 LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 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

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 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 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 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.```
Date
April 2012

Definition at line 565 of file cgbsvxx.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( * )
579  COMPLEX 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, * ), rwork( * )
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, cla_gbrpvgrw
608  LOGICAL lsame
609  REAL slamch, cla_gbrpvgrw
610 * ..
611 * .. External Subroutines ..
612  EXTERNAL cgbequb, cgbtrf, cgbtrs, clacpy, claqgb,
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 CGBRFSX.
638 *
639  rpvgrw = zero
640 *
641 * Test the input parameters. PARAMS is not tested until SGERFSX.
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( 'CGBSVXX', -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 cgbequb( 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 claqgb( 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 clascl2( n, nrhs, r, b, ldb )
743  ELSE
744  IF( colequ ) CALL clascl2( 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 cgbtrf( 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 = cla_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 = cla_gbrpvgrw( n, kl, ku, n, ab, ldab, afb, ldafb )
775 *
776 * Compute the solution matrix X.
777 *
778  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
779  CALL cgbtrs( 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 cgbrfsx( 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, rwork, info )
789
790 *
791 * Scale solutions.
792 *
793  IF ( colequ .AND. notran ) THEN
794  CALL clascl2( n, nrhs, c, x, ldx )
795  ELSE IF ( rowequ .AND. .NOT.notran ) THEN
796  CALL clascl2( n, nrhs, r, x, ldx )
797  END IF
798 *
799  RETURN
800 *
801 * End of CGBSVXX
802 *
subroutine cgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
CGBTRF
Definition: cgbtrf.f:146
subroutine cgbequb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
CGBEQUB
Definition: cgbequb.f:163
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:442
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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:162
subroutine clascl2(M, N, D, X, LDX)
CLASCL2 performs diagonal scaling on a vector.
Definition: clascl2.f:93
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
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...
Definition: cla_gbrpvgrw.f:119
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine cgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
CGBTRS
Definition: cgbtrs.f:140

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