Introduction to Scientific Computing A Matrix-Vector Approach Using MATLAB

by Van Loan, Charles F.

• ISBN13: 9780139491573
• ISBN10: 0139491570
• Edition: 2nd
• Format: Paperback
• Copyright: 1999-07-07
• Publisher: Pearson

Summary

Unique in content and approach, this book covers all the topics that are usually covered in an introduction to scientific computing--but folds in graphics and matrix-vector manipulation in a way that gets readers to appreciate theconnectionbetween continuous mathematics and computing.MATLAB 5is usedthroughoutto encourage experimentation, and each chapter focuses on a different important theorem--allowing readers to appreciate the rigorous side of scientific computing. In addition to standard topical coverage, each chapter includes 1) a sketch of a "hard" problem that involves ill-conditioning, high dimension, etc.; 2)at least one theorem with both a rigorous proof and a "proof by MATLAB" experiment to bolster intuition; 3)at least one recursive algorithm; and 4)at least one connection to a real-world application. The book revolves around examples that are packaged in 200+ M-files, which, collectively, communicate all the key mathematical ideas and an appreciation for the subtleties of numerical computing.Power Tools of the Trade. Polynomial Interpolation. Piecewise Polynomial Interpolation. Numerical Integration. Matrix Computations. Linear Systems. The QR and Cholesky Factorizations. Nonlinear Equations and Optimization. The Initial Value Problem.For engineers and mathematicians.

Table of Contents

Preface to the Second Edition	ix	(1)
Preface to the First Edition	x	(1)
Software	xi
1 Power Tools of the Trade	1	(73)
1.1 Vectors and Plotting	2	(13)
1.2 More Vectors, More Plotting, and Now Matrices	15	(14)
1.3 Building Exploratory Environments	29	(12)
1.4 Error	41	(7)
1.5 Designing Functions	48	(11)
1.6 Structure Arrays and Cell Arrays	59	(5)
1.7 More Refined Graphics	64	(8)
M-Files and References	72	(2)
2 Polynomial Interpolation	74	(30)
2.1 The Vandermonde Approach	76	(7)
2.2 The Newton Representation	83	(6)
2.3 Properties	89	(3)
2.4 Special Topics	92	(11)
M-Files and References	103	(1)
3 Piecewise Polynomial Interpolation	104	(32)
3.1 Piecewise Linear Interpolation	104	(11)
3.2 Piecewise Cubic Hermite Interpolation	115	(8)
3.3 Cubic Splines	123	(11)
M-Files and References	134	(2)
4 Numerical Integration	136	(32)
4.1 The Newton-Cotes Rules	137	(8)
4.2 Composite Rules	145	(4)
4.3 Adaptive Quadrature	149	(6)
4.4 Special Topics	155	(5)
4.5 Shared Memory Adaptive Quadrature	160	(6)
M-Files and References	166	(2)
5 Matrix Computations	168	(41)
5.1 Setting Up Matrix Problems	168	(9)
5.2 Matrix Operations	177	(11)
5.3 Once Again, Setting Up Matrix Problems	188	(6)
5.4 Recursive Matrix Operations	194	(7)
5.5 Distributed Memory Matrix Multiplication	201	(6)
M-Files and References	207	(2)
6 Linear Systems	209	(31)
6.1 Triangular Problems	209	(6)
6.2 Banded Problems	215	(6)
6.3 General Problems	221	(13)
6.4 Analysis	234	(4)
M-Files and References	238	(2)
7 The QR and Cholesky Factorizations	240	(35)
7.1 Least Squares Fitting	240	(7)
7.2 The QR factorization	247	(7)
7.3 The Cholesky Factorization	254	(14)
7.4 High-Performance Cholesky	268	(6)
M-Files and References	274	(1)
8 Nonlinear Equations and Optimization	275	(52)
8.1 Finding Roots	276	(17)
8.2 Minimizing a Function of A Single Variable	293	(12)
8.3 Minimizing Multivariate Functions	305	(8)
8.4 Solving Systems of Nonlinear Equations	313	(12)
M-Files and References	325	(2)
9 The Initial Value Problem	327	(34)
9.1 Basic Concepts	328	(10)
9.2 The Runge-Kutta Methods	338	(11)
9.3 The Adams Methods	349	(10)
M-Files and References	359	(2)
Bibliography	361	(2)
Index	363