NAME

SYNOPSIS

  SUBROUTINE ZGBSVX( FACT, TRANSA, N, NSUB, NSUPER, NRHS, A, LDA, AF, 
 *      LDAF, IPIVOT, EQUED, ROWSC, COLSC, B, LDB, X, LDX, RCOND, FERR, 
 *      BERR, WORK, WORK2, INFO)
  CHARACTER * 1 FACT, TRANSA, EQUED
  DOUBLE COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
  INTEGER N, NSUB, NSUPER, NRHS, LDA, LDAF, LDB, LDX, INFO
  INTEGER IPIVOT(*)
  DOUBLE PRECISION RCOND
  DOUBLE PRECISION ROWSC(*), COLSC(*), FERR(*), BERR(*), WORK2(*)
 
  SUBROUTINE ZGBSVX_64( FACT, TRANSA, N, NSUB, NSUPER, NRHS, A, LDA, 
 *      AF, LDAF, IPIVOT, EQUED, ROWSC, COLSC, B, LDB, X, LDX, RCOND, 
 *      FERR, BERR, WORK, WORK2, INFO)
  CHARACTER * 1 FACT, TRANSA, EQUED
  DOUBLE COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
  INTEGER*8 N, NSUB, NSUPER, NRHS, LDA, LDAF, LDB, LDX, INFO
  INTEGER*8 IPIVOT(*)
  DOUBLE PRECISION RCOND
  DOUBLE PRECISION ROWSC(*), COLSC(*), FERR(*), BERR(*), WORK2(*)
 

F95 INTERFACE

  SUBROUTINE GBSVX( FACT, [TRANSA], [N], NSUB, NSUPER, [NRHS], A, [LDA], 
 *       AF, [LDAF], IPIVOT, EQUED, ROWSC, COLSC, B, [LDB], X, [LDX], 
 *       RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
  CHARACTER(LEN=1) :: FACT, TRANSA, EQUED
  COMPLEX(8), DIMENSION(:) :: WORK
  COMPLEX(8), DIMENSION(:,:) :: A, AF, B, X
  INTEGER :: N, NSUB, NSUPER, NRHS, LDA, LDAF, LDB, LDX, INFO
  INTEGER, DIMENSION(:) :: IPIVOT
  REAL(8) :: RCOND
  REAL(8), DIMENSION(:) :: ROWSC, COLSC, FERR, BERR, WORK2
 
  SUBROUTINE GBSVX_64( FACT, [TRANSA], [N], NSUB, NSUPER, [NRHS], A, 
 *       [LDA], AF, [LDAF], IPIVOT, EQUED, ROWSC, COLSC, B, [LDB], X, [LDX], 
 *       RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
  CHARACTER(LEN=1) :: FACT, TRANSA, EQUED
  COMPLEX(8), DIMENSION(:) :: WORK
  COMPLEX(8), DIMENSION(:,:) :: A, AF, B, X
  INTEGER(8) :: N, NSUB, NSUPER, NRHS, LDA, LDAF, LDB, LDX, INFO
  INTEGER(8), DIMENSION(:) :: IPIVOT
  REAL(8) :: RCOND
  REAL(8), DIMENSION(:) :: ROWSC, COLSC, FERR, BERR, WORK2
 

C INTERFACE

void zgbsvx(char fact, char transa, int n, int nsub, int nsuper, int nrhs, doublecomplex *a, int lda, doublecomplex *af, int ldaf, int *ipivot, char equed, double *rowsc, double *colsc, doublecomplex *b, int ldb, doublecomplex *x, int ldx, double *rcond, double *ferr, double *berr, int *info);

void zgbsvx_64(char fact, char transa, long n, long nsub, long nsuper, long nrhs, doublecomplex *a, long lda, doublecomplex *af, long ldaf, long *ipivot, char equed, double *rowsc, double *colsc, doublecomplex *b, long ldb, doublecomplex *x, long ldx, double *rcond, double *ferr, double *berr, long *info);

PURPOSE

Error bounds on the solution and a condition estimate are also provided.

The following steps are performed by this subroutine:

1. If FACT = 'E', real scaling factors are computed to equilibrate the system:

      TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
      TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
      TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
   Whether or not the system will be equilibrated depends on the
   scaling of the matrix A, but if equilibration is used, A is
   overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
   or diag(C)*B (if TRANS = 'T' or 'C').

2. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as

      A = L * U,
   where L is a product of permutation and unit lower triangular
   matrices with KL subdiagonals, and U is upper triangular with
   KL+KU superdiagonals.

3. If some U(i,i)=0, so that U is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.

4. The system of equations is solved for X using the factored form of A.

5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

6. If equilibration was used, the matrix X is premultiplied by diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so that it solves the original system before equilibration.

ARGUMENTS

* FACT (input)

Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored.

* TRANSA (input)

Specifies the form of the system of equations.

* N (input)

The number of linear equations, i.e., the order of the matrix A. N >= 0.

* NSUB (input)

The number of subdiagonals within the band of A. NSUB >= 0.

* NSUPER (input)

The number of superdiagonals within the band of A. NSUPER >= 0.

* NRHS (input)

The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.

* A (input/output)

On entry, the matrix A in band storage, in rows 1 to NSUB+NSUPER+1. The j-th column of A is stored in the j-th column of the array A as follows: A(NSUPER+1+i-j,j) = A(i,j) for max(1,j-NSUPER)<=i<=min(N,j+kl)

If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in ROWSC and/or COLSC. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A is scaled as follows: EQUED = 'ROWSC': A := diag(ROWSC) * A

EQUED = 'COLSC': A := A * diag(COLSC)

EQUED = 'B': A := diag(ROWSC) * A * diag(COLSC).

* LDA (input)

The leading dimension of the array A. LDA >= NSUB+NSUPER+1.

* AF (input/output)

If FACT = 'F', then AF is an input argument and on entry contains details of the LU factorization of the band matrix A, as computed by CGBTRF. U is stored as an upper triangular band matrix with NSUB+NSUPER superdiagonals in rows 1 to NSUB+NSUPER+1, and the multipliers used during the factorization are stored in rows NSUB+NSUPER+2 to 2*NSUB+NSUPER+1. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix A.

If FACT = 'N', then AF is an output argument and on exit returns details of the LU factorization of A.

If FACT = 'E', then AF is an output argument and on exit returns details of the LU factorization of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix).

* LDAF (input)

The leading dimension of the array AF. LDAF >= 2*NSUB+NSUPER+1.

* IPIVOT (input)

If FACT = 'F', then IPIVOT is an input argument and on entry contains the pivot indices from the factorization A = L*U as computed by CGBTRF; row i of the matrix was interchanged with row IPIVOT(i).

If FACT = 'N', then IPIVOT is an output argument and on exit contains the pivot indices from the factorization A = L*U of the original matrix A.

If FACT = 'E', then IPIVOT is an output argument and on exit contains the pivot indices from the factorization A = L*U of the equilibrated matrix A.

* EQUED (input)

Specifies the form of equilibration that was done.

* ROWSC (input/output)

The row scale factors for A. If EQUED = 'ROWSC' or 'B', A is multiplied on the left by diag(ROWSC); if EQUED = 'N' or 'COLSC', ROWSC is not accessed. ROWSC is an input argument if FACT = 'F'; otherwise, ROWSC is an output argument. If FACT = 'F' and EQUED = 'ROWSC' or 'B', each element of ROWSC must be positive.

* COLSC (input/output)

The column scale factors for A. If EQUED = 'COLSC' or 'B', A is multiplied on the right by diag(COLSC); if EQUED = 'N' or 'ROWSC', COLSC is not accessed. COLSC is an input argument if FACT = 'F'; otherwise, COLSC is an output argument. If FACT = 'F' and EQUED = 'COLSC' or 'B', each element of COLSC must be positive.

* B (input/output)

On entry, the right hand side matrix B. On exit, if EQUED = 'N', B is not modified; if TRANSA = 'N' and EQUED = 'ROWSC' or 'B', B is overwritten by diag(ROWSC)*B; if TRANSA = 'T' or 'COLSC' and EQUED = 'COLSC' or 'B', B is overwritten by diag(COLSC)*B.

* LDB (input)

The leading dimension of the array B. LDB >= max(1,N).

* X (output)

If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to the original system of equations. Note that A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(COLSC))*X if TRANSA = 'N' and EQUED = 'COLSC' or 'B', or inv(diag(ROWSC))*X if TRANSA = 'T' or 'COLSC' and EQUED = 'ROWSC' or 'B'.

* LDX (input)

The leading dimension of the array X. LDX >= max(1,N).

* RCOND (output)

The estimate of the reciprocal condition number of the matrix A after equilibration (if done). If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0.

* FERR (output)

The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.

* BERR (output)

The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).

* WORK (workspace)

dimension(2*N)

* WORK2 (workspace)

On exit, WORK2(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The ``max absolute element'' norm is used. If WORK2(1) is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, condition estimator RCOND, and forward error bound FERR could be unreliable. If factorization fails with 0<INFO<=N, then WORK2(1) contains the reciprocal pivot growth factor for the leading INFO columns of A.

* INFO (output)