LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine zgbsvxx ( character FACT, character TRANS, integer N, integer KL, integer KU, integer NRHS, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldafb, * ) AFB, integer LDAFB, integer, dimension( * ) IPIV, character EQUED, double precision, dimension( * ) R, double precision, dimension( * ) C, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, 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, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO )

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

Purpose:
```    ZGBSVXX uses the LU factorization to compute the solution to a
complex*16 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. ZGBSVXX 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.

ZGBSVXX 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
ZGBSVXX 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 ZGBSVXX 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 COMPLEX*16 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*16 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 ZGBTRF. 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 COMPLEX*16 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*16 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.```
Date
April 2012

Definition at line 562 of file zgbsvxx.f.

562 *
563 * -- LAPACK driver routine (version 3.4.1) --
564 * -- LAPACK is a software package provided by Univ. of Tennessee, --
565 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
566 * April 2012
567 *
568 * .. Scalar Arguments ..
569  CHARACTER equed, fact, trans
570  INTEGER info, ldab, ldafb, ldb, ldx, n, nrhs, nparams,
571  \$ n_err_bnds
572  DOUBLE PRECISION rcond, rpvgrw
573 * ..
574 * .. Array Arguments ..
575  INTEGER ipiv( * )
576  COMPLEX*16 ab( ldab, * ), afb( ldafb, * ), b( ldb, * ),
577  \$ x( ldx , * ),work( * )
578  DOUBLE PRECISION r( * ), c( * ), params( * ), berr( * ),
579  \$ err_bnds_norm( nrhs, * ),
580  \$ err_bnds_comp( nrhs, * ), rwork( * )
581 * ..
582 *
583 * ==================================================================
584 *
585 * .. Parameters ..
586  DOUBLE PRECISION zero, one
587  parameter ( zero = 0.0d+0, one = 1.0d+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  DOUBLE PRECISION amax, bignum, colcnd, rcmax, rcmin,
601  \$ rowcnd, smlnum
602 * ..
603 * .. External Functions ..
604  EXTERNAL lsame, dlamch, zla_gbrpvgrw
605  LOGICAL lsame
606  DOUBLE PRECISION dlamch, zla_gbrpvgrw
607 * ..
608 * .. External Subroutines ..
609  EXTERNAL zgbequb, zgbtrf, zgbtrs, zlacpy, zlaqgb,
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 = dlamch( '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 ZGBRFSX.
635 *
636  rpvgrw = zero
637 *
638 * Test the input parameters. PARAMS is not tested until DGERFSX.
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( 'ZGBSVXX', -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 zgbequb( 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 zlaqgb( 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 ) = 1.0d+0
727  END DO
728  END IF
729  IF ( .NOT.colequ ) THEN
730  DO j = 1, n
731  c( j ) = 1.0d+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 zlascl2( n, nrhs, r, b, ldb )
740  ELSE
741  IF( colequ ) CALL zlascl2( 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 zgbtrf( 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 = zla_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 = zla_gbrpvgrw( n, kl, ku, n, ab, ldab, afb, ldafb )
772 *
773 * Compute the solution matrix X.
774 *
775  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
776  CALL zgbtrs( 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 zgbrfsx( 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 zlascl2( n, nrhs, c, x, ldx )
792  ELSE IF ( rowequ .AND. .NOT.notran ) THEN
793  CALL zlascl2( n, nrhs, r, x, ldx )
794  END IF
795 *
796  RETURN
797 *
798 * End of ZGBSVXX
799 *
subroutine zlaqgb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, EQUED)
ZLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ...
Definition: zlaqgb.f:162
double precision function zla_gbrpvgrw(N, KL, KU, NCOLS, AB, LDAB, AFB, LDAFB)
ZLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix...
Definition: zla_gbrpvgrw.f:119
subroutine zgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
ZGBTRS
Definition: zgbtrs.f:140
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine zgbrfsx(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)
ZGBRFSX
Definition: zgbrfsx.f:442
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zgbequb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
ZGBEQUB
Definition: zgbequb.f:163
subroutine zgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
ZGBTRF
Definition: zgbtrf.f:146
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zlascl2(M, N, D, X, LDX)
ZLASCL2 performs diagonal scaling on a vector.
Definition: zlascl2.f:93

Here is the call graph for this function:

Here is the caller graph for this function: