Overview

This is an advanced textbook based on lectures given at the Moscow Physico-Technical Institute. The lectures are characterized by brevity, logical organization, and occasionally a lighthearted approach. It aims to involve the reader by asking questions, hinting, giving recommendations, comparing different methods, and discussing optomistic and pessemistic approaches to numerical analysis. Since matrix analysis underlies numerical analysis, this text emphasizes methods and algorithms of matrix analysis. Function approximations, methods of solving nonlinear equations, and minimization methods are also considered. As well as considering classical methods, new methods and approaches are discussed - such as spectral distribution theory and what it offers for design and proof of modern preconditioning strategies for large-scale linear algebra problems.

Full Product Details

Author:   Eugene E. Tyrtyshnikov
Publisher:   Birkhauser Boston Inc
Imprint:   Birkhauser Boston Inc
Weight:   0.440kg
ISBN:  

9780817639167

ISBN 10:   0817639160
Pages:   205
Publication Date:   01 July 1997
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   Awaiting stock  
The supplier is currently out of stock of this item. It will be ordered for you and placed on backorder. Once it does come back in stock, we will ship it out for you.

Table of Contents

Lecture 1: metric space; some useful definitions; nested balls; normed space; popular vector norms; matrix norms; equivalent norms; operator norms. Lecture 2: scalar product; length of a vector; isometric matricies; preservation of length and unitary matricies; Schur theorum; normal matricies; positive definite matricies; the singular value decomposition; unitarily invariant norms; a short way to the SVD; approximations of a lower rank; smoothness and ranks. Lecture 3: perturbation theory; condition of a matrix; convergent matricies and series; the simplest iteration method; inverses and series; condition of a linear system; consistency of matrix and right-hand side; eigenvalue perturbations; continuity of the polynomial roots. Lecture 4: diagonal dominance; Gerschgorin disks; small perturbations of eigen values and vectors; condition of a simple eigenvalue; analitic perturbations. Lecture 5: spectral distances; symmetric theorums; Hoffman-Wielandt theorum; permutation vector of a matrix; unnormal extension; eigenvalues of Hermitian matrices; interlacing properties; what are clusters?; singular value clusters; eigenvalue clusters. Lecture 6: floating-point numbers; computer arithmetic axioms; round-off errors for the scalar product; forward and backward analysis; some philosophy; an example of bad operation; one more example; ideal and machine tests; up or down; solving the triangular systems. Lecture 7: direct methods for linear systems; theory of the LU decomposition; round-off errors for the LU decomposition; growth of matrix entries and pivoting; complete pivoting; the Cholesky method; triangular decompositions and linear systems solution; how to refine the solution. Lecture 8: the QR decomposition of a square matrix; the QR decomposition of a rectangular matrix; householder matrices; elimination of elements by reflections; Givens matricies; elimination of elements by rotations; computer realizations of reflections and rotations; orthgonalization method; loss of orthogonality; modified Gram-Schmidt algorithm; bidiagonalization; unitary similarity reduction to the Hessenberg form. Lecture 9: the eigenvalue problem; the power method; subspace iterations; distances between subspaces; subspaces and orthoprojectors; distances and orthoprojectors; subspaces of equal dimension; the CS decomposition; convergence of subspace iterations for the block diagonal matrix; convergance of subspace iterations in the general case. Lecture 10: the QR algorithm; generalised QR algorithm; basic formulas; the QR iteration lemma; convergance of the QR iterations; pessimistric and optimistic; Bruhat decomposition; what if the inverse matrix is not strongly regular; the QR iterations and the subspace iterations. Lecture 11: quadratic convergence; cubic convergence; what makes the QR algorithm efficient; implicit QR iterations; arrangement of computations; how to find the singular value decomposition. Lecture 12: function approximation; (Part contents)

Reviews

The title of this book reflects its structure and style with great exactness... The presentation of the material is very clear and supported by properly chosen exercises of good didactic value added to each lecture. - Zentralblatt Math [The book] is a short and elegant coupling of simplicity and very deep resultsa ][It] can be used both as an introduction and a refinement to a university course. Students [in] mathematics and physics but also other advanced readers and experienced researchers find both the classical fundamental results and new visions and ideas while reading this nice book. - ZAA

The title of this book reflects its structure and style with great exactness... The presentation of the material is very clear and supported by properly chosen exercises of good didactic value added to each lecture. <p>- Zentralblatt Math <p> [The book] is a short and elegant coupling of simplicity and very deep resultsa ][It] can be used both as an introduction and a refinement to a university course. Students [in] mathematics and physics but also other advanced readers and experienced researchers find both the classical fundamental results and new visions and ideas while reading this nice book. <p>- ZAA

Author Information

Tab Content 6

Author Website:  

Customer Reviews

Recent Reviews

No review item found!

Add your own review!

Countries Available

All regions