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(New York:McGraw-Hill), §9.5, p. 437.Pan, V., and Reif, J. 1985, in Proceedings of the Seventeenth Annual ACM Symposium onTheory of Computing (New York: Association for Computing Machinery). [1]Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited.

To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).If we let equation (2.5.7) act on some arbitrary right-hand side b, as one wants a matrix inverseto do, it is obvious that a sufficient condition for convergence is592.6 Singular Value Decomposition2.6 Singular Value Decomposition     =     AU w1 ·w2  ·······VTwN(2.6.1)The matrices U and V are each orthogonal in the sense that their columns areorthonormal,MXUik Uin = δkn1≤k≤N1≤n≤N(2.6.2)Vjk Vjn = δkn1≤k≤N1≤n≤N(2.6.3)i=1NXj=1Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited.

To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).There exists a very powerful set of techniques for dealing with sets of equationsor matrices that are either singular or else numerically very close to singular. In manycases where Gaussian elimination and LU decomposition fail to give satisfactoryresults, this set of techniques, known as singular value decomposition, or SVD,will diagnose for you precisely what the problem is.

In some cases, SVD willnot only diagnose the problem, it will also solve it, in the sense of giving you auseful numerical answer, although, as we shall see, not necessarily “the” answerthat you thought you should get.SVD is also the method of choice for solving most linear least-squares problems.We will outline the relevant theory in this section, but defer detailed discussion ofthe use of SVD in this application to Chapter 15, whose subject is the parametricmodeling of data.SVD methods are based on the following theorem of linear algebra, whose proofis beyond our scope: Any M × N matrix A whose number of rows M is greater thanor equal to its number of columns N , can be written as the product of an M × Ncolumn-orthogonal matrix U, an N × N diagonal matrix W with positive or zeroelements (the singular values), and the transpose of an N × N orthogonal matrix V.The various shapes of these matrices will be made clearer by the following tableau:.

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