NAME

SYNOPSIS

  SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIVOT, B, 
 *      LDB, X, LDX, RCOND, FERR, BERR, WORK, LDWORK, WORK2, INFO)
  CHARACTER * 1 FACT, UPLO
  INTEGER N, NRHS, LDA, LDAF, LDB, LDX, LDWORK, INFO
  INTEGER IPIVOT(*), WORK2(*)
  DOUBLE PRECISION RCOND
  DOUBLE PRECISION A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)
 
  SUBROUTINE DSYSVX_64( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIVOT, 
 *      B, LDB, X, LDX, RCOND, FERR, BERR, WORK, LDWORK, WORK2, INFO)
  CHARACTER * 1 FACT, UPLO
  INTEGER*8 N, NRHS, LDA, LDAF, LDB, LDX, LDWORK, INFO
  INTEGER*8 IPIVOT(*), WORK2(*)
  DOUBLE PRECISION RCOND
  DOUBLE PRECISION A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)
 

F95 INTERFACE

  SUBROUTINE SYSVX( FACT, UPLO, N, NRHS, A, [LDA], AF, [LDAF], IPIVOT, 
 *       B, [LDB], X, [LDX], RCOND, FERR, BERR, [WORK], [LDWORK], [WORK2], 
 *       [INFO])
  CHARACTER(LEN=1) :: FACT, UPLO
  INTEGER :: N, NRHS, LDA, LDAF, LDB, LDX, LDWORK, INFO
  INTEGER, DIMENSION(:) :: IPIVOT, WORK2
  REAL(8) :: RCOND
  REAL(8), DIMENSION(:) :: FERR, BERR, WORK
  REAL(8), DIMENSION(:,:) :: A, AF, B, X
 
  SUBROUTINE SYSVX_64( FACT, UPLO, N, NRHS, A, [LDA], AF, [LDAF], 
 *       IPIVOT, B, [LDB], X, [LDX], RCOND, FERR, BERR, [WORK], [LDWORK], 
 *       [WORK2], [INFO])
  CHARACTER(LEN=1) :: FACT, UPLO
  INTEGER(8) :: N, NRHS, LDA, LDAF, LDB, LDX, LDWORK, INFO
  INTEGER(8), DIMENSION(:) :: IPIVOT, WORK2
  REAL(8) :: RCOND
  REAL(8), DIMENSION(:) :: FERR, BERR, WORK
  REAL(8), DIMENSION(:,:) :: A, AF, B, X
 

C INTERFACE

void dsysvx(char fact, char uplo, int n, int nrhs, double *a, int lda, double *af, int ldaf, int *ipivot, double *b, int ldb, double *x, int ldx, double *rcond, double *ferr, double *berr, int *info);

void dsysvx_64(char fact, char uplo, long n, long nrhs, double *a, long lda, double *af, long ldaf, long *ipivot, double *b, long ldb, double *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:

1. If FACT = 'N', the diagonal pivoting method is used to factor A. The form of the factorization is

      A = U * D * U**T,  if UPLO = 'U', or
      A = L * D * L**T,  if UPLO = 'L',
   where U (or L) is a product of permutation and unit upper (lower)
   triangular matrices, and D is symmetric and block diagonal with
   1-by-1 and 2-by-2 diagonal blocks.

2. If some D(i,i)=0, so that D 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.

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

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

ARGUMENTS

* FACT (input)

Specifies whether or not the factored form of A has been supplied on entry.

* UPLO (input)

* N (input)

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

The symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

* LDA (input)

The leading dimension of the array A. LDA >= max(1,N).

* AF (input/output)

If FACT = 'F', then AF is an input argument and on entry contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by SSYTRF.

If FACT = 'N', then AF is an output argument and on exit returns the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T.

* LDAF (input)

The leading dimension of the array AF. LDAF >= max(1,N).

* IPIVOT (input)

If FACT = 'F', then IPIVOT is an input argument and on entry contains details of the interchanges and the block structure of D, as determined by SSYTRF. If IPIVOT(k) > 0, then rows and columns k and IPIVOT(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIVOT(k) = IPIVOT(k-1) < 0, then rows and columns k-1 and -IPIVOT(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIVOT(k) = IPIVOT(k+1) < 0, then rows and columns k+1 and -IPIVOT(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

If FACT = 'N', then IPIVOT is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by SSYTRF.

* B (input)

The N-by-NRHS right hand side matrix 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.

* 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. 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)

On exit, if INFO = 0, WORK(1) returns the optimal LDWORK.

* LDWORK (input)

The length of WORK. LDWORK >= 3*N, and for best performance LDWORK >= N*NB, where NB is the optimal blocksize for SSYTRF.

If LDWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LDWORK is issued by XERBLA.

* WORK2 (workspace)

* INFO (output)