Scientific Computing With Matlab and Octave

by Quarteroni, Alfio; Saleri, Fausto; Gervasio, Paola

• ISBN13: 9783642124297
• ISBN10: 3642124291
• Edition: 3rd
• Format: Hardcover
• Copyright: 2010-08-27
• Publisher: Springer Verlag

Summary

This textbook is an introduction to Scientific Computing, in which several numerical methods for the computer-based solution of certain classes of mathematical problems are illustrated. The authors show how to compute the zeros or the integrals of continuous functions, solve linear systems, approximate functions using polynomials and construct accurate approximations for the solution of ordinary and partial differential equations. To make the format concrete and appealing, the programming environments Matlab and Octave are adopted as faithful companions. The book contains the solutions to several problems posed in exercises and examples, often originating from important applications. At the end of each chapter, a specific section is devoted to subjects which were not addressed in the book and contains bibliographical references for a more comprehensive treatment of the material.

Table of Contents

What can't be ignored	p. 1
The MATLAB and Octave environments	p. 1
Real numbers	p. 3
How we represent them	p. 3
How we operate with floating-point numbers	p. 6
Complex numbers	p. 8
Matrices	p. 10
Vectors	p. 14
Real functions	p. 16
The zeros	p. 18
Polynomials	p. 20
Integration and differentiation	p. 22
To err is not only human	p. 25
Talking about costs	p. 29
The MATLAB language	p. 30
MATLAB statements	p. 32
Programming in MATLAB	p. 34
Examples of differences between MATLAB and Octave languages	p. 37
What we haven't told you	p. 38
Exercises	p. 38
Nonlinear equations	p. 41
Some representative problems	p. 41
The bisection method	p. 43
The Newton method	p. 47
How to terminate Newton's iterations	p. 49
The Newton method for systems of nonlinear equations	p. 51
Fixed point iterations	p. 54
How to terminate fixed point iterations	p. 60
Acceleration using Aitken's method	p. 60
Algebraic polynomials	p. 65
Hörner's algorithm	p. 66
The Newton-Hörner method	p. 68
What we haven't told you	p. 70
Exercises	p. 72
Approximation of functions and data	p. 75
Some representative problems	p. 75
Approximation by Taylor's polynomials	p. 77
Interpolation	p. 78
Lagrangian polynomial interpolation	p. 79
Stability of polynomial interpolation	p. 84
Interpolation at Chebyshev nodes	p. 86
Trigonometric interpolation and FFT	p. 88
Piecewise linear interpolation	p. 93
Approximation by spline functions	p. 94
The least-squares method	p. 99
What we haven't told you	p. 103
Exercises	p. 105
Numerical differentiation and integration	p. 107
Some representative problems	p. 107
Approximation of function derivatives	p. 109
Numerical integration	p. 111
Midpoint formula	p. 112
Trapezoidal formula	p. 114
Simpson formula	p. 115
Interpolatory quadratures	p. 117
Simpson adaptive formula	p. 121
What we haven't told you	p. 125
Exercises	p. 126
Linear systems	p. 129
Some representative problems	p. 129
Linear system and complexity	p. 134
The LU factorization method	p. 135
The pivoting technique	p. 144
How accurate is the solution of a linear system?	p. 147
How to solve a tridiagonal system	p. 150
Overdetermined systems	p. 152
What is hidden behind the MATLAB command	p. 154
Iterative methods	p. 157
How to construct an iterative method	p. 158
Richardson and gradient methods	p. 162
The conjugate gradient method	p. 166
When should an iterative method be stopped?	p. 169
To wrap-up: direct or iterative?	p. 171
What we haven't told you	p. 177
Exercises	p. 177
Eigenvalues and eigenvectors	p. 181
Some representative problems	p. 182
The power method	p. 184
Convergence analysis	p. 187
Generalization of the power method	p. 188
How to compute the shift	p. 190
Computation of all the eigenvalues	p. 193
What we haven't told you	p. 197
Exercises	p. 197
Ordinary differential equations	p. 201
Some representative problems	p. 201
The Cauchy problem	p. 204
Euler methods	p. 205
Convergence analysis	p. 208
The Crank-Nicolson method	p. 212
Zero-stability	p. 214
Stability on unbounded intervals	p. 216
The region of absolute stability	p. 219
Absolute stability controls perturbations	p. 220
High order methods	p. 228
The predictor-corrector methods	p. 234
Systems of differential equations	p. 236
Some examples	p. 242
The spherical pendulum	p. 242
The three-body problem	p. 246
Some stiff problems	p. 248
What we haven't told you	p. 252
Exercises	p. 252
Numerical approximation of boundary-value problems	p. 255
Some representative problems	p. 256
Approximation of boundary-value problems	p. 258
Finite difference approximation of the one-dimensional Poisson problem	p. 259
Finite difference approximation of a convection-dominated problem	p. 262
Finite element approximation of the one-dimensional Poisson problem	p. 263
Finite difference approximation of the two-dimensional Poisson problem	p. 267
Consistency and convergence of finite difference discretization of the Poisson problem	p. 272
Finite difference approximation of the one-dimensional heat equation	p. 274
Finite element approximation of the one-dimensional heat equation	p. 278
Hyperbolic equations: a scalar pure advection problem	p. 281
Finite difference discretization of the scalar transport equation	p. 283
Finite difference analysis for the scalar transport equation	p. 285
Finite element space discretization of the scalar advection equation	p. 292
The wave equation	p. 293
Finite difference approximation of the wave equation	p. 295
What we haven't told you	p. 299
Exercises	p. 300
Solutions of the exercises	p. 303
	p. 303
	p. 306
	p. 312
	p. 315
	p. 320
	p. 327
	p. 330
	p. 339
References	p. 347
Index	p. 353
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