NAME

SYNOPSIS

  SUBROUTINE ZSPSVX( FACT, UPLO, N, NRHS, A, AF, IPIVOT, B, LDB, X, 
 *      LDX, RCOND, FERR, BERR, WORK, WORK2, INFO)
  CHARACTER * 1 FACT, UPLO
  DOUBLE COMPLEX A(*), AF(*), B(LDB,*), X(LDX,*), WORK(*)
  INTEGER N, NRHS, LDB, LDX, INFO
  INTEGER IPIVOT(*)
  DOUBLE PRECISION RCOND
  DOUBLE PRECISION FERR(*), BERR(*), WORK2(*)
 
  SUBROUTINE ZSPSVX_64( FACT, UPLO, N, NRHS, A, AF, IPIVOT, B, LDB, X, 
 *      LDX, RCOND, FERR, BERR, WORK, WORK2, INFO)
  CHARACTER * 1 FACT, UPLO
  DOUBLE COMPLEX A(*), AF(*), B(LDB,*), X(LDX,*), WORK(*)
  INTEGER*8 N, NRHS, LDB, LDX, INFO
  INTEGER*8 IPIVOT(*)
  DOUBLE PRECISION RCOND
  DOUBLE PRECISION FERR(*), BERR(*), WORK2(*)
 

F95 INTERFACE

  SUBROUTINE SPSVX( FACT, UPLO, N, NRHS, A, AF, IPIVOT, B, [LDB], X, 
 *       [LDX], RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
  CHARACTER(LEN=1) :: FACT, UPLO
  COMPLEX(8), DIMENSION(:) :: A, AF, WORK
  COMPLEX(8), DIMENSION(:,:) :: B, X
  INTEGER :: N, NRHS, LDB, LDX, INFO
  INTEGER, DIMENSION(:) :: IPIVOT
  REAL(8) :: RCOND
  REAL(8), DIMENSION(:) :: FERR, BERR, WORK2
 
  SUBROUTINE SPSVX_64( FACT, UPLO, N, NRHS, A, AF, IPIVOT, B, [LDB], 
 *       X, [LDX], RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
  CHARACTER(LEN=1) :: FACT, UPLO
  COMPLEX(8), DIMENSION(:) :: A, AF, WORK
  COMPLEX(8), DIMENSION(:,:) :: B, X
  INTEGER(8) :: N, NRHS, LDB, LDX, INFO
  INTEGER(8), DIMENSION(:) :: IPIVOT
  REAL(8) :: RCOND
  REAL(8), DIMENSION(:) :: FERR, BERR, WORK2
 

C INTERFACE

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

void zspsvx_64(char fact, char uplo, long n, long nrhs, doublecomplex *a, doublecomplex *af, long *ipivot, 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:

1. If FACT = 'N', the diagonal pivoting method is used to factor A as 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 upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array A as follows: if UPLO = 'U', A(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', A(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. See below for further details.

* 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 CSPTRF, stored as a packed triangular matrix in the same storage format as A.

If FACT = 'N', then AF is an output argument and on exit 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 CSPTRF, stored as a packed triangular matrix in the same storage format as A.

* 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 CSPTRF. 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 CSPTRF.

* 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)

dimension(2*N)

* WORK2 (workspace)

* INFO (output)