path: root/SRC/csysvx.f

      SUBROUTINE CSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
     $                   LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
     $                   RWORK, INFO )
*
*  -- LAPACK driver routine (version 3.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2006
*
*     .. Scalar Arguments ..
      CHARACTER          FACT, UPLO
      INTEGER            INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
      REAL               RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      REAL               BERR( * ), FERR( * ), RWORK( * )
      COMPLEX            A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   WORK( * ), X( LDX, * )
*     ..
*
*  Purpose
*  =======
*
*  CSYSVX 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.
*
*  Arguments
*  =========
*
*  FACT    (input) 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.
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The number of linear equations, i.e., the order of the
*          matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrices B and X.  NRHS >= 0.
*
*  A       (input) COMPLEX 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  AF      (input or output) COMPLEX 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 CSYTRF.
*
*          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) INTEGER
*          The leading dimension of the array AF.  LDAF >= max(1,N).
*
*  IPIV    (input or output) 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 CSYTRF.
*          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 CSYTRF.
*
*  B       (input) COMPLEX array, dimension (LDB,NRHS)
*          The N-by-NRHS right hand side matrix B.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  X       (output) COMPLEX array, dimension (LDX,NRHS)
*          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
*
*  LDX     (input) INTEGER
*          The leading dimension of the array X.  LDX >= max(1,N).
*
*  RCOND   (output) REAL
*          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) REAL 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.
*
*  BERR    (output) REAL 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).
*
*  WORK    (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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 CSYTRF.
*
*          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.
*
*  RWORK   (workspace) REAL array, dimension (N)
*
*  INFO    (output) 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.
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO
      PARAMETER          ( ZERO = 0.0E+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, NOFACT
      INTEGER            LWKOPT, NB
      REAL               ANORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      REAL               CLANSY, SLAMCH
      EXTERNAL           ILAENV, LSAME, CLANSY, SLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           CLACPY, CSYCON, CSYRFS, CSYTRF, CSYTRS, XERBLA
*     ..
*     .. 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, 'CSYTRF', UPLO, N, -1, -1, -1 )
            LWKOPT = MAX( LWKOPT, N*NB )
         END IF
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'CSYSVX', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
      IF( NOFACT ) THEN
*
*        Compute the factorization A = U*D*U' or A = L*D*L'.
*
         CALL CLACPY( UPLO, N, N, A, LDA, AF, LDAF )
         CALL CSYTRF( 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 = CLANSY( 'I', UPLO, N, A, LDA, RWORK )
*
*     Compute the reciprocal of the condition number of A.
*
      CALL CSYCON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK, INFO )
*
*     Compute the solution vectors X.
*
      CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL CSYTRS( 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 CSYRFS( 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.SLAMCH( 'Epsilon' ) )
     $   INFO = N + 1
*
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of CSYSVX
*
      END