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### IMSL_SVDCOMP

IMSL_SVDCOMP

The IMSL_SVDCOMP function computes the singular value decomposition (SVD), A = USVT, of a real or complex rectangular matrix A. An estimate of the rank of A also can be computed.

The IMSL_SVDCOMP function computes the singular value decomposition of a real or complex matrix A. It reduces the matrix A to a bidiagonal matrix B by pre- and post-multiplying Householder transformations, then, it computes singular value decomposition of B using the implicit-shifted QR algorithm. An estimate of the rank of the matrix A is obtained by finding the smallest integer k such that:

sk,k ≤ TOL_RANK or sk,k ≤ TOL_RANK * ||A||infinity

Since si + 1, i + 1 ≤ s i,i , it follows that all the s i,i satisfy the same inequality for i = k, ..., min(m, n) – 2. The rank is set to the value k. If A = USVT, its generalized inverse is A+ = VS+UT. Here, S+ = diag (s–1 0,0,..., s–1 i,i, 0, ..., 0). Only singular values that are not negligible are reciprocated. If the keyword INVERSE is specified, the function first computes the singular value decomposition of the matrix A, then computes the generalized inverse. The IMSL_SVDCOMP function fails if the QR algorithm does not converge after 30 iterations.

## Examples

### Example 1

This example computes the singular values of a 6-by-4 real matrix.

`RM, a, 6, 4`
` `
`; Define the matrix.`
`row 0: 1 2 1 4`
`row 1: 3 2 1 3`
`row 2: 4 3 1 4`
`row 3: 2 1 3 1`
`row 4: 1 5 2 2`
`row 5: 1 2 2 3`
` `
`; Call IMSL_SVDCOMP and output the results.`
`singvals = IMSL_SVDCOMP(a)`
`PM, singvals`
`  11.4850`
`  3.26975`
`  2.65336`
`  2.08873`

### Example 2

This example computes the singular value decomposition of the 6-by-4 real matrix A. Matrices U and V are returned using keywords U and V.

`RM, a, 6, 4`
` `
`; Define the matrix.`
`row 0: 1 2 1 4`
`row 1: 3 2 1 3`
`row 2: 4 3 1 4`
`row 3: 2 1 3 1`
`row 4: 1 5 2 2`
`row 5: 1 2 2 3`
` `
`; Call IMSL_SVDCOMP with keywords U and V and output the results.`
`singvals = IMSL_SVDCOMP(a, U = u, V = v)`
`PM, singvals, Title = 'Singular values', Format = '(f12.6)'`
`Singular values`
`  11.485018`
`  3.269752`
`  2.653356`
`  2.088730`
`PM, u, Title = 'Left singular vectors, U', Format = '(4f12.6)'`
`Left singular vectors, U`
`-0.3804755863D, -0.1196709926D, -0.4390828244D,  0.5653995859D`
`-0.4037537132D, -0.3451108371D,  0.0565761853D, -0.2147755765D`
`-0.5451204862D, -0.4292648935D, -0.0513926928D, -0.4321441628D`
`-0.2647842940D,  0.0683195253D,  0.8838608674D,  0.2152536982D`
`-0.4463101123D,  0.8168276233D, -0.1418996751D, -0.3212695843D`
`-0.3546286566D,  0.1021473992D,  0.0043184440D,  0.5458002219D`
`PM, v, Title = 'Right singular vectors, V', Format = '(4f12.6)'`
`Right singular vectors, V`
`-0.4442941288D, -0.5555312578D,  0.4353789667D, -0.5517543874D`
`-0.5580672382D,  0.6542987401D, -0.2774569005D, -0.4283360652D`
`-0.3243861032D,  0.3513606456D,  0.7320995334D,  0.4851284633D`
`-0.6212385538D, -0.3739303104D, -0.4444019542D,  0.5260662366`

## Errors

### Warning Errors

MATH_SLOWCONVERGENT_MATRIX: Convergence cannot be reached after 30 iterations.

## Syntax

Result = IMSL_SVDCOMP(A [, /DOUBLE] [, INVERSE=variable] [, RANK=variable] [, TOL_RANK=variable] [, U=variable] [, V=variable])

## Return Value

One-dimensional array containing ordered singular values of A.

## Arguments

### A

Two-dimensional matrix containing the coefficient matrix. Element A (i, j) contains the j-th coefficient of the i-th equation.

## Keywords

### DOUBLE (optional)

If present and nonzero, double precision is used.

### INVERSE (optional)

Named variable into which the generalized inverse of the matrix A is stored.

### RANK (optional)

Named variable into which an estimate of the rank of A is stored.

### TOL_RANK (optional)

Named variable containing the tolerance used to determine when a singular value is negligible and replaced by the value zero. If TOL_RANK > 0, then a singular value si,i is considered negligible if si,i ≤ TOL_RANK. If TOL_RANK < 0, then a singular value si,i is considered negligible if si,i ≤ TOL_RANK * ||A||infinity.

In this case, |TOL_RANK| should be an estimate of relative error or uncertainty in the data.

### U (optional)

Named variable into which the left-singular vectors of A are stored.

### V (optional)

Named variable into which the right-singular vectors of A are stored.

## Version History

 6.4 Introduced