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

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

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

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

Purpose:
```    CGESVXX 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. CGESVXX 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.

CGESVXX 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
CGESVXX 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 CGESVXX 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)``` [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 COMPLEX 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 COMPLEX 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 CGETRF. 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 CGETRF; 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.```

Definition at line 538 of file cgesvxx.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( * )
556 COMPLEX 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, * ), RWORK( * )
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,
581 \$ ROWCND, SMLNUM
582* ..
583* .. External Functions ..
584 EXTERNAL lsame, slamch, cla_gerpvgrw
585 LOGICAL LSAME
586 REAL SLAMCH, CLA_GERPVGRW
587* ..
588* .. External Subroutines ..
589 EXTERNAL cgeequb, cgetrf, cgetrs, clacpy, claqge,
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 CGERFSX.
615*
616 rpvgrw = zero
617*
618* Test the input parameters. PARAMS is not tested until CGERFSX.
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( 'CGESVXX', -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 cgeequb( 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 claqge( 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 clascl2( n, nrhs, r, b, ldb )
716 ELSE
717 IF( colequ ) CALL clascl2( 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 clacpy( 'Full', n, n, a, lda, af, ldaf )
725 CALL cgetrf( 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 = cla_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 = cla_gerpvgrw( n, n, a, lda, af, ldaf )
743*
744* Compute the solution matrix X.
745*
746 CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
747 CALL cgetrs( 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 cgerfsx( 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, rwork, info )
756*
757* Scale solutions.
758*
759 IF ( colequ .AND. notran ) THEN
760 CALL clascl2 ( n, nrhs, c, x, ldx )
761 ELSE IF ( rowequ .AND. .NOT.notran ) THEN
762 CALL clascl2 ( n, nrhs, r, x, ldx )
763 END IF
764*
765 RETURN
766*
767* End of CGESVXX
768*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine claqge(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
CLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
Definition: claqge.f:143
subroutine cgerfsx(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, RWORK, INFO)
CGERFSX
Definition: cgerfsx.f:414
real function cla_gerpvgrw(N, NCOLS, A, LDA, AF, LDAF)
CLA_GERPVGRW multiplies a square real matrix by a complex matrix.
Definition: cla_gerpvgrw.f:98
subroutine cgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
CGETRS
Definition: cgetrs.f:121
subroutine cgetrf(M, N, A, LDA, IPIV, INFO)
CGETRF
Definition: cgetrf.f:108
subroutine cgeequb(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
CGEEQUB
Definition: cgeequb.f:147
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
subroutine clascl2(M, N, D, X, LDX)
CLASCL2 performs diagonal scaling on a matrix.
Definition: clascl2.f:91
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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