Computational matrix analysis
type: Item
instance of: scholarly article
zbMATH Open document ID: 1252.65065
publication date: 2012-05-15T00:00:00Z
Mathematics Subject Classification ID: 65Fxx
Mathematics Subject Classification ID: 65-01
Mathematics Subject Classification ID: 00A06
Mathematics Subject Classification ID: 65Y04
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A16
Mathematics Subject Classification ID: 15A12
Mathematics Subject Classification ID: 65Y20
zbMATH DE Number: 6035286
zbMATH Keywords: floating point arithmetic
zbMATH Keywords: finite arithmetic
zbMATH Keywords: rounding error
zbMATH Keywords: Gaussian elimination
zbMATH Keywords: eigenvalues
zbMATH Keywords: QR algorithm
zbMATH Keywords: singular value decomposition
zbMATH Keywords: textbook
zbMATH Keywords: LU and QR factorization
zbMATH Keywords: least squares solution
zbMATH Keywords: generalized inverse
zbMATH Keywords: software
zbMATH Keywords: numerical stability
zbMATH Keywords: ill conditioning
zbMATH Keywords: complexity
zbMATH Keywords: perturbation analysis
zbMATH Keywords: conditioning
zbMATH Keywords: matrix function
zbMATH Keywords: matrix equation
zbMATH Keywords: numerical examples
author: Alan J. Laub
describes a project that uses: Matlab
MaRDI profile type: Publication
title: Computational matrix analysis
review text: This booklet is not a numerical linear algebra course. It assumes that the basic notions and algorithms are known (Gaussian elimination, LU and QR factorization, least squares solution, generalized inverse, eigenvalues, singular value decomposition (SVD), etc.). It only adds to this the particular aspects of the computations when the algorithms are implemented in finite arithmetic. Obviously, the use of Matlab or the likes is necessary for a proper assimilation of the material, but it does not introduce the code either. What it does contain are the notions and the formulas to write good software, or rather to analyse existing software and understand why it is implemented as it is.NEWLINENEWLINEThe first chapters are devoted to generalities like floating point arithmetic, numerical stability of a method and ill conditioning of a problem, complexity by counting flops, and the importance of row or column oriented processing of a matrix. After that the particularities of the algorithms are analysed. Gaussian elimination and the solution of linear systems, (generalized) linear least squares (QR, Gram-Schmidt, Householder, Givens), eigenvalues (double-shift QR), and the Golub-Kahan-Reinsch SVD algorithm. For the latter more advanced algorithms, the reader is referred to the literature for further details, but the basics are given together with an analysis of the condition (perturbation analysis). Of particular interest is the statistical condition estimation, as opposed to worst case conditioning. The book ends with some applications; computing a matrix function, in particular \(e^A\), and the solution of some matrix equations (Lyapunov, Sylvester, Riccati).NEWLINENEWLINEBecause the analysis is never very deep, it should be accessible to many students from mathematics, sciences, and engineering. It stays on a basic level throughout, so there is no discussion of large-scale problems, sparsity, advanced structured matrices, etc. This lines up with the detailed step-by-step …
reviewed by: Adhemar Bultheel
description: scientific article; zbMATH DE number 6035286