LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ sgbsvxx()

 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

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

Definition at line 558 of file sgbsvxx.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( * ), IWORK( * )
576 REAL 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, * )
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, sla_gbrpvgrw
605 LOGICAL LSAME
606 REAL SLAMCH, SLA_GBRPVGRW
607* ..
608* .. External Subroutines ..
609 EXTERNAL sgbequb, sgbtrf, sgbtrs, slacpy, slaqgb,
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 SGBRFSX.
635*
636 rpvgrw = zero
637*
638* Test the input parameters. PARAMS is not tested until SGBRFSX.
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( 'SGBSVXX', -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 sgbequb( 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 slaqgb( 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 slascl2(n, nrhs, r, b, ldb)
740 ELSE
741 IF( colequ ) CALL slascl2(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 sgbtrf( 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 = sla_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 = sla_gbrpvgrw( n, kl, ku, n, ab, ldab, afb, ldafb )
772*
773* Compute the solution matrix X.
774*
775 CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
776 CALL sgbtrs( 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 sgbrfsx( 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, iwork, info )
786*
787* Scale solutions.
788*
789 IF ( colequ .AND. notran ) THEN
790 CALL slascl2 ( n, nrhs, c, x, ldx )
791 ELSE IF ( rowequ .AND. .NOT.notran ) THEN
792 CALL slascl2 ( n, nrhs, r, x, ldx )
793 END IF
794*
795 RETURN
796*
797* End of SGBSVXX
798*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine sgbequb(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info)
SGBEQUB
Definition sgbequb.f:160
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:440
subroutine sgbtrf(m, n, kl, ku, ab, ldab, ipiv, info)
SGBTRF
Definition sgbtrf.f:144
subroutine sgbtrs(trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
SGBTRS
Definition sgbtrs.f:138
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.
subroutine slacpy(uplo, m, n, a, lda, b, ldb)
SLACPY copies all or part of one two-dimensional array to another.
Definition slacpy.f:103
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
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:159
subroutine slascl2(m, n, d, x, ldx)
SLASCL2 performs diagonal scaling on a matrix.
Definition slascl2.f:90
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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