomials     249
Differentiation Using Curve Fitting     250
Use of MATLAB Built-In Functions for Numerical Differentiation     250
Richardson's Extrapolation     252
Error in Numerical Differentiation     255
Numerical Partial Differentiation     257
Problems     260
Numerical Integration     267
Background     267
Overview of Approaches in Numerical Integration     268
Rectangle and Midpoint Methods     270
Trapezoidal Method     272
Composite Trapezoidal Method     273
Simpson's Methods     276
Simpson's 1/3 Method     276
Simpson's 3/8 Method     279
Gauss Quadrature     281
Evaluation of Multiple Integrals     287
Use of MATLAB Built-In Functions for Integration     288
Estimation of Error in Numerical Integration      290
Richardson's Extrapolation     292
Romberg Integration     295
Improper Integrals     298
Integrals with Singularities     298
Integrals with Unbounded Limits     299
Problems     300
Ordinary Differential Equations: Initial- Value Problems     307
Background     307
Euler's Methods     312
Euler's Explicit Method     312
Analysis of Truncation Error in Euler's Explicit Method     316
Euler's Implicit Method     320
Modified Euler's Method     323
Midpoint Method     326
Runge-Kutta Methods     327
Second-Order Runge-Kutta Methods     328
Third-Order Runge-Kutta Methods     332
Fourth-Order Runge-Kutta Methods     333
Multistep Methods     339
Adams-Bashforth Method     340
Adams-Moulton Method     341
Predictor-Corrector Methods     342
System of First-Order Ordinary Differential Equations     344
Solving a System of First-Order ODEs Using Euler's Explicit Method     346
Solving a System of First-Order ODEs Using Second-Order Runge-Kutta Method (Modified Euler Version)     346
Solving a System of First-Order ODEs Using the Classical Fourth-Order Runge-Kutta Method     353
Solving a Higher-Order Initial Value Problem     354
Use of MATLAB Built-In Functions for Solving Initial-Value Problems     359
Solving a Single First-Order ODE Using MATLAB     360
Solving a System of First-Order ODEs Using MATLAB     366
Local Truncation Error in Second-Order Range-Kutta Method     369
Step Size For Desired Accuracy     370
Stability     374
Stiff Ordinary Differential Equations     376
Problems     379
Ordinary Differential Equations: Boundary-Value Problems     387
Background     387
The Shooting Method     390
Finite Difference Method     398
Use of MATLAB Built-In Functions for Solving Boundary Value Problems     408
Error and Stability in Numerical Solution of Boundary Value Problems     413
Problems     415
Introductory MATLAB     421
Background     421
Starting with MATLAB     421
Arrays     426
Mathematical Operations with Arrays     431
Script Files     435
Function Files     438
Programming in MATLAB      440
Relational and Logical Operators     440
Conditional Statements, if-else Structures     442
Loops     444
Plotting     445
Problems     447
MATLAB Programs     451
Index     455