LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ cgbsvxx()

subroutine cgbsvxx ( character  fact,
character  trans,
integer  n,
integer  kl,
integer  ku,
integer  nrhs,
complex, dimension( ldab, * )  ab,
integer  ldab,
complex, dimension( ldafb, * )  afb,
integer  ldafb,
integer, dimension( * )  ipiv,
character  equed,
real, dimension( * )  r,
real, dimension( * )  c,
complex, dimension( ldb, * )  b,
integer  ldb,
complex, 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,
complex, dimension( * )  work,
real, dimension( * )  rwork,
integer  info 
)

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

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

Purpose:
    CGBSVXX uses the LU factorization to compute the solution to a
    complex 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. CGBSVXX 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.

    CGBSVXX 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
    CGBSVXX 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 CGBSVXX 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 COMPLEX 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 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 CGBTRF.  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 CGETRF; 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 COMPLEX 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 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 CGESVX, this quantity is
     returned in RWORK(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 < 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 <= 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 < 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 COMPLEX array, dimension (2*N)
[out]RWORK
          RWORK is REAL 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 558 of file cgbsvxx.f.

563*
564* -- LAPACK driver routine --
565* -- LAPACK is a software package provided by Univ. of Tennessee, --
566* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
567*
568* .. Scalar Arguments ..
569 CHARACTER EQUED, FACT, TRANS
570 INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
571 $ N_ERR_BNDS
572 REAL RCOND, RPVGRW
573* ..
574* .. Array Arguments ..
575 INTEGER IPIV( * )
576 COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
577 $ X( LDX , * ),WORK( * )
578 REAL R( * ), C( * ), PARAMS( * ), BERR( * ),
579 $ ERR_BNDS_NORM( NRHS, * ),
580 $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
581* ..
582*
583* ==================================================================
584*
585* .. Parameters ..
586 REAL ZERO, ONE
587 parameter( zero = 0.0e+0, one = 1.0e+0 )
588 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
589 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
590 INTEGER CMP_ERR_I, PIV_GROWTH_I
591 parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
592 $ berr_i = 3 )
593 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
594 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
595 $ piv_growth_i = 9 )
596* ..
597* .. Local Scalars ..
598 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
599 INTEGER INFEQU, I, J, KL, KU
600 REAL AMAX, BIGNUM, COLCND, RCMAX, RCMIN,
601 $ ROWCND, SMLNUM
602* ..
603* .. External Functions ..
604 EXTERNAL lsame, slamch, cla_gbrpvgrw
605 LOGICAL LSAME
606 REAL SLAMCH, CLA_GBRPVGRW
607* ..
608* .. External Subroutines ..
609 EXTERNAL cgbequb, cgbtrf, cgbtrs, clacpy, claqgb,
611* ..
612* .. Intrinsic Functions ..
613 INTRINSIC max, min
614* ..
615* .. Executable Statements ..
616*
617 info = 0
618 nofact = lsame( fact, 'N' )
619 equil = lsame( fact, 'E' )
620 notran = lsame( trans, 'N' )
621 smlnum = slamch( 'Safe minimum' )
622 bignum = one / smlnum
623 IF( nofact .OR. equil ) THEN
624 equed = 'N'
625 rowequ = .false.
626 colequ = .false.
627 ELSE
628 rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
629 colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
630 END IF
631*
632* Default is failure. If an input parameter is wrong or
633* factorization fails, make everything look horrible. Only the
634* pivot growth is set here, the rest is initialized in CGBRFSX.
635*
636 rpvgrw = zero
637*
638* Test the input parameters. PARAMS is not tested until CGERFSX.
639*
640 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
641 $ lsame( fact, 'F' ) ) THEN
642 info = -1
643 ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
644 $ lsame( trans, 'C' ) ) THEN
645 info = -2
646 ELSE IF( n.LT.0 ) THEN
647 info = -3
648 ELSE IF( kl.LT.0 ) THEN
649 info = -4
650 ELSE IF( ku.LT.0 ) THEN
651 info = -5
652 ELSE IF( nrhs.LT.0 ) THEN
653 info = -6
654 ELSE IF( ldab.LT.kl+ku+1 ) THEN
655 info = -8
656 ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
657 info = -10
658 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
659 $ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
660 info = -12
661 ELSE
662 IF( rowequ ) THEN
663 rcmin = bignum
664 rcmax = zero
665 DO 10 j = 1, n
666 rcmin = min( rcmin, r( j ) )
667 rcmax = max( rcmax, r( j ) )
668 10 CONTINUE
669 IF( rcmin.LE.zero ) THEN
670 info = -13
671 ELSE IF( n.GT.0 ) THEN
672 rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
673 ELSE
674 rowcnd = one
675 END IF
676 END IF
677 IF( colequ .AND. info.EQ.0 ) THEN
678 rcmin = bignum
679 rcmax = zero
680 DO 20 j = 1, n
681 rcmin = min( rcmin, c( j ) )
682 rcmax = max( rcmax, c( j ) )
683 20 CONTINUE
684 IF( rcmin.LE.zero ) THEN
685 info = -14
686 ELSE IF( n.GT.0 ) THEN
687 colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
688 ELSE
689 colcnd = one
690 END IF
691 END IF
692 IF( info.EQ.0 ) THEN
693 IF( ldb.LT.max( 1, n ) ) THEN
694 info = -15
695 ELSE IF( ldx.LT.max( 1, n ) ) THEN
696 info = -16
697 END IF
698 END IF
699 END IF
700*
701 IF( info.NE.0 ) THEN
702 CALL xerbla( 'CGBSVXX', -info )
703 RETURN
704 END IF
705*
706 IF( equil ) THEN
707*
708* Compute row and column scalings to equilibrate the matrix A.
709*
710 CALL cgbequb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
711 $ amax, infequ )
712 IF( infequ.EQ.0 ) THEN
713*
714* Equilibrate the matrix.
715*
716 CALL claqgb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
717 $ amax, equed )
718 rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
719 colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
720 END IF
721*
722* If the scaling factors are not applied, set them to 1.0.
723*
724 IF ( .NOT.rowequ ) THEN
725 DO j = 1, n
726 r( j ) = 1.0
727 END DO
728 END IF
729 IF ( .NOT.colequ ) THEN
730 DO j = 1, n
731 c( j ) = 1.0
732 END DO
733 END IF
734 END IF
735*
736* Scale the right-hand side.
737*
738 IF( notran ) THEN
739 IF( rowequ ) CALL clascl2( n, nrhs, r, b, ldb )
740 ELSE
741 IF( colequ ) CALL clascl2( n, nrhs, c, b, ldb )
742 END IF
743*
744 IF( nofact .OR. equil ) THEN
745*
746* Compute the LU factorization of A.
747*
748 DO 40, j = 1, n
749 DO 30, i = kl+1, 2*kl+ku+1
750 afb( i, j ) = ab( i-kl, j )
751 30 CONTINUE
752 40 CONTINUE
753 CALL cgbtrf( n, n, kl, ku, afb, ldafb, ipiv, info )
754*
755* Return if INFO is non-zero.
756*
757 IF( info.GT.0 ) THEN
758*
759* Pivot in column INFO is exactly 0
760* Compute the reciprocal pivot growth factor of the
761* leading rank-deficient INFO columns of A.
762*
763 rpvgrw = cla_gbrpvgrw( n, kl, ku, info, ab, ldab, afb,
764 $ ldafb )
765 RETURN
766 END IF
767 END IF
768*
769* Compute the reciprocal pivot growth factor RPVGRW.
770*
771 rpvgrw = cla_gbrpvgrw( n, kl, ku, n, ab, ldab, afb, ldafb )
772*
773* Compute the solution matrix X.
774*
775 CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
776 CALL cgbtrs( trans, n, kl, ku, nrhs, afb, ldafb, ipiv, x, ldx,
777 $ info )
778*
779* Use iterative refinement to improve the computed solution and
780* compute error bounds and backward error estimates for it.
781*
782 CALL cgbrfsx( trans, equed, n, kl, ku, nrhs, ab, ldab, afb, ldafb,
783 $ ipiv, r, c, b, ldb, x, ldx, rcond, berr,
784 $ n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params,
785 $ work, rwork, info )
786
787*
788* Scale solutions.
789*
790 IF ( colequ .AND. notran ) THEN
791 CALL clascl2( n, nrhs, c, x, ldx )
792 ELSE IF ( rowequ .AND. .NOT.notran ) THEN
793 CALL clascl2( n, nrhs, r, x, ldx )
794 END IF
795*
796 RETURN
797*
798* End of CGBSVXX
799*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine cgbequb(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info)
CGBEQUB
Definition cgbequb.f:161
subroutine cgbrfsx(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)
CGBRFSX
Definition cgbrfsx.f:440
subroutine cgbtrf(m, n, kl, ku, ab, ldab, ipiv, info)
CGBTRF
Definition cgbtrf.f:144
subroutine cgbtrs(trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
CGBTRS
Definition cgbtrs.f:138
real function cla_gbrpvgrw(n, kl, ku, ncols, ab, ldab, afb, ldafb)
CLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.
subroutine clacpy(uplo, m, n, a, lda, b, ldb)
CLACPY copies all or part of one two-dimensional array to another.
Definition clacpy.f:103
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
subroutine claqgb(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, equed)
CLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ.
Definition claqgb.f:160
subroutine clascl2(m, n, d, x, ldx)
CLASCL2 performs diagonal scaling on a matrix.
Definition clascl2.f:91
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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