 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dgesvxx()

 subroutine dgesvxx ( character FACT, character TRANS, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, 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 )

DGESVXX computes the solution to system of linear equations A * X = B for GE matrices

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

Purpose:
```    DGESVXX 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. DGESVXX 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.

DGESVXX 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
DGESVXX 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 DGESVXX 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] 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] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in R and/or C. A 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] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in,out] AF ``` AF is DOUBLE PRECISION array, dimension (LDAF,N) If FACT = 'F', then AF is an input argument and on entry contains the factors L and U from the factorization A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then AF 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] LDAF ``` LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).``` [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 535 of file dgesvxx.f.

540 *
541 * -- LAPACK driver routine --
542 * -- LAPACK is a software package provided by Univ. of Tennessee, --
543 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
544 *
545 * .. Scalar Arguments ..
546  CHARACTER EQUED, FACT, TRANS
547  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
548  \$ N_ERR_BNDS
549  DOUBLE PRECISION RCOND, RPVGRW
550 * ..
551 * .. Array Arguments ..
552  INTEGER IPIV( * ), IWORK( * )
553  DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
554  \$ X( LDX , * ),WORK( * )
555  DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
556  \$ ERR_BNDS_NORM( NRHS, * ),
557  \$ ERR_BNDS_COMP( NRHS, * )
558 * ..
559 *
560 * =====================================================================
561 *
562 * .. Parameters ..
563  DOUBLE PRECISION ZERO, ONE
564  parameter( zero = 0.0d+0, one = 1.0d+0 )
565  INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
566  INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
567  INTEGER CMP_ERR_I, PIV_GROWTH_I
568  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
569  \$ berr_i = 3 )
570  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
571  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
572  \$ piv_growth_i = 9 )
573 * ..
574 * .. Local Scalars ..
575  LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
576  INTEGER INFEQU, J
577  DOUBLE PRECISION AMAX, BIGNUM, COLCND, RCMAX, RCMIN, ROWCND,
578  \$ SMLNUM
579 * ..
580 * .. External Functions ..
581  EXTERNAL lsame, dlamch, dla_gerpvgrw
582  LOGICAL LSAME
583  DOUBLE PRECISION DLAMCH, DLA_GERPVGRW
584 * ..
585 * .. External Subroutines ..
586  EXTERNAL dgeequb, dgetrf, dgetrs, dlacpy, dlaqge,
588 * ..
589 * .. Intrinsic Functions ..
590  INTRINSIC max, min
591 * ..
592 * .. Executable Statements ..
593 *
594  info = 0
595  nofact = lsame( fact, 'N' )
596  equil = lsame( fact, 'E' )
597  notran = lsame( trans, 'N' )
598  smlnum = dlamch( 'Safe minimum' )
599  bignum = one / smlnum
600  IF( nofact .OR. equil ) THEN
601  equed = 'N'
602  rowequ = .false.
603  colequ = .false.
604  ELSE
605  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
606  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
607  END IF
608 *
609 * Default is failure. If an input parameter is wrong or
610 * factorization fails, make everything look horrible. Only the
611 * pivot growth is set here, the rest is initialized in DGERFSX.
612 *
613  rpvgrw = zero
614 *
615 * Test the input parameters. PARAMS is not tested until DGERFSX.
616 *
617  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
618  \$ lsame( fact, 'F' ) ) THEN
619  info = -1
620  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
621  \$ lsame( trans, 'C' ) ) THEN
622  info = -2
623  ELSE IF( n.LT.0 ) THEN
624  info = -3
625  ELSE IF( nrhs.LT.0 ) THEN
626  info = -4
627  ELSE IF( lda.LT.max( 1, n ) ) THEN
628  info = -6
629  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
630  info = -8
631  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
632  \$ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
633  info = -10
634  ELSE
635  IF( rowequ ) THEN
636  rcmin = bignum
637  rcmax = zero
638  DO 10 j = 1, n
639  rcmin = min( rcmin, r( j ) )
640  rcmax = max( rcmax, r( j ) )
641  10 CONTINUE
642  IF( rcmin.LE.zero ) THEN
643  info = -11
644  ELSE IF( n.GT.0 ) THEN
645  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
646  ELSE
647  rowcnd = one
648  END IF
649  END IF
650  IF( colequ .AND. info.EQ.0 ) THEN
651  rcmin = bignum
652  rcmax = zero
653  DO 20 j = 1, n
654  rcmin = min( rcmin, c( j ) )
655  rcmax = max( rcmax, c( j ) )
656  20 CONTINUE
657  IF( rcmin.LE.zero ) THEN
658  info = -12
659  ELSE IF( n.GT.0 ) THEN
660  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
661  ELSE
662  colcnd = one
663  END IF
664  END IF
665  IF( info.EQ.0 ) THEN
666  IF( ldb.LT.max( 1, n ) ) THEN
667  info = -14
668  ELSE IF( ldx.LT.max( 1, n ) ) THEN
669  info = -16
670  END IF
671  END IF
672  END IF
673 *
674  IF( info.NE.0 ) THEN
675  CALL xerbla( 'DGESVXX', -info )
676  RETURN
677  END IF
678 *
679  IF( equil ) THEN
680 *
681 * Compute row and column scalings to equilibrate the matrix A.
682 *
683  CALL dgeequb( n, n, a, lda, r, c, rowcnd, colcnd, amax,
684  \$ infequ )
685  IF( infequ.EQ.0 ) THEN
686 *
687 * Equilibrate the matrix.
688 *
689  CALL dlaqge( n, n, a, lda, r, c, rowcnd, colcnd, amax,
690  \$ equed )
691  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
692  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
693  END IF
694 *
695 * If the scaling factors are not applied, set them to 1.0.
696 *
697  IF ( .NOT.rowequ ) THEN
698  DO j = 1, n
699  r( j ) = 1.0d+0
700  END DO
701  END IF
702  IF ( .NOT.colequ ) THEN
703  DO j = 1, n
704  c( j ) = 1.0d+0
705  END DO
706  END IF
707  END IF
708 *
709 * Scale the right-hand side.
710 *
711  IF( notran ) THEN
712  IF( rowequ ) CALL dlascl2( n, nrhs, r, b, ldb )
713  ELSE
714  IF( colequ ) CALL dlascl2( n, nrhs, c, b, ldb )
715  END IF
716 *
717  IF( nofact .OR. equil ) THEN
718 *
719 * Compute the LU factorization of A.
720 *
721  CALL dlacpy( 'Full', n, n, a, lda, af, ldaf )
722  CALL dgetrf( n, n, af, ldaf, ipiv, info )
723 *
724 * Return if INFO is non-zero.
725 *
726  IF( info.GT.0 ) THEN
727 *
728 * Pivot in column INFO is exactly 0
729 * Compute the reciprocal pivot growth factor of the
730 * leading rank-deficient INFO columns of A.
731 *
732  rpvgrw = dla_gerpvgrw( n, info, a, lda, af, ldaf )
733  RETURN
734  END IF
735  END IF
736 *
737 * Compute the reciprocal pivot growth factor RPVGRW.
738 *
739  rpvgrw = dla_gerpvgrw( n, n, a, lda, af, ldaf )
740 *
741 * Compute the solution matrix X.
742 *
743  CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
744  CALL dgetrs( trans, n, nrhs, af, ldaf, ipiv, x, ldx, info )
745 *
746 * Use iterative refinement to improve the computed solution and
747 * compute error bounds and backward error estimates for it.
748 *
749  CALL dgerfsx( trans, equed, n, nrhs, a, lda, af, ldaf,
750  \$ ipiv, r, c, b, ldb, x, ldx, rcond, berr,
751  \$ n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params,
752  \$ work, iwork, info )
753 *
754 * Scale solutions.
755 *
756  IF ( colequ .AND. notran ) THEN
757  CALL dlascl2 ( n, nrhs, c, x, ldx )
758  ELSE IF ( rowequ .AND. .NOT.notran ) THEN
759  CALL dlascl2 ( n, nrhs, r, x, ldx )
760  END IF
761 *
762  RETURN
763 *
764 * End of DGESVXX
765
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 dlaqge(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
DLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
Definition: dlaqge.f:142
subroutine dgetrf(M, N, A, LDA, IPIV, INFO)
DGETRF
Definition: dgetrf.f:108
subroutine dgeequb(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
DGEEQUB
Definition: dgeequb.f:146
subroutine dgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DGETRS
Definition: dgetrs.f:121
double precision function dla_gerpvgrw(N, NCOLS, A, LDA, AF, LDAF)
DLA_GERPVGRW
Definition: dla_gerpvgrw.f:100
subroutine dgerfsx(TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
DGERFSX
Definition: dgerfsx.f:414
subroutine dlascl2(M, N, D, X, LDX)
DLASCL2 performs diagonal scaling on a vector.
Definition: dlascl2.f:90
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