IDL Analyst Reference Guide: Linear Systems

IMSL_SVDCOMP

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

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

If present and nonzero, double precision is used.

INVERSE

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

RANK

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

TOL_RANK

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 s_i,i is considered negligible if s_i,i £ TOL_RANK. If TOL_RANK < 0, then a singular value s_i,i is considered negligible if s_i,i £ TOL_RANK * ||A||_infinity.

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

U

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

V

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

Discussion

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 s_k,k £ TOL_RANK or s_k,k £ TOL_RANK * ||A||_infinity.

Since s_{i + 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 = USV^T, its generalized inverse is A⁺ = VS⁺U^T. 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  

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  
   -0.380476    0.119671    0.439083   -0.565399  
   -0.403754    0.345111   -0.056576    0.214776  
   -0.545120    0.429265    0.051392    0.432144  
   -0.264784   -0.068320   -0.883861   -0.215254  
   -0.446310   -0.816828    0.141900    0.321270  
   -0.354629   -0.102147   -0.004318   -0.545800  
PM, v, Title = 'Right singular vectors, V', Format = '(4f12.6)'  
Right singular vectors, V  
   -0.444294    0.555531   -0.435379    0.551754  
   -0.558067   -0.654299    0.277457    0.428336  
   -0.324386   -0.351361   -0.732099   -0.485129  
   -0.621239    0.373931    0.444402   -0.526066