 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ sgesvxx()

 subroutine sgesvxx ( character FACT, character TRANS, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, 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 )

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

Purpose:
```    SGESVXX 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. SGESVXX 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.

SGESVXX 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
SGESVXX 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 SGESVXX 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] 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 REAL 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 REAL 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 SGETRF. 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 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 REAL 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 REAL 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.```

Definition at line 538 of file sgesvxx.f.

543 *
544 * -- LAPACK driver routine --
545 * -- LAPACK is a software package provided by Univ. of Tennessee, --
546 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
547 *
548 * .. Scalar Arguments ..
549  CHARACTER EQUED, FACT, TRANS
550  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
551  \$ N_ERR_BNDS
552  REAL RCOND, RPVGRW
553 * ..
554 * .. Array Arguments ..
555  INTEGER IPIV( * ), IWORK( * )
556  REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
557  \$ X( LDX , * ),WORK( * )
558  REAL R( * ), C( * ), PARAMS( * ), BERR( * ),
559  \$ ERR_BNDS_NORM( NRHS, * ),
560  \$ ERR_BNDS_COMP( NRHS, * )
561 * ..
562 *
563 * ==================================================================
564 *
565 * .. Parameters ..
566  REAL ZERO, ONE
567  parameter( zero = 0.0e+0, one = 1.0e+0 )
568  INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
569  INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
570  INTEGER CMP_ERR_I, PIV_GROWTH_I
571  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
572  \$ berr_i = 3 )
573  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
574  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
575  \$ piv_growth_i = 9 )
576 * ..
577 * .. Local Scalars ..
578  LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
579  INTEGER INFEQU, J
580  REAL AMAX, BIGNUM, COLCND, RCMAX, RCMIN, ROWCND,
581  \$ SMLNUM
582 * ..
583 * .. External Functions ..
584  EXTERNAL lsame, slamch, sla_gerpvgrw
585  LOGICAL LSAME
586  REAL SLAMCH, SLA_GERPVGRW
587 * ..
588 * .. External Subroutines ..
589  EXTERNAL sgeequb, sgetrf, sgetrs, slacpy, slaqge,
591 * ..
592 * .. Intrinsic Functions ..
593  INTRINSIC max, min
594 * ..
595 * .. Executable Statements ..
596 *
597  info = 0
598  nofact = lsame( fact, 'N' )
599  equil = lsame( fact, 'E' )
600  notran = lsame( trans, 'N' )
601  smlnum = slamch( 'Safe minimum' )
602  bignum = one / smlnum
603  IF( nofact .OR. equil ) THEN
604  equed = 'N'
605  rowequ = .false.
606  colequ = .false.
607  ELSE
608  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
609  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
610  END IF
611 *
612 * Default is failure. If an input parameter is wrong or
613 * factorization fails, make everything look horrible. Only the
614 * pivot growth is set here, the rest is initialized in SGERFSX.
615 *
616  rpvgrw = zero
617 *
618 * Test the input parameters. PARAMS is not tested until SGERFSX.
619 *
620  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
621  \$ lsame( fact, 'F' ) ) THEN
622  info = -1
623  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
624  \$ lsame( trans, 'C' ) ) THEN
625  info = -2
626  ELSE IF( n.LT.0 ) THEN
627  info = -3
628  ELSE IF( nrhs.LT.0 ) THEN
629  info = -4
630  ELSE IF( lda.LT.max( 1, n ) ) THEN
631  info = -6
632  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
633  info = -8
634  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
635  \$ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
636  info = -10
637  ELSE
638  IF( rowequ ) THEN
639  rcmin = bignum
640  rcmax = zero
641  DO 10 j = 1, n
642  rcmin = min( rcmin, r( j ) )
643  rcmax = max( rcmax, r( j ) )
644  10 CONTINUE
645  IF( rcmin.LE.zero ) THEN
646  info = -11
647  ELSE IF( n.GT.0 ) THEN
648  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
649  ELSE
650  rowcnd = one
651  END IF
652  END IF
653  IF( colequ .AND. info.EQ.0 ) THEN
654  rcmin = bignum
655  rcmax = zero
656  DO 20 j = 1, n
657  rcmin = min( rcmin, c( j ) )
658  rcmax = max( rcmax, c( j ) )
659  20 CONTINUE
660  IF( rcmin.LE.zero ) THEN
661  info = -12
662  ELSE IF( n.GT.0 ) THEN
663  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
664  ELSE
665  colcnd = one
666  END IF
667  END IF
668  IF( info.EQ.0 ) THEN
669  IF( ldb.LT.max( 1, n ) ) THEN
670  info = -14
671  ELSE IF( ldx.LT.max( 1, n ) ) THEN
672  info = -16
673  END IF
674  END IF
675  END IF
676 *
677  IF( info.NE.0 ) THEN
678  CALL xerbla( 'SGESVXX', -info )
679  RETURN
680  END IF
681 *
682  IF( equil ) THEN
683 *
684 * Compute row and column scalings to equilibrate the matrix A.
685 *
686  CALL sgeequb( n, n, a, lda, r, c, rowcnd, colcnd, amax,
687  \$ infequ )
688  IF( infequ.EQ.0 ) THEN
689 *
690 * Equilibrate the matrix.
691 *
692  CALL slaqge( n, n, a, lda, r, c, rowcnd, colcnd, amax,
693  \$ equed )
694  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
695  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
696  END IF
697 *
698 * If the scaling factors are not applied, set them to 1.0.
699 *
700  IF ( .NOT.rowequ ) THEN
701  DO j = 1, n
702  r( j ) = 1.0
703  END DO
704  END IF
705  IF ( .NOT.colequ ) THEN
706  DO j = 1, n
707  c( j ) = 1.0
708  END DO
709  END IF
710  END IF
711 *
712 * Scale the right-hand side.
713 *
714  IF( notran ) THEN
715  IF( rowequ ) CALL slascl2( n, nrhs, r, b, ldb )
716  ELSE
717  IF( colequ ) CALL slascl2( n, nrhs, c, b, ldb )
718  END IF
719 *
720  IF( nofact .OR. equil ) THEN
721 *
722 * Compute the LU factorization of A.
723 *
724  CALL slacpy( 'Full', n, n, a, lda, af, ldaf )
725  CALL sgetrf( n, n, af, ldaf, ipiv, info )
726 *
727 * Return if INFO is non-zero.
728 *
729  IF( info.GT.0 ) THEN
730 *
731 * Pivot in column INFO is exactly 0
732 * Compute the reciprocal pivot growth factor of the
733 * leading rank-deficient INFO columns of A.
734 *
735  rpvgrw = sla_gerpvgrw( n, info, a, lda, af, ldaf )
736  RETURN
737  END IF
738  END IF
739 *
740 * Compute the reciprocal pivot growth factor RPVGRW.
741 *
742  rpvgrw = sla_gerpvgrw( n, n, a, lda, af, ldaf )
743 *
744 * Compute the solution matrix X.
745 *
746  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
747  CALL sgetrs( trans, n, nrhs, af, ldaf, ipiv, x, ldx, info )
748 *
749 * Use iterative refinement to improve the computed solution and
750 * compute error bounds and backward error estimates for it.
751 *
752  CALL sgerfsx( trans, equed, n, nrhs, a, lda, af, ldaf,
753  \$ ipiv, r, c, b, ldb, x, ldx, rcond, berr,
754  \$ n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params,
755  \$ work, iwork, info )
756 *
757 * Scale solutions.
758 *
759  IF ( colequ .AND. notran ) THEN
760  CALL slascl2 ( n, nrhs, c, x, ldx )
761  ELSE IF ( rowequ .AND. .NOT.notran ) THEN
762  CALL slascl2 ( n, nrhs, r, x, ldx )
763  END IF
764 *
765  RETURN
766 *
767 * End of SGESVXX
768
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
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine slaqge(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
SLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
Definition: slaqge.f:142
subroutine sgetrf(M, N, A, LDA, IPIV, INFO)
SGETRF
Definition: sgetrf.f:108
subroutine sgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SGETRS
Definition: sgetrs.f:121
real function sla_gerpvgrw(N, NCOLS, A, LDA, AF, LDAF)
SLA_GERPVGRW
Definition: sla_gerpvgrw.f:97
subroutine sgerfsx(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)
SGERFSX
Definition: sgerfsx.f:414
subroutine sgeequb(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
SGEEQUB
Definition: sgeequb.f:146
subroutine slascl2(M, N, D, X, LDX)
SLASCL2 performs diagonal scaling on a vector.
Definition: slascl2.f:90
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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