Exploring Numerical Methods: An Introduction to Scientific Computing Using Matlab

by Linz, Peter; Wang, Richard

• ISBN13: 9780763714994
• ISBN10: 0763714992
• Format: Hardcover
• Copyright: 2003-05-01
• Publisher: Jones & Bartlett

Summary

Exploring Numerical Methods is designed to provide beginning engineering and science students, as well as upper-level mathematics students, with an introduction to numerical analysis that emphasizes insight and hands-on experience. To serve the needs of both the younger and the more experienced audience, each chapter begins with an intuitive presentation of motivation and simple algorithms. Topics are developed progressively within each chapter and the advanced material, which reveals underlying theory and discusses complicated methods, is clearly marked. The text takes a focused approach to introducing the more important numerical algorithms and exposes students to partial differential equations by using simple prototypes. This text provides a strong experiential basis for future study.

Table of Contents

Numerical Analysis and Scientific Computing	1	(10)
Computing with Numbers	11	(20)
Number Representation	11	(2)
Floating-Point Representation	13	(4)
Rounding Errors: A Practical View	17	(8)
Theories of Rounding Error*	25	(6)
The Solution of Systems of Linear Equations	31	(26)
The Gauss Elimination Method	32	(7)
Computing the Inverse of a Matrix	39	(3)
Least Squares Solutions	42	(4)
Testing the Accuracy of a Solution	46	(2)
Error Analysis for Matrix Problems*	48	(9)
Polynomials and Polynomial Approximations	57	(36)
Some Important Properties of Polynomials	57	(4)
Polynomial Approximation by Interpolation	61	(8)
Polynomial Approximation by Least Squares	69	(2)
Least Squares Approximation and Orthogonol Polynomials*	71	(8)
Legendre Polynomials	72	(1)
Chebyshev Polynomials of the First Kind	72	(1)
Chebyshev Polynomials of the Second Kind	73	(1)
Laguerre Polynomials	73	(1)
Hermite Polynomials	74	(5)
Fourier Approximations*	79	(5)
Best Approximations in the Maximum Norm*	84	(9)
Piecewise Polynomials and Data Fitting	93	(32)
Piecewise Polynomial Functions	95	(4)
Bases for Piecewise Polynomials	99	(6)
Approximation of Functions by Piecewise Polynomials	105	(10)
Data Fitting*	115	(10)
Numerical Differentiation and Integration	125	(46)
Finite Differences and Numerical Evaluation of Derivatives	126	(5)
Numerical Integration by Interpolation	131	(7)
The Rectangular and Midpoint Rules	132	(1)
The Trapezoidal Rule	132	(1)
The Simpson Method	133	(1)
The Three-Eights Rule	133	(5)
Error Analysis for Interpolatory Quadratures	138	(5)
Adaptive Quadrature	143	(5)
Gaussian Quadrature*	148	(8)
Dealing with Difficult Integrands*	156	(9)
Multi-Dimensional Integration*	165	(6)
The Solution of Nonlinear Equations	171	(38)
Some Simple Root-Finding Methods	172	(7)
Convergence Rates of Root-Finding Methods	179	(3)
Difficulties with Root-Finding	182	(3)
The Solution of Simultaneous Equations by Newton's Method	185	(4)
The Method of Successive Substitution*	189	(6)
Minimization Methods*	195	(5)
Roots of Polynomials*	200	(5)
Selecting a Root-Finding Method*	205	(4)
Advanced Matrix Algorithms	209	(28)
The LU and Cholesky Factorizations	209	(8)
The QR Decomposition and Least Squares Solutions	217	(6)
The Singular Value Decomposition*	223	(6)
Sparse Matrices*	229	(8)
Eigenvalue Problems	237	(30)
Some Useful Results	239	(6)
The Power Method	245	(4)
The Jacobi and Householder Methods for Symmetric Matrices*	249	(7)
Eigenvalues of Nonsymmetric Matrices*	256	(2)
Finding Eigenvectors by Inverse Iteration*	258	(3)
Error Estimates for Symmetric Matrices*	261	(6)
Initial Value Problems for Ordinary Differential Equations	267	(46)
Existence and Uniqueness of the Solution	269	(4)
Some Simple Numerical Methods	273	(4)
Multistep and Predictor-Corrector Methods	277	(5)
Runge-Kutta Methods	282	(4)
Systems and Equations of Higher Order	286	(3)
Convergence of Approximations*	289	(8)
Stability*	297	(8)
Stiff Equations*	305	(4)
Error Estimation and Step Size Selection*	309	(4)
Boundary and Eigenvalue Problems for Ordinary Differential Equations	313	(30)
Shooting Methods	315	(6)
Finite Difference Methods*	321	(8)
The Collocation, Least Squares, and Galerkin's Methods*	329	(7)
Eigenvalue Problems for Differential Equations*	336	(7)
Initial Value Problems for Partial Differential Equations*	343	(22)
Finite Difference Methods for the Heat Equation	345	(11)
Finite Difference Methods for the Wave Equation	356	(3)
Semi-Discretization and the Method of Lines	359	(2)
Distinction between Parabolic and Hyperbolic Equations	361	(4)
Boundary Value Problems for Partial Differential Equations*	365	(18)
Finite Difference Methods	366	(4)
Error Analysis for the Finite Difference Method	370	(2)
Solving Finite Difference Systems	372	(4)
The Finite Element Method	376	(3)
Multidimensional Equations	379	(4)
Ill-Conditioned and Ill-Posed Problems*	383	(30)
The Numerical Rank of a Matrix	384	(3)
Computing Generalized Solutions	387	(2)
Inverse and Ill-Posed Problems	389	(4)
Regularization	393	(3)
The Tikhonov Method	396	(6)
Choosing the Regularization Parameters	402	(5)
The Numerical Solution of Integral Equations of the First Kind	407	(4)
Conclusions	411	(2)
Numerical Computing in MATLAB	413	(14)
MATLAB Arithmetic and Standard Functions	413	(3)
Solving Linear Systems	416	(1)
Polynomials and Polynomial Approximations	417	(1)
Interpolation and Data Fitting	418	(1)
Numerical Integration	419	(2)
Root-Finding	421	(1)
Matrix Functions in MATLAB	422	(2)
Software for Eigensolutions	424	(1)
Software for ODE Initial Value Problems	425	(2)
NASOFT User Manual and Example Scripts	427	(22)
Piecewise Polynomial Approximations	427	(3)
hat	427	(1)
quad_q	428	(1)
quad_r	428	(1)
spl_b3	429	(1)
spl_b31	429	(1)
spl_b32	430	(1)
Data Fitting	430	(2)
lsq_lin	431	(1)
lsq_quad	431	(1)
lsq_cspl	432	(1)
Numerical Integration	432	(4)
gauss	432	(1)
quad_cg	433	(1)
quad_dbl	434	(1)
quad_sim	435	(1)
Solving Problems of Nonlinear Equations	436	(3)
muller	436	(2)
newton	438	(1)
Eigenvalue Problems	439	(2)
eigpower	439	(1)
jacobi	440	(1)
hholder	440	(1)
Solution of Ordinary Differential Equations---Initial Value Problems	441	(2)
oderk4	441	(1)
odepc	442	(1)
Linear Two-Point Boundary Value Problems	443	(2)
bvp_fd	443	(1)
bvp_coll	444	(1)
Partial Differential Equations	445	(2)
heat	445	(1)
wave	446	(1)
Ill-Conditioned Linear Systems	447	(1)
tik_reg	447	(1)
Example Scripts	448	(1)
NASOFT	448	(1)
Explorations	449	(16)
Floating-Point Representation, Rounding, and Accuracy of Elementary Functions	449	(2)
Solving Linear Systems	451	(1)
Approximation by Polynomials	452	(1)
Interpolation and Data Fitting with Piecewise Polynomials	453	(2)
Approximating Derivatives and Integrals	455	(2)
Solving Nonlinear Equations	457	(1)
Matrix Methods	458	(1)
Computing Eigensystems of Matrices	458	(1)
ODE Initial Value Problems	459	(2)
ODE Boundary and Eigenvalue Problems	461	(1)
The Heat and Wave Equations	462	(1)
PDE Boundary Value Problems	462	(1)
Ill-Posed Problems and Regularization	463	(2)
References	465	(2)
Subject Index	467	(2)
Index	469