SVDC

The SVDC procedure computes the Singular Value Decomposition (SVD) of a square (n x n) or non-square (n x m) array as the product of orthogonal and diagonal arrays. SVD is a very powerful tool for the solution of linear systems, and is often used when a solution cannot be determined by other numerical algorithms.

The SVD of an (n x m) non-square array A is computed as the product of an (n x m) column orthogonal array U, an (n x n) diagonal array SV, composed of the singular values, and the transpose of an (n x n) orthogonal array V: A = U SV V^T

Note: If you are working with complex inputs, use the LA_SVD procedure instead.

Examples

To find the singular values of an array A:

; Define the array A:
A = [[1.0, 2.0, -1.0, 2.5], $
     [1.5, 3.3, -0.5, 2.0], $
     [3.1, 0.7,  2.2, 0.0], $
     [0.0, 0.3, -2.0, 5.3], $
     [2.1, 1.0,  4.3, 2.2], $
     [0.0, 5.5,  3.8, 0.2]]
; Compute the Singular Value Decomposition:
SVDC, A, W, U, V
; Print the singular values:
PRINT, W

IDL prints:

8.81973      2.65502      4.30598      6.84484

To verify the decomposition, use the relationship A = U ## SV ## TRANSPOSE(V), where SV is a diagonal array created from the output vector W:

sv = FLTARR(4, 4)
FOR K = 0, 3 DO sv[K,K] = W[K]
result = U ## sv ## TRANSPOSE(V)
PRINT, result

IDL prints:

      1.00000      2.00000     -1.00000      2.50000

      1.50000      3.30000    -0.500001      2.00000

      3.10000     0.700000      2.20000      0.00000

  2.23517e-08     0.300000     -2.00000      5.30000

      2.10000     0.999999      4.30000      2.20000

 -3.91155e-07      5.50000      3.80000     0.200000

This is the input array, to within machine precision.

Syntax

SVDC, A, W, U, V [, /COLUMN] [, /DOUBLE] [, ITMAX=value]

Arguments

A

The square (n x n) or non-square (n x m) single- or double-precision floating-point array to decompose.

W

On output, W is an n-element output vector containing the “singular values.”

U

On output, U is an n-column, m-row orthogonal array used in the decomposition of A.

V

On output, V is an n-column, n-row orthogonal array used in the decomposition of A.

Keywords

COLUMN

Set this keyword if the input array A is in column-major format (composed of column vectors) rather than in row-major format (composed of row vectors).

DOUBLE

Set this keyword to force the computation to be done in double-precision arithmetic.

ITMAX

Set this keyword to specify the maximum number of iterations. The default value is 30.

Version History

4.0	Introduced

Resources and References

SVDC is based on the routine svdcmp described in section 2.6 of Numerical Recipes in C: The Art of Scientific Computing (Second Edition), published by Cambridge University Press, and is used by permission.