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

SGES, DGES, CGES, and ZGES--General Matrix, Its Transpose, or Its Conjugate Transpose Solve

These subroutines solve the system Ax = b for x, where A is a general matrix and x and b are vectors. Using the iopt argument, they can also solve the real system A^Tx = b or the complex system A^Hx = b for x. These subroutines use the results of the factorization of matrix A, produced by a preceding call to SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, or ZGEF, respectively.

Table 88. Data Types

Note:

The input to these solve subroutines must be the output from the factorization subroutines SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, and ZGEF, respectively.

Syntax

On Entry

a

is the factorization of matrix A, produced by a preceding call to SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, or ZGEF, respectively. Specified as: an lda by (at least) n array, containing numbers of the data type indicated in Table 88.

lda

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

n

is the order of matrix A. Specified as: a fullword integer; 0 <= n <= lda.

ipvt

is the integer vector ipvt of length n, containing the pivot indices produced by a preceding call to SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, or ZGEF, respectively. Specified as: a one-dimensional array of (at least) length n, containing fullword integers.

bx

is the vector b of length n, containing the right-hand side of the system. Specified as: a one-dimensional array of (at least) length n, containing numbers of the data type indicated in Table 88.

iopt

determines the type of computation to be performed, where:

If iopt = 0, A is used in the computation.

If iopt = 1, A^T is used in SGES and DGES. A^H is used in CGES and ZGES.

Note:

No data should be moved to form A^T or A^H; that is, the matrix A should always be stored in its untransposed form.

Specified as: a fullword integer; iopt = 0 or 1.

On Return

bx

is the solution vector x of length n, containing the results of the computation. Returned as: a one-dimensional array, containing numbers of the data type indicated in Table 88.

Notes

1. The scalar data specified for input arguments lda and n for these subroutines must be the same as the corresponding input arguments specified for SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, and ZGEF, respectively.
2. The array data specified for input arguments a and ipvt for these subroutines must be the same as the corresponding output arguments for SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, and ZGEF, respectively.
3. The vectors and matrices used in this computation must have no common elements; otherwise, results are unpredictable. See Concepts.

Function

The system Ax = b is solved for x, where A is a general matrix and x and b are vectors. Using the iopt argument, this subroutine can also solve the real system A^Tx = b or the complex system A^Hx = b for x. These subroutines use the results of the factorization of matrix A, produced by a preceding call to SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, or ZGEF, respectively. For a description of how A is factored, see SGEF, DGEF, CGEF, and ZGEF--General Matrix Factorization.

If n is 0, no computation is performed. See references [36] and [38].

Error Conditions

Computational Errors

None

Note:

If the factorization performed by SGEF, DGEF, CGEF, ZGEF, SGEFCD, DGEFCD, or DGEFP failed because a pivot element is zero, the results returned by this subroutine are unpredictable, and there may be a divide-by-zero program exception message.

Input-Argument Errors

1. lda <= 0
2. n < 0
3. n > lda
4. iopt <> 0 or 1

Example 1

Part 1

This part of the example shows how to solve the system Ax = b, where matrix A is the same matrix factored in the Example 1 for SGEF and DGEF.

Call Statement and Input

           A  LDA  N   IPVT   BX  IOPT
           |   |   |    |     |    |
CALL SGES( A , 9 , 9 , IPVT , BX , 0  )

Output

BX       =  (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)

Part 2

This part of the example shows how to solve the system A^Tx = b, where matrix A is the input matrix factored in Example 1 for SGEF and DGEF. Most of the input is the same in Part 2 as in Part 1.

Call Statement and Input

           A  LDA  N   IPVT   BX  IOPT
           |   |   |    |     |    |
CALL SGES( A , 9 , 9 , IPVT , BX , 1  )

Output

BX       =  (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)

Example 2

Part 1

This part of the example shows how to solve the system Ax = b, where matrix A is the same matrix factored in the Example 2 for CGEF and ZGEF.

Call Statement and Input

           A  LDA  N   IPVT   BX  IOPT
           |   |   |    |     |    |
CALL CGES( A , 4 , 4 , IPVT , BX , 0  )

Output

BX       =  ((9.0, 0.0), (5.0, 1.0), (1.0, 6.0), (3.0, 4.0))

Part 2

This part of the example shows how to solve the system A^Hx = b, where matrix A is the input matrix factored in Example 2 for CGEF and ZGEF. Most of the input is the same in Part 2 as in Part 1.

Call Statement and Input

           A  LDA  N   IPVT   BX  IOPT
           |   |   |    |     |    |
CALL CGES( A , 4 , 4 , IPVT , BX , 1  )

Output

BX       =  ((9.0, 0.0), (5.0, 1.0), (1.0, 6.0), (3.0, 4.0))