path: root/SRC/zhesvxx.f

      SUBROUTINE ZHESVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
     $                    EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
     $                    N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
     $                    NPARAMS, PARAMS, WORK, RWORK, INFO )
*
*     -- LAPACK driver routine (version 3.2.1)                          --
*     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
*     -- Jason Riedy of Univ. of California Berkeley.                 --
*     -- April 2009                                                   --
*
*     -- LAPACK is a software package provided by Univ. of Tennessee, --
*     -- Univ. of California Berkeley and NAG Ltd.                    --
*
      IMPLICIT NONE
*     ..
*     .. Scalar Arguments ..
      CHARACTER          EQUED, FACT, UPLO
      INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
     $                   N_ERR_BNDS
      DOUBLE PRECISION   RCOND, RPVGRW
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      COMPLEX*16         A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   WORK( * ), X( LDX, * )
      DOUBLE PRECISION   S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
     $                   ERR_BNDS_NORM( NRHS, * ),
     $                   ERR_BNDS_COMP( NRHS, * )
*     ..
*
*     Purpose
*     =======
*
*     ZHESVXX uses the diagonal pivoting factorization to compute the
*     solution to a complex*16 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.
*
*     If requested, both normwise and maximum componentwise error bounds
*     are returned. ZHESVXX will return a solution with a tiny
*     guaranteed error (O(eps) where eps is the working machine
*     precision) unless the matrix is very ill-conditioned, in which
*     case a warning is returned. Relevant condition numbers also are
*     calculated and returned.
*
*     ZHESVXX accepts user-provided factorizations and equilibration
*     factors; see the definitions of the FACT and EQUED options.
*     Solving with refinement and using a factorization from a previous
*     ZHESVXX call will also produce a solution with either O(eps)
*     errors or warnings, but we cannot make that claim for general
*     user-provided factorizations and equilibration factors if they
*     differ from what ZHESVXX would itself produce.
*
*     Description
*     ===========
*
*     The following steps are performed:
*
*     1. If FACT = 'E', double precision scaling factors are computed to equilibrate
*     the system:
*
*       diag(S)*A*diag(S)     *inv(diag(S))*X = diag(S)*B
*
*     Whether or not the system will be equilibrated depends on the
*     scaling of the matrix A, but if equilibration is used, A is
*     overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
*
*     2. If FACT = 'N' or 'E', the LU decomposition is used to factor
*     the matrix A (after equilibration if FACT = 'E') 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.
*
*     3. 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 (see
*     argument RCOND).  If the reciprocal of the condition number is
*     less than machine precision, the routine still goes on to solve
*     for X and compute error bounds as described below.
*
*     4. The system of equations is solved for X using the factored form
*     of A.
*
*     5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
*     the routine will use iterative refinement to try to get a small
*     error and error bounds.  Refinement calculates the residual to at
*     least twice the working precision.
*
*     6. If equilibration was used, the matrix X is premultiplied by
*     diag(R) so that it solves the original system before
*     equilibration.
*
*     Arguments
*     =========
*
*     Some optional parameters are bundled in the PARAMS array.  These
*     settings determine how refinement is performed, but often the
*     defaults are acceptable.  If the defaults are acceptable, users
*     can pass NPARAMS = 0 which prevents the source code from accessing
*     the PARAMS argument.
*
*     FACT    (input) CHARACTER*1
*     Specifies whether or not the factored form of the matrix A is
*     supplied on entry, and if not, whether the matrix A should be
*     equilibrated before it is factored.
*       = 'F':  On entry, AF and IPIV contain the factored form of A.
*               If EQUED is not 'N', the matrix A has been
*               equilibrated with scaling factors given by S.
*               A, AF, and IPIV are not modified.
*       = 'N':  The matrix A will be copied to AF and factored.
*       = 'E':  The matrix A will be equilibrated if necessary, then
*               copied to AF and factored.
*
*     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/output) 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.
*
*     On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
*     diag(S)*A*diag(S).
*
*     LDA     (input) INTEGER
*     The leading dimension of the array A.  LDA >= max(1,N).
*
*     AF      (input or output) 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 DSYTRF.
*
*     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 ZHETRF.  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 ZHETRF.
*
*     EQUED   (input or output) CHARACTER*1
*     Specifies the form of equilibration that was done.
*       = 'N':  No equilibration (always true if FACT = 'N').
*       = 'Y':  Both row and column equilibration, i.e., A has been
*               replaced by diag(S) * A * diag(S).
*     EQUED is an input argument if FACT = 'F'; otherwise, it is an
*     output argument.
*
*     S       (input or output) DOUBLE PRECISION array, dimension (N)
*     The scale factors for A.  If EQUED = 'Y', A is multiplied on
*     the left and right by diag(S).  S is an input argument if FACT =
*     'F'; otherwise, S is an output argument.  If FACT = 'F' and EQUED
*     = 'Y', each element of S must be positive.  If S is output, each
*     element of S is a power of the radix. If S is input, each element
*     of S should be a power of the radix to ensure a reliable solution
*     and error estimates. Scaling by powers of the radix does not cause
*     rounding errors unless the result underflows or overflows.
*     Rounding errors during scaling lead to refining with a matrix that
*     is not equivalent to the input matrix, producing error estimates
*     that may not be reliable.
*
*     B       (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
*     On entry, the N-by-NRHS right hand side matrix B.
*     On exit,
*     if EQUED = 'N', B is not modified;
*     if EQUED = 'Y', B is overwritten by diag(S)*B;
*
*     LDB     (input) INTEGER
*     The leading dimension of the array B.  LDB >= max(1,N).
*
*     X       (output) COMPLEX*16 array, dimension (LDX,NRHS)
*     If INFO = 0, the N-by-NRHS solution matrix X to the original
*     system of equations.  Note that A and B are modified on exit if
*     EQUED .ne. 'N', and the solution to the equilibrated system is
*     inv(diag(S))*X.
*
*     LDX     (input) INTEGER
*     The leading dimension of the array X.  LDX >= max(1,N).
*
*     RCOND   (output) DOUBLE PRECISION
*     Reciprocal scaled condition number.  This is an estimate of the
*     reciprocal Skeel condition number of the matrix A after
*     equilibration (if done).  If this is less than the machine
*     precision (in particular, if it is zero), the matrix is singular
*     to working precision.  Note that the error may still be small even
*     if this number is very small and the matrix appears ill-
*     conditioned.
*
*     RPVGRW  (output) DOUBLE PRECISION
*     Reciprocal pivot growth.  On exit, this contains the reciprocal
*     pivot growth factor norm(A)/norm(U). The "max absolute element"
*     norm is used.  If this is much less than 1, then the stability of
*     the LU factorization of the (equilibrated) matrix A could be poor.
*     This also means that the solution X, estimated condition numbers,
*     and error bounds could be unreliable. If factorization fails with
*     0<INFO<=N, then this contains the reciprocal pivot growth factor
*     for the leading INFO columns of A.
*
*     BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
*     Componentwise relative backward error.  This is 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).
*
*     N_ERR_BNDS (input) INTEGER
*     Number of error bounds to return for each right hand side
*     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
*     ERR_BNDS_COMP below.
*
*     ERR_BNDS_NORM  (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
*     For each right-hand side, this array contains information about
*     various error bounds and condition numbers corresponding to the
*     normwise relative error, which is defined as follows:
*
*     Normwise relative error in the ith solution vector:
*             max_j (abs(XTRUE(j,i) - X(j,i)))
*            ------------------------------
*                  max_j abs(X(j,i))
*
*     The array is indexed by the type of error information as described
*     below. There currently are up to three pieces of information
*     returned.
*
*     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
*     right-hand side.
*
*     The second index in ERR_BNDS_NORM(:,err) contains the following
*     three fields:
*     err = 1 "Trust/don't trust" boolean. Trust the answer if the
*              reciprocal condition number is less than the threshold
*              sqrt(n) * dlamch('Epsilon').
*
*     err = 2 "Guaranteed" error bound: The estimated forward error,
*              almost certainly within a factor of 10 of the true error
*              so long as the next entry is greater than the threshold
*              sqrt(n) * dlamch('Epsilon'). This error bound should only
*              be trusted if the previous boolean is true.
*
*     err = 3  Reciprocal condition number: Estimated normwise
*              reciprocal condition number.  Compared with the threshold
*              sqrt(n) * dlamch('Epsilon') to determine if the error
*              estimate is "guaranteed". These reciprocal condition
*              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
*              appropriately scaled matrix Z.
*              Let Z = S*A, where S scales each row by a power of the
*              radix so all absolute row sums of Z are approximately 1.
*
*     See Lapack Working Note 165 for further details and extra
*     cautions.
*
*     ERR_BNDS_COMP  (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
*     For each right-hand side, this array contains information about
*     various error bounds and condition numbers corresponding to the
*     componentwise relative error, which is defined as follows:
*
*     Componentwise relative error in the ith solution vector:
*                    abs(XTRUE(j,i) - X(j,i))
*             max_j ----------------------
*                         abs(X(j,i))
*
*     The array is indexed by the right-hand side i (on which the
*     componentwise relative error depends), and the type of error
*     information as described below. There currently are up to three
*     pieces of information returned for each right-hand side. If
*     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
*     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
*     the first (:,N_ERR_BNDS) entries are returned.
*
*     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
*     right-hand side.
*
*     The second index in ERR_BNDS_COMP(:,err) contains the following
*     three fields:
*     err = 1 "Trust/don't trust" boolean. Trust the answer if the
*              reciprocal condition number is less than the threshold
*              sqrt(n) * dlamch('Epsilon').
*
*     err = 2 "Guaranteed" error bound: The estimated forward error,
*              almost certainly within a factor of 10 of the true error
*              so long as the next entry is greater than the threshold
*              sqrt(n) * dlamch('Epsilon'). This error bound should only
*              be trusted if the previous boolean is true.
*
*     err = 3  Reciprocal condition number: Estimated componentwise
*              reciprocal condition number.  Compared with the threshold
*              sqrt(n) * dlamch('Epsilon') to determine if the error
*              estimate is "guaranteed". These reciprocal condition
*              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
*              appropriately scaled matrix Z.
*              Let Z = S*(A*diag(x)), where x is the solution for the
*              current right-hand side and S scales each row of
*              A*diag(x) by a power of the radix so all absolute row
*              sums of Z are approximately 1.
*
*     See Lapack Working Note 165 for further details and extra
*     cautions.
*
*     NPARAMS (input) INTEGER
*     Specifies the number of parameters set in PARAMS.  If .LE. 0, the
*     PARAMS array is never referenced and default values are used.
*
*     PARAMS  (input / output) DOUBLE PRECISION array, dimension NPARAMS
*     Specifies algorithm parameters.  If an entry is .LT. 0.0, then
*     that entry will be filled with default value used for that
*     parameter.  Only positions up to NPARAMS are accessed; defaults
*     are used for higher-numbered parameters.
*
*       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
*            refinement or not.
*         Default: 1.0D+0
*            = 0.0 : No refinement is performed, and no error bounds are
*                    computed.
*            = 1.0 : Use the extra-precise refinement algorithm.
*              (other values are reserved for future use)
*
*       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
*            computations allowed for refinement.
*         Default: 10
*         Aggressive: Set to 100 to permit convergence using approximate
*                     factorizations or factorizations other than LU. If
*                     the factorization uses a technique other than
*                     Gaussian elimination, the guarantees in
*                     err_bnds_norm and err_bnds_comp may no longer be
*                     trustworthy.
*
*       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
*            will attempt to find a solution with small componentwise
*            relative error in the double-precision algorithm.  Positive
*            is true, 0.0 is false.
*         Default: 1.0 (attempt componentwise convergence)
*
*     WORK    (workspace) COMPLEX*16 array, dimension (2*N)
*
*     RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N)
*
*     INFO    (output) INTEGER
*       = 0:  Successful exit. The solution to every right-hand side is
*         guaranteed.
*       < 0:  If INFO = -i, the i-th argument had an illegal value
*       > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
*         has been completed, but the factor U is exactly singular, so
*         the solution and error bounds could not be computed. RCOND = 0
*         is returned.
*       = N+J: The solution corresponding to the Jth right-hand side is
*         not guaranteed. The solutions corresponding to other right-
*         hand sides K with K > J may not be guaranteed as well, but
*         only the first such right-hand side is reported. If a small
*         componentwise error is not requested (PARAMS(3) = 0.0) then
*         the Jth right-hand side is the first with a normwise error
*         bound that is not guaranteed (the smallest J such
*         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
*         the Jth right-hand side is the first with either a normwise or
*         componentwise error bound that is not guaranteed (the smallest
*         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
*         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
*         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
*         about all of the right-hand sides check ERR_BNDS_NORM or
*         ERR_BNDS_COMP.
*
*     ==================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      INTEGER            FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
      INTEGER            RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
      INTEGER            CMP_ERR_I, PIV_GROWTH_I
      PARAMETER          ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
     $                   BERR_I = 3 )
      PARAMETER          ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
      PARAMETER          ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
     $                   PIV_GROWTH_I = 9 )
*     ..
*     .. Local Scalars ..
      LOGICAL            EQUIL, NOFACT, RCEQU
      INTEGER            INFEQU, J
      DOUBLE PRECISION   AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
*     ..
*     .. External Functions ..
      EXTERNAL           LSAME, DLAMCH,  ZLA_HERPVGRW
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, ZLA_HERPVGRW
*     ..
*     .. External Subroutines ..
      EXTERNAL           ZHECON, ZHEEQUB, ZHETRF, ZHETRS, ZLACPY,
     $                   ZLAQHE, XERBLA, ZLASCL2, ZHERFSX
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      NOFACT = LSAME( FACT, 'N' )
      EQUIL = LSAME( FACT, 'E' )
      SMLNUM = DLAMCH( 'Safe minimum' )
      BIGNUM = ONE / SMLNUM
      IF( NOFACT .OR. EQUIL ) THEN
         EQUED = 'N'
         RCEQU = .FALSE.
      ELSE
         RCEQU = LSAME( EQUED, 'Y' )
      ENDIF
*
*     Default is failure.  If an input parameter is wrong or
*     factorization fails, make everything look horrible.  Only the
*     pivot growth is set here, the rest is initialized in ZHERFSX.
*
      RPVGRW = ZERO
*
*     Test the input parameters.  PARAMS is not tested until ZHERFSX.
*
      IF( .NOT.NOFACT .AND. .NOT.EQUIL .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( LSAME( FACT, 'F' ) .AND. .NOT.
     $        ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
         INFO = -9
      ELSE
         IF ( RCEQU ) THEN
            SMIN = BIGNUM
            SMAX = ZERO
            DO 10 J = 1, N
               SMIN = MIN( SMIN, S( J ) )
               SMAX = MAX( SMAX, S( J ) )
 10         CONTINUE
            IF( SMIN.LE.ZERO ) THEN
               INFO = -10
            ELSE IF( N.GT.0 ) THEN
               SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
            ELSE
               SCOND = ONE
            END IF
         END IF
         IF( INFO.EQ.0 ) THEN
            IF( LDB.LT.MAX( 1, N ) ) THEN
               INFO = -12
            ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
               INFO = -14
            END IF
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'ZHESVXX', -INFO )
         RETURN
      END IF
*
      IF( EQUIL ) THEN
*
*     Compute row and column scalings to equilibrate the matrix A.
*
         CALL ZHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFEQU )
         IF( INFEQU.EQ.0 ) THEN
*
*     Equilibrate the matrix.
*
            CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
            RCEQU = LSAME( EQUED, 'Y' )
         END IF
      END IF
*
*     Scale the right-hand side.
*
      IF( RCEQU ) CALL ZLASCL2( N, NRHS, S, B, LDB )
*
      IF( NOFACT .OR. EQUIL ) THEN
*
*        Compute the LU factorization of A.
*
         CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF )
         CALL ZHETRF( UPLO, N, AF, LDAF, IPIV, WORK, 5*MAX(1,N), INFO )
*
*        Return if INFO is non-zero.
*
         IF( INFO.GT.0 ) THEN
*
*           Pivot in column INFO is exactly 0
*           Compute the reciprocal pivot growth factor of the
*           leading rank-deficient INFO columns of A.
*
            IF( N.GT.0 )
     $           RPVGRW = ZLA_HERPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF,
     $           IPIV, RWORK )
            RETURN
         END IF
      END IF
*
*     Compute the reciprocal pivot growth factor RPVGRW.
*
      IF( N.GT.0 )
     $     RPVGRW = ZLA_HERPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF, IPIV,
     $     RWORK )
*
*     Compute the solution matrix X.
*
      CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL ZHETRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
*
*     Use iterative refinement to improve the computed solution and
*     compute error bounds and backward error estimates for it.
*
      CALL ZHERFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
     $     S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
     $     ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )
*
*     Scale solutions.
*
      IF ( RCEQU ) THEN
         CALL ZLASCL2 ( N, NRHS, S, X, LDX )
      END IF
*
      RETURN
*
*     End of ZHESVXX
*
      END