LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dgbsvxx()

 subroutine dgbsvxx ( character FACT, character TRANS, integer N, integer KL, integer KU, integer NRHS, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldafb, * ) AFB, integer LDAFB, integer, dimension( * ) IPIV, character EQUED, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision, dimension( ldb, * ) B, integer LDB, double precision, 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, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

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

Purpose:
```    DGBSVXX uses the LU factorization to compute the solution to a
double precision 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. DGBSVXX 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.

DGBSVXX 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
DGBSVXX 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 DGBSVXX 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DGBTRF. 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 DGETRF; 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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.```

Definition at line 555 of file dgbsvxx.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, KL, KU
569 DOUBLE PRECISION RCOND, RPVGRW
570* ..
571* .. Array Arguments ..
572 INTEGER IPIV( * ), IWORK( * )
573 DOUBLE PRECISION 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, * )
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
597 DOUBLE PRECISION AMAX, BIGNUM, COLCND, RCMAX, RCMIN,
598 \$ ROWCND, SMLNUM
599* ..
600* .. External Functions ..
601 EXTERNAL lsame, dlamch, dla_gbrpvgrw
602 LOGICAL LSAME
603 DOUBLE PRECISION DLAMCH, DLA_GBRPVGRW
604* ..
605* .. External Subroutines ..
606 EXTERNAL dgbequb, dgbtrf, dgbtrs, dlacpy, dlaqgb,
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 DGBRFSX.
632*
633 rpvgrw = zero
634*
635* Test the input parameters. PARAMS is not tested until DGBRFSX.
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( 'DGBSVXX', -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 dgbequb( 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 dlaqgb( 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 dlascl2(n, nrhs, r, b, ldb)
737 ELSE
738 IF( colequ ) CALL dlascl2(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 dgbtrf( 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 = dla_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 = dla_gbrpvgrw( n, kl, ku, n, ab, ldab, afb, ldafb )
769*
770* Compute the solution matrix X.
771*
772 CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
773 CALL dgbtrs( 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 dgbrfsx( 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, iwork, info )
783*
784* Scale solutions.
785*
786 IF ( colequ .AND. notran ) THEN
787 CALL dlascl2 ( n, nrhs, c, x, ldx )
788 ELSE IF ( rowequ .AND. .NOT.notran ) THEN
789 CALL dlascl2 ( n, nrhs, r, x, ldx )
790 END IF
791*
792 RETURN
793*
794* End of DGBSVXX
795*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dlaqgb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, EQUED)
DLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ.
Definition: dlaqgb.f:159
double precision function dla_gbrpvgrw(N, KL, KU, NCOLS, AB, LDAB, AFB, LDAFB)
DLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.
Definition: dla_gbrpvgrw.f:117
subroutine dgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
DGBTRS
Definition: dgbtrs.f:138
subroutine dgbequb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
DGBEQUB
Definition: dgbequb.f:160
subroutine dgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
DGBTRF
Definition: dgbtrf.f:144
subroutine dgbrfsx(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)
DGBRFSX
Definition: dgbrfsx.f:440
subroutine dlascl2(M, N, D, X, LDX)
DLASCL2 performs diagonal scaling on a matrix.
Definition: dlascl2.f:90
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