LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ zgbsvxx()

subroutine zgbsvxx ( character  FACT,
character  TRANS,
integer  N,
integer  KL,
integer  KU,
integer  NRHS,
complex*16, dimension( ldab, * )  AB,
integer  LDAB,
complex*16, dimension( ldafb, * )  AFB,
integer  LDAFB,
integer, dimension( * )  IPIV,
character  EQUED,
double precision, dimension( * )  R,
double precision, dimension( * )  C,
complex*16, dimension( ldb, * )  B,
integer  LDB,
complex*16, dimension( ldx , * )  X,
integer  LDX,
double precision  RCOND,
double precision  RPVGRW,
double precision, dimension( * )  BERR,
integer  N_ERR_BNDS,
double precision, dimension( nrhs, * )  ERR_BNDS_NORM,
double precision, dimension( nrhs, * )  ERR_BNDS_COMP,
integer  NPARAMS,
double precision, dimension( * )  PARAMS,
complex*16, dimension( * )  WORK,
double precision, dimension( * )  RWORK,
integer  INFO 
)

ZGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

Download ZGBSVXX + dependencies [TGZ] [ZIP] [TXT]

Purpose:
    ZGBSVXX uses the LU factorization to compute the solution to a
    complex*16 system of linear equations  A * X = B,  where A is an
    N-by-N matrix and X and B are N-by-NRHS matrices.

    If requested, both normwise and maximum componentwise error bounds
    are returned. ZGBSVXX 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.

    ZGBSVXX 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
    ZGBSVXX 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 ZGBSVXX would itself produce.
Description:
    The following steps are performed:

    1. If FACT = 'E', double precision scaling factors are computed to equilibrate
    the system:

      TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
      TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
      TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
    or diag(C)*B (if TRANS = 'T' or 'C').

    2. If FACT = 'N' or 'E', the LU decomposition is used to factor
    the matrix A (after equilibration if FACT = 'E') as

      A = P * L * U,

    where P is a permutation matrix, L is a unit lower triangular
    matrix, and U is upper triangular.

    3. If some U(i,i)=0, so that U 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
    that it solves the original system before equilibration.
     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.
Parameters
[in]FACT
          FACT is 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 R and C.
               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.
[in]TRANS
          TRANS is CHARACTER*1
     Specifies the form of the system of equations:
       = 'N':  A * X = B     (No transpose)
       = 'T':  A**T * X = B  (Transpose)
       = 'C':  A**H * X = B  (Conjugate Transpose = Transpose)
[in]N
          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.
[in]KL
          KL is INTEGER
     The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
     The number of superdiagonals within the band of A.  KU >= 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,out]AB
          AB is COMPLEX*16 array, dimension (LDAB,N)
     On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
     The j-th column of A is stored in the j-th column of the
     array AB as follows:
     AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

     If FACT = 'F' and EQUED is not 'N', then AB must have been
     equilibrated by the scaling factors in R and/or C.  AB is not
     modified if FACT = 'F' or 'N', or if FACT = 'E' and
     EQUED = 'N' on exit.

     On exit, if EQUED .ne. 'N', A is scaled as follows:
     EQUED = 'R':  A := diag(R) * A
     EQUED = 'C':  A := A * diag(C)
     EQUED = 'B':  A := diag(R) * A * diag(C).
[in]LDAB
          LDAB is INTEGER
     The leading dimension of the array AB.  LDAB >= KL+KU+1.
[in,out]AFB
          AFB is COMPLEX*16 array, dimension (LDAFB,N)
     If FACT = 'F', then AFB is an input argument and on entry
     contains details of the LU factorization of the band matrix
     A, as computed by ZGBTRF.  U is stored as an upper triangular
     band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
     and the multipliers used during the factorization are stored
     in rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB is
     the factored form of the equilibrated matrix A.

     If FACT = 'N', then AF is an output argument and on exit
     returns the factors L and U from the factorization A = P*L*U
     of the original matrix A.

     If FACT = 'E', then AF is an output argument and on exit
     returns the factors L and U from the factorization A = P*L*U
     of the equilibrated matrix A (see the description of A for
     the form of the equilibrated matrix).
[in]LDAFB
          LDAFB is INTEGER
     The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
[in,out]IPIV
          IPIV is INTEGER array, dimension (N)
     If FACT = 'F', then IPIV is an input argument and on entry
     contains the pivot indices from the factorization A = P*L*U
     as computed by ZGETRF; row i of the matrix was interchanged
     with row IPIV(i).

     If FACT = 'N', then IPIV is an output argument and on exit
     contains the pivot indices from the factorization A = P*L*U
     of the original matrix A.

     If FACT = 'E', then IPIV is an output argument and on exit
     contains the pivot indices from the factorization A = P*L*U
     of the equilibrated matrix A.
[in,out]EQUED
          EQUED is CHARACTER*1
     Specifies the form of equilibration that was done.
       = 'N':  No equilibration (always true if FACT = 'N').
       = 'R':  Row equilibration, i.e., A has been premultiplied by
               diag(R).
       = 'C':  Column equilibration, i.e., A has been postmultiplied
               by diag(C).
       = 'B':  Both row and column equilibration, i.e., A has been
               replaced by diag(R) * A * diag(C).
     EQUED is an input argument if FACT = 'F'; otherwise, it is an
     output argument.
[in,out]R
          R is DOUBLE PRECISION array, dimension (N)
     The row scale factors for A.  If EQUED = 'R' or 'B', A is
     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
     is not accessed.  R is an input argument if FACT = 'F';
     otherwise, R is an output argument.  If FACT = 'F' and
     EQUED = 'R' or 'B', each element of R must be positive.
     If R is output, each element of R is a power of the radix.
     If R is input, each element of R 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.
[in,out]C
          C is DOUBLE PRECISION array, dimension (N)
     The column scale factors for A.  If EQUED = 'C' or 'B', A is
     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
     is not accessed.  C is an input argument if FACT = 'F';
     otherwise, C is an output argument.  If FACT = 'F' and
     EQUED = 'C' or 'B', each element of C must be positive.
     If C is output, each element of C is a power of the radix.
     If C is input, each element of C 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.
[in,out]B
          B is 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 TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
        diag(R)*B;
     if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
        overwritten by diag(C)*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, 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(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
     inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
[in]LDX
          LDX is INTEGER
     The leading dimension of the array X.  LDX >= max(1,N).
[out]RCOND
          RCOND is 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.
[out]RPVGRW
          RPVGRW is 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.  In ZGESVX, this quantity is
     returned in RWORK(1).
[out]BERR
          BERR is 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).
[in]N_ERR_BNDS
          N_ERR_BNDS is 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.
[out]ERR_BNDS_NORM
          ERR_BNDS_NORM is 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.
[out]ERR_BNDS_COMP
          ERR_BNDS_COMP is 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 < 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.
[in]NPARAMS
          NPARAMS is INTEGER
     Specifies the number of parameters set in PARAMS.  If <= 0, the
     PARAMS array is never referenced and default values are used.
[in,out]PARAMS
          PARAMS is DOUBLE PRECISION array, dimension NPARAMS
     Specifies algorithm parameters.  If an entry is < 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)
[out]WORK
          WORK is COMPLEX*16 array, dimension (2*N)
[out]RWORK
          RWORK is DOUBLE PRECISION array, dimension (2*N)
[out]INFO
          INFO is 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 555 of file zgbsvxx.f.

560 *
561 * -- LAPACK driver routine --
562 * -- LAPACK is a software package provided by Univ. of Tennessee, --
563 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
564 *
565 * .. Scalar Arguments ..
566  CHARACTER EQUED, FACT, TRANS
567  INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
568  $ N_ERR_BNDS
569  DOUBLE PRECISION RCOND, RPVGRW
570 * ..
571 * .. Array Arguments ..
572  INTEGER IPIV( * )
573  COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
574  $ X( LDX , * ),WORK( * )
575  DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
576  $ ERR_BNDS_NORM( NRHS, * ),
577  $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
578 * ..
579 *
580 * ==================================================================
581 *
582 * .. Parameters ..
583  DOUBLE PRECISION ZERO, ONE
584  parameter( zero = 0.0d+0, one = 1.0d+0 )
585  INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
586  INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
587  INTEGER CMP_ERR_I, PIV_GROWTH_I
588  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
589  $ berr_i = 3 )
590  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
591  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
592  $ piv_growth_i = 9 )
593 * ..
594 * .. Local Scalars ..
595  LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
596  INTEGER INFEQU, I, J, KL, KU
597  DOUBLE PRECISION AMAX, BIGNUM, COLCND, RCMAX, RCMIN,
598  $ ROWCND, SMLNUM
599 * ..
600 * .. External Functions ..
601  EXTERNAL lsame, dlamch, zla_gbrpvgrw
602  LOGICAL LSAME
603  DOUBLE PRECISION DLAMCH, ZLA_GBRPVGRW
604 * ..
605 * .. External Subroutines ..
606  EXTERNAL zgbequb, zgbtrf, zgbtrs, zlacpy, zlaqgb,
608 * ..
609 * .. Intrinsic Functions ..
610  INTRINSIC max, min
611 * ..
612 * .. Executable Statements ..
613 *
614  info = 0
615  nofact = lsame( fact, 'N' )
616  equil = lsame( fact, 'E' )
617  notran = lsame( trans, 'N' )
618  smlnum = dlamch( 'Safe minimum' )
619  bignum = one / smlnum
620  IF( nofact .OR. equil ) THEN
621  equed = 'N'
622  rowequ = .false.
623  colequ = .false.
624  ELSE
625  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
626  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
627  END IF
628 *
629 * Default is failure. If an input parameter is wrong or
630 * factorization fails, make everything look horrible. Only the
631 * pivot growth is set here, the rest is initialized in ZGBRFSX.
632 *
633  rpvgrw = zero
634 *
635 * Test the input parameters. PARAMS is not tested until ZGERFSX.
636 *
637  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
638  $ lsame( fact, 'F' ) ) THEN
639  info = -1
640  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
641  $ lsame( trans, 'C' ) ) THEN
642  info = -2
643  ELSE IF( n.LT.0 ) THEN
644  info = -3
645  ELSE IF( kl.LT.0 ) THEN
646  info = -4
647  ELSE IF( ku.LT.0 ) THEN
648  info = -5
649  ELSE IF( nrhs.LT.0 ) THEN
650  info = -6
651  ELSE IF( ldab.LT.kl+ku+1 ) THEN
652  info = -8
653  ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
654  info = -10
655  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
656  $ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
657  info = -12
658  ELSE
659  IF( rowequ ) THEN
660  rcmin = bignum
661  rcmax = zero
662  DO 10 j = 1, n
663  rcmin = min( rcmin, r( j ) )
664  rcmax = max( rcmax, r( j ) )
665  10 CONTINUE
666  IF( rcmin.LE.zero ) THEN
667  info = -13
668  ELSE IF( n.GT.0 ) THEN
669  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
670  ELSE
671  rowcnd = one
672  END IF
673  END IF
674  IF( colequ .AND. info.EQ.0 ) THEN
675  rcmin = bignum
676  rcmax = zero
677  DO 20 j = 1, n
678  rcmin = min( rcmin, c( j ) )
679  rcmax = max( rcmax, c( j ) )
680  20 CONTINUE
681  IF( rcmin.LE.zero ) THEN
682  info = -14
683  ELSE IF( n.GT.0 ) THEN
684  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
685  ELSE
686  colcnd = one
687  END IF
688  END IF
689  IF( info.EQ.0 ) THEN
690  IF( ldb.LT.max( 1, n ) ) THEN
691  info = -15
692  ELSE IF( ldx.LT.max( 1, n ) ) THEN
693  info = -16
694  END IF
695  END IF
696  END IF
697 *
698  IF( info.NE.0 ) THEN
699  CALL xerbla( 'ZGBSVXX', -info )
700  RETURN
701  END IF
702 *
703  IF( equil ) THEN
704 *
705 * Compute row and column scalings to equilibrate the matrix A.
706 *
707  CALL zgbequb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
708  $ amax, infequ )
709  IF( infequ.EQ.0 ) THEN
710 *
711 * Equilibrate the matrix.
712 *
713  CALL zlaqgb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
714  $ amax, equed )
715  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
716  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
717  END IF
718 *
719 * If the scaling factors are not applied, set them to 1.0.
720 *
721  IF ( .NOT.rowequ ) THEN
722  DO j = 1, n
723  r( j ) = 1.0d+0
724  END DO
725  END IF
726  IF ( .NOT.colequ ) THEN
727  DO j = 1, n
728  c( j ) = 1.0d+0
729  END DO
730  END IF
731  END IF
732 *
733 * Scale the right-hand side.
734 *
735  IF( notran ) THEN
736  IF( rowequ ) CALL zlascl2( n, nrhs, r, b, ldb )
737  ELSE
738  IF( colequ ) CALL zlascl2( n, nrhs, c, b, ldb )
739  END IF
740 *
741  IF( nofact .OR. equil ) THEN
742 *
743 * Compute the LU factorization of A.
744 *
745  DO 40, j = 1, n
746  DO 30, i = kl+1, 2*kl+ku+1
747  afb( i, j ) = ab( i-kl, j )
748  30 CONTINUE
749  40 CONTINUE
750  CALL zgbtrf( n, n, kl, ku, afb, ldafb, ipiv, info )
751 *
752 * Return if INFO is non-zero.
753 *
754  IF( info.GT.0 ) THEN
755 *
756 * Pivot in column INFO is exactly 0
757 * Compute the reciprocal pivot growth factor of the
758 * leading rank-deficient INFO columns of A.
759 *
760  rpvgrw = zla_gbrpvgrw( n, kl, ku, info, ab, ldab, afb,
761  $ ldafb )
762  RETURN
763  END IF
764  END IF
765 *
766 * Compute the reciprocal pivot growth factor RPVGRW.
767 *
768  rpvgrw = zla_gbrpvgrw( n, kl, ku, n, ab, ldab, afb, ldafb )
769 *
770 * Compute the solution matrix X.
771 *
772  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
773  CALL zgbtrs( trans, n, kl, ku, nrhs, afb, ldafb, ipiv, x, ldx,
774  $ info )
775 *
776 * Use iterative refinement to improve the computed solution and
777 * compute error bounds and backward error estimates for it.
778 *
779  CALL zgbrfsx( trans, equed, n, kl, ku, nrhs, ab, ldab, afb, ldafb,
780  $ ipiv, r, c, b, ldb, x, ldx, rcond, berr,
781  $ n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params,
782  $ work, rwork, info )
783 
784 *
785 * Scale solutions.
786 *
787  IF ( colequ .AND. notran ) THEN
788  CALL zlascl2( n, nrhs, c, x, ldx )
789  ELSE IF ( rowequ .AND. .NOT.notran ) THEN
790  CALL zlascl2( n, nrhs, r, x, ldx )
791  END IF
792 *
793  RETURN
794 *
795 * End of ZGBSVXX
796 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zlaqgb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, EQUED)
ZLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ.
Definition: zlaqgb.f:160
subroutine zgbequb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
ZGBEQUB
Definition: zgbequb.f:161
subroutine zgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
ZGBTRF
Definition: zgbtrf.f:144
double precision function zla_gbrpvgrw(N, KL, KU, NCOLS, AB, LDAB, AFB, LDAFB)
ZLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.
Definition: zla_gbrpvgrw.f:117
subroutine zgbrfsx(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
ZGBRFSX
Definition: zgbrfsx.f:440
subroutine zgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
ZGBTRS
Definition: zgbtrs.f:138
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zlascl2(M, N, D, X, LDX)
ZLASCL2 performs diagonal scaling on a vector.
Definition: zlascl2.f:91
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