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### SVDC

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  VT

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 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.