Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference

SGESVS and DGESVS--Linear Least Squares Solution for a General Matrix Using the Singular Value Decomposition

These subroutines compute the minimal norm linear least squares solution of AX is congruent to B, where A is a general matrix, using the singular value decomposition computed by SGESVF or DGESVF.

Table 118. Data Types

Syntax

On Entry

v

is the orthogonal matrix V of order n in the singular value decomposition of matrix A. It is produced by a preceding call to SGESVF or DGESVF, where it corresponds to output argument a.

Specified as: an ldv by (at least) n array, containing numbers of the data type indicated in Table 118.

ldv

is the leading dimension of the array specified for v. Specified as: a fullword integer; ldv > 0 and ldv >= n.

ub

is an n by nb matrix, containing U^TB. It is produced by a preceding call to SGESVF or DGESVF, where it corresponds to output argument b. On output, U^TB is overwritten; that is, the original input is not preserved.

Specified as: an ldub by (at least) nb array, containing numbers of the data type indicated in Table 118.

ldub

is the leading dimension of the array specified for ub. Specified as: a fullword integer; ldub > 0 and ldub >= n.

nb

is the number of columns in matrices X and U^TB. Specified as: a fullword integer; nb > 0.

s

is the vector s of length n, containing the singular values of matrix A. It is produced by a preceding call to SGESVF or DGESVF, where it corresponds to output argument s.

Specified as: a one-dimensional array of (at least) length n, containing numbers of the data type indicated in Table 118; s_i >= 0.

x

See On Return.

ldx

is the leading dimension of the array specified for x. Specified as: a fullword integer; ldx > 0 and ldx >= n.

m

is the number of rows in matrix A. Specified as: a fullword integer; m >= 0.

n

is the number of columns in matrix A, the order of matrix V, the number of elements in vector s, the number of rows in matrix UB, and the number of rows in matrix X. Specified as: a fullword integer; n >= 0.

tau

is the error tolerance tau. Any singular values in vector s that are less than tau are treated as zeros when computing matrix X. Specified as: a number of the data type indicated in Table 118; tau >= 0. For more information on the values for tau, see Notes.

On Return

x

is an n by nb matrix, containing the minimal norm linear least solutions of AX is congruent to B. The nb column vectors of X contain minimal norm solution vectors for nb distinct linear least squares problems.

Returned as: an ldx by (at least) nb array, containing numbers of the data type indicated in Table 118.

Notes

1. V, X, s, and U^TB can have no common elements; otherwise the results are unpredictable.
2. In problems involving experimental data, tau should reflect the absolute accuracy of the matrix elements:

   tau >= max(|DELTA_ij|)

   where DELTA_ij are the errors in a_ij. In problems where the matrix elements are known exactly or are only affected by roundoff errors:

   
   
   
   

   where:

   epsilon is equal to 0.11920E-06 for SGESVS and 0.22204D-15 for DGESVS.

   s is a vector containing the singular values of matrix A.

   For more information, see references [13], [58], [78], and pages 134 to 151 in reference [99].

Function

The minimal norm linear least squares solution of AX is congruent to B, where A is a real general matrix, is computed using the singular value decomposition, produced by a preceding call to SGESVF or DGESVF. From SGESVF or DGESVF, the singular value decomposition of A is given by the following:

A = USigmaV^T

The linear least squares of solution X, for AX is congruent to B, is given by the following formula:

X = VSigma⁺U^TB

where:

If m or n is equal to 0, no computation is performed. See references [13], [58], [78], and pages 134 to 151 in reference [99]. These algorithms have a tendency to generate underflows that may hurt overall performance. The system default is to mask underflow, which improves the performance of these subroutines.

Error Conditions

Computational Errors

None

Input-Argument Errors

1. ldv <= 0
2. n > ldv
3. ldub <= 0
4. n > ldub
5. ldx <= 0
6. n > ldx
7. nb <= 0
8. m < 0
9. n < 0
10. tau < 0

Example 1

This example finds the linear least squares solution for the underdetermined system AX is congruent to B, using the singular value decomposition computed by DGESVF. Matrix A is:

                       *               *
                       | 1.0  2.0  2.0 |
                       | 2.0  4.0  5.0 |
                       *               *

and matrix B is:

                           *     *
                           | 1.0 |
                           | 4.0 |
                           *     *

On output, matrix U^TB is overwritten.

Note:

This example corresponds to Example 3 of DGESVF on page "Example 3".

Call Statement and Input

             V  LDV  UB  LDUB  NB  S   X  LDX  M   N   TAU
             |   |   |    |    |   |   |   |   |   |    |
CALL DGESVS( V , 3 , UB , 3  , 1 , S , X , 3 , 2 , 3 , TAU )

        *                        *
        | -0.304  -0.894   0.328 |
V    =  | -0.608   0.447   0.656 |
        | -0.733   0.000  -0.680 |
        *                        *

        *        *
        | -4.061 |
UB   =  |  0.000 |
        | -0.714 |
        *        *
 
S        =  (7.342, 0.000, 0.305)
TAU      =  0.3993D-14

Output

        *        *
        | -0.600 |
X    =  | -1.200 |
        |  2.000 |
        *        *

Example 2

This example finds the linear least squares solution for the overdetermined system AX is congruent to B, using the singular value decomposition computed by DGESVF. Matrix A is:

                          *          *
                          | 1.0  4.0 |
                          | 2.0  5.0 |
                          | 3.0  6.0 |
                          *          *

and where B is:

                          *           *
                          | 7.0  10.0 |
                          | 8.0  11.0 |
                          | 9.0  12.0 |
                          *           *

On output, matrix U^TB is overwritten.

Note:

This example corresponds to Example 4 of DGESVF on page "Example 4".

Call Statement

             V  LDV  UB  LDUB  NB  S   X  LDX  M   N   TAU
             |   |   |    |    |   |   |   |   |   |    |
CALL DGESVS( V , 3 , UB , 3  , 2 , S , X , 2 , 3 , 2 , TAU )

Input

        *                *
        |  0.922  -0.386 |
V    =  | -0.386  -0.922 |
        |   .       .    |
        *                *

        *                  *
        |  -1.310   -2.321 |
UB   =  | -13.867  -18.963 |
        |    .        .    |
        *                  *
 
S        =  (0.773, 9.508)
TAU      =  0.5171D-14

Output

        *                *
X    =  | -1.000  -2.000 |
        |  2.000   3.000 |
        *                *