SVD Procedure

Performs a singular value decomposition on a matrix.

SVD, a, W [, u [, v]]

a The two-dimensional matrix to be decomposed. It has m rows and n columns, and m must be greater than or equal to n.

W An n-element vector of "singular values" equal to the diagonal elements of w: w_{i, j} = W_j, w_{i, j} = 0 for i j.

u (optional) The m-by-n column matrix of the decomposition of a.

v (optional) The n-by-n orthogonal matrix of the decomposition of a.

None.

SVD performs a singular value decomposition on a matrix a. It is useful for solving linear least-squares equations. SVD is able to deal with matrices that are singular or very close to singular.

The SVD procedure function transforms an m-by-n matrix a to the product of an m-by-n column orthogonal matrix u, an n-by-n diagonal matrix w, and the transpose of an n-by-n orthogonal matrix v. In other words, u, w, and v are matrices that are calculated by SVD.

By definition, the product of these three matrices (u, w, v) is equal to a:

a = uwv^t

where the superscript t denotes the matrix transpose.

In a linear system of equations,

Ax = b

where there are at least as many equations as unknowns, the least-squares solution vector x is given by:

x = v . [diag(1/w_j)](u^t . b)

A PV-WAVE function that returns x, given A and b is:

Function SVD_SOLVE, A, b

; Return the vector x, the solution of the system of equations Ax = b.

; A is an (m, n) matrix, where m  n.

SVD, A, w, u, v

; Call SVD to decompose A.

n = N_ELEMENTS(w)

; Make the diagonal matrix Wi,i = 1/wi.

wp = FLTARR(n,n)
FOR i=0, n-1 DO IF w(i) NE 0 THEN wp(i,i) = 1./w(i) 
RETURN, v # wp # (TRANSPOSE(u) # b)

; Calculate x from equation and return.

END

SVBKSB

The SVD function is based on the subroutine SVDCMP in Numerical Recipes in C: The Art of Scientific Computing, by Flannery, Press, Teukolsky, and Vetterling, Cambridge University Press, Cambridge, MA, 1988, and is used by permission.

PV-WAVE's SVD is a translation of the ALGOL procedure SVD described in Linear Algebra, Vol. II of The Handbook for Automatic Computation, by Wilkinson and Reinsch, Springer-Verlag, New York, NY, 1971.

For more information on SVD, also consult Computer Models for Mathematical Computations, by Forsythe, Malcom, and Moler, Prentice Hall, Englewood Cliffs, NJ, 1977.