Solves a complex Hermitian system of linear equations with iterative refinement.

Required Arguments

A — Complex N by N matrix containing the coefficient matrix of the Hermitian linear system.   (Input)
Only the upper triangle of A is referenced.

B — Complex vector of length N containing the right-hand side of the linear system.   (Input)

X — Complex vector of length N containing the solution to the linear system.   (Output)

Optional Arguments

N — Number of equations.   (Input)
Default: N = size (A,2).

LDA — Leading dimension of A exactly as specified in the dimension statement of the calling program.   (Input)
Default: LDA  = size (A,1).

FORTRAN 90 Interface

Generic:                              CALL LSAHF (A, B, X [,…])

Specific:                             The specific interface names are S_LSAHF and D_LSAHF.

FORTRAN 77 Interface

Single:                                CALL LSAHF (N, A, LDA, B, X)

Double:                              The double precision name is DLSAHF.

Description

Routine LSAHF solves systems of linear algebraic equations having a complex Hermitian indefinite coefficient matrix. It first uses the routine LFCHF to compute a U DUH factorization of the coefficient matrix and to estimate the condition number of the matrix. D is a block diagonal matrix with blocks of order 1 or 2 and U is a matrix composed of the product of a permutation matrix and a unit upper triangular matrix. The solution of the linear system is then found using the iterative refinement routine LFIHF.

LSAHF fails if a block in D is singular or if the iterative refinement algorithm fails to converge. These errors occur only if A is singular or very close to a singular matrix.

If the estimated condition number is greater than 1/ε (where ε is machine precision), a warning error is issued. This indicates that very small changes in A can cause very large changes in the solution x. Iterative refinement can sometimes find the solution to such a system. LSAHF solves the problem that is represented in the computer; however, this problem may differ from the problem whose solution is desired.

Comments

1.         Workspace may be explicitly provided, if desired, by use of L2AHF/DL2AHF. The reference is:

CALL L2AHF (N, A, LDA, B, X, FACT, IPVT, CWK)

The additional arguments are as follows:

FACT — Complex work vector of length N² containing information about the U DUH factorization of A on output.

IPVT — Integer work vector of length N containing the pivoting information for the factorization of A on output.

CWK — Complex work vector of length N.

2.         Informational errors

Type   Code

3           1                  The input matrix is algorithmically singular.

3           4                  The input matrix is not Hermitian. It has a diagonal entry with a small imaginary part.

4           2                  The input matrix singular.

4           4                  The input matrix is not Hermitian. It has a diagonal entry with an imaginary part.

3.         Integer Options with Chapter 11 Options Manager

16       This option uses four values to solve memory bank conflict (access inefficiency) problems. In routine L2AHF the leading dimension of FACT is increased by IVAL(3) when N is a multiple of IVAL(4). The values IVAL(3) and IVAL(4) are temporarily replaced by IVAL(1) and IVAL(2), respectively, in LSAHF. Additional memory allocation for FACT and option value restoration are done automatically in LSAHF. Users directly calling L2AHF can allocate additional space for FACT and set IVAL(3) and IVAL(4) so that memory bank conflicts no longer cause inefficiencies. There is no requirement that users change existing applications that use LSAHF or L2AHF. Default values for the option are
IVAL(*) = 1, 16, 0, 1.

17       This option has two values that determine if the L1 condition number is to be computed. Routine LSAHF temporarily replaces IVAL(2) by IVAL(1). The routine L2CHF computes the condition number if IVAL(2) = 2. Otherwise L2CHF skips this computation. LSAHF restores the option. Default values for the option are
IVAL(*) = 1, 2.

Example

A system of three linear equations is solved. The coefficient matrix has complex Hermitian form and the right-hand-side vector b has three elements.

 

      USE LSAHF_INT
      USE WRCRN_INT

!                                 Declare variables

      INTEGER    LDA, N

      PARAMETER  (LDA=3, N=3)

      COMPLEX    A(LDA,LDA), B(N), X(N)

!

!                               Set values for A and B

!

!                               A = ( 3.0+0.0i   1.0-1.0i   4.0+0.0i )

!                                   ( 1.0+1.0i   2.0+0.0i  -5.0+1.0i )

!                                   ( 4.0+0.0i  -5.0-1.0i  -2.0+0.0i )

!

!                               B = ( 7.0+32.0i -39.0-21.0i 51.0+9.0i )

!

      DATA A/(3.0,0.0), (1.0,1.0), (4.0,0.0), (1.0,-1.0), (2.0,0.0),&

            (-5.0,-1.0), (4.0,0.0), (-5.0,1.0), (-2.0,0.0)/

      DATA B/(7.0,32.0), (-39.0,-21.0), (51.0,9.0)/

!

      CALL LSAHF (A, B, X)

!                                 Print results

      CALL WRCRN ('X', X, 1, N, 1)

      END

Output

 

                        X
              1                2                3
(  2.00,  1.00)  (-10.00, -1.00)  (  3.00,  5.00)

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