chesvx.f

          SUBROUTINE CHESVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
         $                   LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
         $                   RWORK, INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley 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
    *  =======
    *
    *  CHESVX 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 Hermitian 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**H,  if UPLO = 'U', or
    *        A = L * D * L**H,  if UPLO = 'L',
    *     where U (or L) is a product of permutation and unit upper (lower)
    *     triangular matrices, and D is Hermitian 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 Hermitian 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**H or A = L*D*L**H as computed by CHETRF.
    *
    *          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**H or A = L*D*L**H.
    *
    *  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 CHETRF.
    *          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 CHETRF.
    *
    *  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 CHETRF.
    *
    *          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               CLANHE, SLAMCH
          EXTERNAL           ILAENV, LSAME, CLANHE, SLAMCH
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           CHECON, CHERFS, CHETRF, CHETRS, CLACPY, 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, 'CHETRF', UPLO, N, -1, -1, -1 )
                LWKOPT = MAX( LWKOPT, N*NB )
             END IF
             WORK( 1 ) = LWKOPT
          END IF
    *
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'CHESVX', -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 CHETRF( 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 = CLANHE( 'I', UPLO, N, A, LDA, RWORK )
    *
    *     Compute the reciprocal of the condition number of A.
    *
          CALL CHECON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK, INFO )
    *
    *     Compute the solution vectors X.
    *
          CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
          CALL CHETRS( 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 CHERFS( 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 CHESVX
    *
          END