SVD Procedure

Performs a singular value decomposition on a matrix.

Usage

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

Input Parameters

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.

Output Parameters

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 ¹ 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.

Keywords

None.

Discussion

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)

Example

The following PV‑WAVE function returns x, given A and b. A is an (m, n) matrix, where m ³ n.

Function SVD_SOLVE, A, b

   ; Call SVD to decompose A.

   SVD, A, w, u, v

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

   n = N_ELEMENTS(w)

   wp = FLTARR(n,n)

   ; Calculate x from equation and return.

   FOR i=0L, n-1 DO IF w(i) NE 0 THEN wp(i,i) = 1./w(i)

      RETURN, v # wp # (TRANSPOSE(u) # b)

END