LAPACK  3.6.1
LAPACK: Linear Algebra PACKage
subroutine sgbsvxx ( character  FACT,
character  TRANS,
integer  N,
integer  KL,
integer  KU,
integer  NRHS,
real, dimension( ldab, * )  AB,
integer  LDAB,
real, dimension( ldafb, * )  AFB,
integer  LDAFB,
integer, dimension( * )  IPIV,
character  EQUED,
real, dimension( * )  R,
real, dimension( * )  C,
real, dimension( ldb, * )  B,
integer  LDB,
real, dimension( ldx , * )  X,
integer  LDX,
real  RCOND,
real  RPVGRW,
real, dimension( * )  BERR,
integer  N_ERR_BNDS,
real, dimension( nrhs, * )  ERR_BNDS_NORM,
real, dimension( nrhs, * )  ERR_BNDS_COMP,
integer  NPARAMS,
real, dimension( * )  PARAMS,
real, dimension( * )  WORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

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

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

Purpose:
    SGBSVXX uses the LU factorization to compute the solution to a
    real 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. SGBSVXX 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.

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

    1. If FACT = 'E', real 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 REAL 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 REAL 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 SGBTRF.  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 SGETRF; 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL
     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 REAL
     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 SGESVX, this quantity is
     returned in WORK(1).
[out]BERR
          BERR is REAL 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 REAL 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) * slamch('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) * slamch('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) * slamch('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 REAL 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) * slamch('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) * slamch('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) * slamch('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 .LE. 0, the
     PARAMS array is never referenced and default values are used.
[in,out]PARAMS
          PARAMS is REAL 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: 1.0
            = 0.0 : No refinement is performed, and no error bounds are
                    computed.
            = 1.0 : Use the double-precision refinement algorithm,
                    possibly with doubled-single computations if the
                    compilation environment does not support DOUBLE
                    PRECISION.
              (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 REAL array, dimension (4*N)
[out]IWORK
          IWORK is INTEGER array, dimension (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.
Date
April 2012

Definition at line 565 of file sgbsvxx.f.

565 *
566 * -- LAPACK driver routine (version 3.4.1) --
567 * -- LAPACK is a software package provided by Univ. of Tennessee, --
568 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
569 * April 2012
570 *
571 * .. Scalar Arguments ..
572  CHARACTER equed, fact, trans
573  INTEGER info, ldab, ldafb, ldb, ldx, n, nrhs, nparams,
574  $ n_err_bnds
575  REAL rcond, rpvgrw
576 * ..
577 * .. Array Arguments ..
578  INTEGER ipiv( * ), iwork( * )
579  REAL ab( ldab, * ), afb( ldafb, * ), b( ldb, * ),
580  $ x( ldx , * ),work( * )
581  REAL r( * ), c( * ), params( * ), berr( * ),
582  $ err_bnds_norm( nrhs, * ),
583  $ err_bnds_comp( nrhs, * )
584 * ..
585 *
586 * ==================================================================
587 *
588 * .. Parameters ..
589  REAL zero, one
590  parameter ( zero = 0.0e+0, one = 1.0e+0 )
591  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
592  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
593  INTEGER cmp_err_i, piv_growth_i
594  parameter ( final_nrm_err_i = 1, final_cmp_err_i = 2,
595  $ berr_i = 3 )
596  parameter ( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
597  parameter ( cmp_rcond_i = 7, cmp_err_i = 8,
598  $ piv_growth_i = 9 )
599 * ..
600 * .. Local Scalars ..
601  LOGICAL colequ, equil, nofact, notran, rowequ
602  INTEGER infequ, i, j, kl, ku
603  REAL amax, bignum, colcnd, rcmax, rcmin,
604  $ rowcnd, smlnum
605 * ..
606 * .. External Functions ..
607  EXTERNAL lsame, slamch, sla_gbrpvgrw
608  LOGICAL lsame
609  REAL slamch, sla_gbrpvgrw
610 * ..
611 * .. External Subroutines ..
612  EXTERNAL sgbequb, sgbtrf, sgbtrs, slacpy, slaqgb,
614 * ..
615 * .. Intrinsic Functions ..
616  INTRINSIC max, min
617 * ..
618 * .. Executable Statements ..
619 *
620  info = 0
621  nofact = lsame( fact, 'N' )
622  equil = lsame( fact, 'E' )
623  notran = lsame( trans, 'N' )
624  smlnum = slamch( 'Safe minimum' )
625  bignum = one / smlnum
626  IF( nofact .OR. equil ) THEN
627  equed = 'N'
628  rowequ = .false.
629  colequ = .false.
630  ELSE
631  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
632  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
633  END IF
634 *
635 * Default is failure. If an input parameter is wrong or
636 * factorization fails, make everything look horrible. Only the
637 * pivot growth is set here, the rest is initialized in SGBRFSX.
638 *
639  rpvgrw = zero
640 *
641 * Test the input parameters. PARAMS is not tested until SGBRFSX.
642 *
643  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
644  $ lsame( fact, 'F' ) ) THEN
645  info = -1
646  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
647  $ lsame( trans, 'C' ) ) THEN
648  info = -2
649  ELSE IF( n.LT.0 ) THEN
650  info = -3
651  ELSE IF( kl.LT.0 ) THEN
652  info = -4
653  ELSE IF( ku.LT.0 ) THEN
654  info = -5
655  ELSE IF( nrhs.LT.0 ) THEN
656  info = -6
657  ELSE IF( ldab.LT.kl+ku+1 ) THEN
658  info = -8
659  ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
660  info = -10
661  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
662  $ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
663  info = -12
664  ELSE
665  IF( rowequ ) THEN
666  rcmin = bignum
667  rcmax = zero
668  DO 10 j = 1, n
669  rcmin = min( rcmin, r( j ) )
670  rcmax = max( rcmax, r( j ) )
671  10 CONTINUE
672  IF( rcmin.LE.zero ) THEN
673  info = -13
674  ELSE IF( n.GT.0 ) THEN
675  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
676  ELSE
677  rowcnd = one
678  END IF
679  END IF
680  IF( colequ .AND. info.EQ.0 ) THEN
681  rcmin = bignum
682  rcmax = zero
683  DO 20 j = 1, n
684  rcmin = min( rcmin, c( j ) )
685  rcmax = max( rcmax, c( j ) )
686  20 CONTINUE
687  IF( rcmin.LE.zero ) THEN
688  info = -14
689  ELSE IF( n.GT.0 ) THEN
690  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
691  ELSE
692  colcnd = one
693  END IF
694  END IF
695  IF( info.EQ.0 ) THEN
696  IF( ldb.LT.max( 1, n ) ) THEN
697  info = -15
698  ELSE IF( ldx.LT.max( 1, n ) ) THEN
699  info = -16
700  END IF
701  END IF
702  END IF
703 *
704  IF( info.NE.0 ) THEN
705  CALL xerbla( 'SGBSVXX', -info )
706  RETURN
707  END IF
708 *
709  IF( equil ) THEN
710 *
711 * Compute row and column scalings to equilibrate the matrix A.
712 *
713  CALL sgbequb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
714  $ amax, infequ )
715  IF( infequ.EQ.0 ) THEN
716 *
717 * Equilibrate the matrix.
718 *
719  CALL slaqgb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
720  $ amax, equed )
721  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
722  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
723  END IF
724 *
725 * If the scaling factors are not applied, set them to 1.0.
726 *
727  IF ( .NOT.rowequ ) THEN
728  DO j = 1, n
729  r( j ) = 1.0
730  END DO
731  END IF
732  IF ( .NOT.colequ ) THEN
733  DO j = 1, n
734  c( j ) = 1.0
735  END DO
736  END IF
737  END IF
738 *
739 * Scale the right hand side.
740 *
741  IF( notran ) THEN
742  IF( rowequ ) CALL slascl2(n, nrhs, r, b, ldb)
743  ELSE
744  IF( colequ ) CALL slascl2(n, nrhs, c, b, ldb)
745  END IF
746 *
747  IF( nofact .OR. equil ) THEN
748 *
749 * Compute the LU factorization of A.
750 *
751  DO 40, j = 1, n
752  DO 30, i = kl+1, 2*kl+ku+1
753  afb( i, j ) = ab( i-kl, j )
754  30 CONTINUE
755  40 CONTINUE
756  CALL sgbtrf( n, n, kl, ku, afb, ldafb, ipiv, info )
757 *
758 * Return if INFO is non-zero.
759 *
760  IF( info.GT.0 ) THEN
761 *
762 * Pivot in column INFO is exactly 0
763 * Compute the reciprocal pivot growth factor of the
764 * leading rank-deficient INFO columns of A.
765 *
766  rpvgrw = sla_gbrpvgrw( n, kl, ku, info, ab, ldab, afb,
767  $ ldafb )
768  RETURN
769  END IF
770  END IF
771 *
772 * Compute the reciprocal pivot growth factor RPVGRW.
773 *
774  rpvgrw = sla_gbrpvgrw( n, kl, ku, n, ab, ldab, afb, ldafb )
775 *
776 * Compute the solution matrix X.
777 *
778  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
779  CALL sgbtrs( trans, n, kl, ku, nrhs, afb, ldafb, ipiv, x, ldx,
780  $ info )
781 *
782 * Use iterative refinement to improve the computed solution and
783 * compute error bounds and backward error estimates for it.
784 *
785  CALL sgbrfsx( trans, equed, n, kl, ku, nrhs, ab, ldab, afb, ldafb,
786  $ ipiv, r, c, b, ldb, x, ldx, rcond, berr,
787  $ n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params,
788  $ work, iwork, info )
789 *
790 * Scale solutions.
791 *
792  IF ( colequ .AND. notran ) THEN
793  CALL slascl2 ( n, nrhs, c, x, ldx )
794  ELSE IF ( rowequ .AND. .NOT.notran ) THEN
795  CALL slascl2 ( n, nrhs, r, x, ldx )
796  END IF
797 *
798  RETURN
799 *
800 * End of SGBSVXX
801 *
real function sla_gbrpvgrw(N, KL, KU, NCOLS, AB, LDAB, AFB, LDAFB)
SLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix...
Definition: sla_gbrpvgrw.f:119
subroutine slascl2(M, N, D, X, LDX)
SLASCL2 performs diagonal scaling on a vector.
Definition: slascl2.f:92
subroutine slaqgb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, EQUED)
SLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ...
Definition: slaqgb.f:161
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine sgbrfsx(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, IWORK, INFO)
SGBRFSX
Definition: sgbrfsx.f:442
subroutine sgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
SGBTRS
Definition: sgbtrs.f:140
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine sgbequb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
SGBEQUB
Definition: sgbequb.f:162
subroutine sgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
SGBTRF
Definition: sgbtrf.f:146

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