zsysvx()

subroutine zsysvx	(	character	fact,
		character	uplo,
		integer	n,
		integer	nrhs,
		complex*16, dimension( lda, * )	a,
		integer	lda,
		complex*16, dimension( ldaf, * )	af,
		integer	ldaf,
		integer, dimension( * )	ipiv,
		complex*16, dimension( ldb, * )	b,
		integer	ldb,
		complex*16, dimension( ldx, * )	x,
		integer	ldx,
		double precision	rcond,
		double precision, dimension( * )	ferr,
		double precision, dimension( * )	berr,
		complex*16, dimension( * )	work,
		integer	lwork,
		double precision, dimension( * )	rwork,
		integer	info )

ZSYSVX computes the solution to system of linear equations A * X = B for SY matrices

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Purpose:

!>
!> ZSYSVX uses the diagonal pivoting factorization to compute the
!> solution to a complex system of linear equations A * X = B,
!> where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
!> matrices.
!>
!> Error bounds on the solution and a condition estimate are also
!> provided.
!> 

Description:

!>
!> 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.
!> 

Parameters

[in]	FACT	!> FACT is CHARACTER*1 !> Specifies whether or not the factored form of A has been !> supplied on entry. !> = 'F': On entry, AF and IPIV contain the factored form !> of A. A, AF and IPIV will not be modified. !> = 'N': The matrix A will be copied to AF and factored. !>
[in]	UPLO	!> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !>
[in]	N	!> N is INTEGER !> The number of linear equations, i.e., the order of the !> matrix A. N >= 0. !>
[in]	NRHS	!> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !>
[in]	A	!> A is COMPLEX*16 array, dimension (LDA,N) !> 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. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !>
[in,out]	AF	!> AF is COMPLEX*16 array, dimension (LDAF,N) !> 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 ZSYTRF. !> !> 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. !>
[in]	LDAF	!> LDAF is INTEGER !> The leading dimension of the array AF. LDAF >= max(1,N). !>
[in,out]	IPIV	!> IPIV is INTEGER array, dimension (N) !> If FACT = 'F', then IPIV is an input argument and on entry !> contains details of the interchanges and the block structure !> of D, as determined by ZSYTRF. !> If IPIV(k) > 0, then rows and columns k and IPIV(k) were !> interchanged and D(k,k) is a 1-by-1 diagonal block. !> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and !> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) !> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = !> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were !> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. !> !> If FACT = 'N', then IPIV is an output argument and on exit !> contains details of the interchanges and the block structure !> of D, as determined by ZSYTRF. !>
[in]	B	!> B is COMPLEX*16 array, dimension (LDB,NRHS) !> The N-by-NRHS right hand side matrix B. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !>
[out]	X	!> X is COMPLEX*16 array, dimension (LDX,NRHS) !> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. !>
[in]	LDX	!> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !>
[out]	RCOND	!> RCOND is DOUBLE PRECISION !> 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. !>
[out]	FERR	!> FERR is DOUBLE PRECISION array, dimension (NRHS) !> 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. !>
[out]	BERR	!> BERR is DOUBLE PRECISION array, dimension (NRHS) !> 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). !>
[out]	WORK	!> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !>
[in]	LWORK	!> LWORK is INTEGER !> The length of WORK. LWORK >= max(1,2*N), and for best !> performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where !> NB is the optimal blocksize for ZSYTRF. !> !> If LWORK = -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 LWORK is issued by XERBLA. !>
[out]	RWORK	!> RWORK is DOUBLE PRECISION array, dimension (N) !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> > 0: if INFO = i, and i is !> <= N: D(i,i) is exactly zero. The factorization !> has been completed but the factor D is exactly !> singular, so the solution and error bounds could !> not be computed. RCOND = 0 is returned. !> = N+1: D is nonsingular, but RCOND is less than machine !> precision, meaning that the matrix is singular !> to working precision. Nevertheless, the !> solution and error bounds are computed because !> there are a number of situations where the !> computed solution can be more accurate than the !> value of RCOND would suggest. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 280 of file zsysvx.f.

*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          FACT, UPLO
      INTEGER            INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
      COMPLEX*16         A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      parameter( zero = 0.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, NOFACT
      INTEGER            LWKOPT, NB
      DOUBLE PRECISION   ANORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, ZLANSY
      EXTERNAL           lsame, ilaenv, dlamch, zlansy
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zlacpy, zsycon, zsyrfs, zsytrf,
     $                   zsytrs
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      nofact = lsame( fact, 'N' )
      lquery = ( lwork.EQ.-1 )
      IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
         info = -1
      ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
     $         .NOT.lsame( uplo, 'L' ) )
     $          THEN
         info = -2
      ELSE IF( n.LT.0 ) THEN
         info = -3
      ELSE IF( nrhs.LT.0 ) THEN
         info = -4
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -6
      ELSE IF( ldaf.LT.max( 1, n ) ) THEN
         info = -8
      ELSE IF( ldb.LT.max( 1, n ) ) THEN
         info = -11
      ELSE IF( ldx.LT.max( 1, n ) ) THEN
         info = -13
      ELSE IF( lwork.LT.max( 1, 2*n ) .AND. .NOT.lquery ) THEN
         info = -18
      END IF
*
      IF( info.EQ.0 ) THEN
         lwkopt = max( 1, 2*n )
         IF( nofact ) THEN
            nb = ilaenv( 1, 'ZSYTRF', uplo, n, -1, -1, -1 )
            lwkopt = max( lwkopt, n*nb )
         END IF
         work( 1 ) = lwkopt
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZSYSVX', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
      IF( nofact ) THEN
*
*        Compute the factorization A = U*D*U**T or A = L*D*L**T.
*
         CALL zlacpy( uplo, n, n, a, lda, af, ldaf )
         CALL zsytrf( uplo, n, af, ldaf, ipiv, work, lwork, info )
*
*        Return if INFO is non-zero.
*
         IF( info.GT.0 )THEN
            rcond = zero
            RETURN
         END IF
      END IF
*
*     Compute the norm of the matrix A.
*
      anorm = zlansy( 'I', uplo, n, a, lda, rwork )
*
*     Compute the reciprocal of the condition number of A.
*
      CALL zsycon( uplo, n, af, ldaf, ipiv, anorm, rcond, work,
     $             info )
*
*     Compute the solution vectors X.
*
      CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
      CALL zsytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
*
*     Use iterative refinement to improve the computed solutions and
*     compute error bounds and backward error estimates for them.
*
      CALL zsyrfs( uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x,
     $             ldx, ferr, berr, work, rwork, info )
*
*     Set INFO = N+1 if the matrix is singular to working precision.
*
      IF( rcond.LT.dlamch( 'Epsilon' ) )
     $   info = n + 1
*
      work( 1 ) = lwkopt
*
      RETURN
*
*     End of ZSYSVX
*

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