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Introduction to Numerical Linear Algebra and Optimisation : Cambridge Texts in Applied Mathematics - Philippe G. Ciarlet

Introduction to Numerical Linear Algebra and Optimisation

Cambridge Texts in Applied Mathematics

Paperback Published: 30th October 1989
ISBN: 9780521339841
Number Of Pages: 452

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Based on courses taught to advanced undergraduate students, this book offers a broad introduction to the methods of numerical linear algebra and optimization. The prerequisites are familiarity with the basic properties of matrices, finite-dimensional vector spaces and advanced calculus, and some exposure to fundamental notions from functional analysis. The book is divided into two parts. The first part deals with numerical linear algebra (numerical analysis of matrices, direct and indirect methods for solving linear systems, calculation of eigenvalues and eigenvectors) and the second, optimizations (general algorithms, linear and nonlinear programming). Summaries of basic mathematics are provided, proof of theorems are complete yet kept as simple as possible, applications from physics and mechanics are discussed, a great many exercises are included, and there is a useful guide to further reading.

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'A valuable and welcome addition to the literature.' Mathematics of Computation 'Thanks to the author's sensible choice of material, clear proofs with many figures and the useful guide to the literature, the book can be wholeheartedly recommended to all those interested in this area.' Austrian Mathematical Society

Preface
Numerical linear algebra
A summary of results on matricesp. 1
Introductionp. 1
Key definitions and notationp. 2
Reduction of matricesp. 10
Special properties of symmetric and Hermitian matricesp. 15
Vector and matrix normsp. 20
Sequences of vectors and matricesp. 32
General results in the numerical analysis of matricesp. 37
Introductionp. 37
The two fundamental problems; general observations on the methods in usep. 37
Condition of a linear systemp. 45
Condition of the eigenvalue problemp. 58
Sources of problems in the numerical analysis of matricesp. 64
Introductionp. 64
The finite-difference method for a one-dimensional boundary-value problemp. 65
The finite-difference method for a two-dimensional boundary-value problemp. 79
The finite-difference method for time-dependent boundary-value problemsp. 86
Variational approximation of a one-dimensional boundary-value problemp. 94
Variational approximation of a two-dimensional boundary-value problemp. 106
Eigenvalue problemsp. 110
Interpolation and approximation problemsp. 115
Direct methods for the solution of linear systemsp. 124
Introductionp. 124
Two remarks concerning the solution of linear systemsp. 125
Gaussian eliminationp. 127
The LU factorisation of a matrixp. 138
The Cholesky factorisation and methodp. 147
The QR factorisation of a matrix and Householder's methodp. 152
Iterative methods for the solution of linear systemsp. 159
Introductionp. 159
General results on iterative methodsp. 159
Description of the methods of Jacobi, Gauss-Seidel and relaxationp. 163
Convergence of the Jacobi, Gauss-Seidel and relaxation methodsp. 171
Methods for the calculation of eigenvalues and eigenvectorsp. 186
Introductionp. 186
The Jacobi methodp. 187
The Givens-Householder methodp. 196
The QR algorithmp. 204
Calculation of eigenvectorsp. 212
Optimisation
A review of differential calculus. Some applicationsp. 216
Introductionp. 216
First and second derivatives of a functionp. 218
Extrema of real functions: Lagrange multipliersp. 232
Extrema of real functions: consideration of the second derivativesp. 240
Extrema of real functions: consideration of convexityp. 241
Newton's methodp. 251
General results on optimisation. Some algorithmsp. 266
Introductionp. 266
The projection theorem; some consequencesp. 268
General results on optimisation problemsp. 277
Examples of optimisation problemsp. 286
Relaxation and gradient methods for unconstrained problemsp. 291
Conjugate gradient methods for unconstrained problemsp. 311
Relaxation, gradient and penalty-function methods for constrained problemsp. 321
Introduction to non-linear programmingp. 330
Introductionp. 330
The Farkas lemmap. 332
The Kuhn-Tucker conditionsp. 336
Lagrangians and saddle points. Introduction to dualityp. 350
Uzawa's methodp. 360
Linear programmingp. 369
Introductionp. 369
General results on linear programmingp. 370
Examples of linear programming problemsp. 374
The simplex methodp. 378
Duality and linear programmingp. 400
Bibliography and commentsp. 411
Main notations usedp. 422
Indexp. 428
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521339841
ISBN-10: 0521339847
Series: Cambridge Texts in Applied Mathematics
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 452
Published: 30th October 1989
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 23.01 x 15.42  x 2.52
Weight (kg): 0.66

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