LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ chesvxx()

 subroutine chesvxx ( character fact, character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, real, dimension( * ) s, 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 )

CHESVXX computes the solution to system of linear equations A * X = B for HE matrices

Purpose:
```    CHESVXX uses the diagonal pivoting factorization to compute the
solution to a complex system of linear equations A * X = B, where
A is an N-by-N Hermitian matrix and X and B are N-by-NRHS
matrices.

If requested, both normwise and maximum componentwise error bounds
are returned. CHESVXX 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.

CHESVXX 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
CHESVXX 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 CHESVXX would itself produce.```
Description:
```    The following steps are performed:

1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:

diag(S)*A*diag(S)     *inv(diag(S))*X = diag(S)*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(S)*A*diag(S) and B by diag(S)*B.

2. If FACT = 'N' or 'E', the LU decomposition is used to factor
the matrix A (after equilibration if FACT = 'E') as

A = U * D * U**T,  if UPLO = 'U', or
A = L * D * L**T,  if UPLO = 'L',

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is Hermitian and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

3. If some D(i,i)=0, so that D 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(R) 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 S. 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] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [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) The Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S).``` [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 block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by CHETRF. If FACT = 'N', then AF is an output argument and on exit returns the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H.``` [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 details of the interchanges and the block structure of D, as determined by CHETRF. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. If FACT = 'N', then IPIV is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by CHETRF.``` [in,out] EQUED ``` EQUED is CHARACTER*1 Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). = 'Y': Both row and column equilibration, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.``` [in,out] S ``` S is REAL array, dimension (N) The scale factors for A. If EQUED = 'Y', A is multiplied on the left and right by diag(S). S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive. If S is output, each element of S is a power of the radix. If S is input, each element of S 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 EQUED = 'Y', B is overwritten by diag(S)*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(S))*X.``` [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 505 of file chesvxx.f.

509*
510* -- LAPACK driver routine --
511* -- LAPACK is a software package provided by Univ. of Tennessee, --
512* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
513*
514* .. Scalar Arguments ..
515 CHARACTER EQUED, FACT, UPLO
516 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
517 \$ N_ERR_BNDS
518 REAL RCOND, RPVGRW
519* ..
520* .. Array Arguments ..
521 INTEGER IPIV( * )
522 COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
523 \$ WORK( * ), X( LDX, * )
524 REAL S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
525 \$ ERR_BNDS_NORM( NRHS, * ),
526 \$ ERR_BNDS_COMP( NRHS, * )
527* ..
528*
529* ==================================================================
530*
531* .. Parameters ..
532 REAL ZERO, ONE
533 parameter( zero = 0.0e+0, one = 1.0e+0 )
534 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
535 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
536 INTEGER CMP_ERR_I, PIV_GROWTH_I
537 parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
538 \$ berr_i = 3 )
539 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
540 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
541 \$ piv_growth_i = 9 )
542* ..
543* .. Local Scalars ..
544 LOGICAL EQUIL, NOFACT, RCEQU
545 INTEGER INFEQU, J
546 REAL AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
547* ..
548* .. External Functions ..
549 EXTERNAL lsame, slamch, cla_herpvgrw
550 LOGICAL LSAME
551 REAL SLAMCH, CLA_HERPVGRW
552* ..
553* .. External Subroutines ..
554 EXTERNAL cheequb, chetrf, chetrs, clacpy,
556* ..
557* .. Intrinsic Functions ..
558 INTRINSIC max, min
559* ..
560* .. Executable Statements ..
561*
562 info = 0
563 nofact = lsame( fact, 'N' )
564 equil = lsame( fact, 'E' )
565 smlnum = slamch( 'Safe minimum' )
566 bignum = one / smlnum
567 IF( nofact .OR. equil ) THEN
568 equed = 'N'
569 rcequ = .false.
570 ELSE
571 rcequ = lsame( equed, 'Y' )
572 ENDIF
573*
574* Default is failure. If an input parameter is wrong or
575* factorization fails, make everything look horrible. Only the
576* pivot growth is set here, the rest is initialized in CHERFSX.
577*
578 rpvgrw = zero
579*
580* Test the input parameters. PARAMS is not tested until CHERFSX.
581*
582 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
583 \$ lsame( fact, 'F' ) ) THEN
584 info = -1
585 ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
586 \$ .NOT.lsame( uplo, 'L' ) ) THEN
587 info = -2
588 ELSE IF( n.LT.0 ) THEN
589 info = -3
590 ELSE IF( nrhs.LT.0 ) THEN
591 info = -4
592 ELSE IF( lda.LT.max( 1, n ) ) THEN
593 info = -6
594 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
595 info = -8
596 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
597 \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
598 info = -9
599 ELSE
600 IF ( rcequ ) THEN
601 smin = bignum
602 smax = zero
603 DO 10 j = 1, n
604 smin = min( smin, s( j ) )
605 smax = max( smax, s( j ) )
606 10 CONTINUE
607 IF( smin.LE.zero ) THEN
608 info = -10
609 ELSE IF( n.GT.0 ) THEN
610 scond = max( smin, smlnum ) / min( smax, bignum )
611 ELSE
612 scond = one
613 END IF
614 END IF
615 IF( info.EQ.0 ) THEN
616 IF( ldb.LT.max( 1, n ) ) THEN
617 info = -12
618 ELSE IF( ldx.LT.max( 1, n ) ) THEN
619 info = -14
620 END IF
621 END IF
622 END IF
623*
624 IF( info.NE.0 ) THEN
625 CALL xerbla( 'CHESVXX', -info )
626 RETURN
627 END IF
628*
629 IF( equil ) THEN
630*
631* Compute row and column scalings to equilibrate the matrix A.
632*
633 CALL cheequb( uplo, n, a, lda, s, scond, amax, work, infequ )
634 IF( infequ.EQ.0 ) THEN
635*
636* Equilibrate the matrix.
637*
638 CALL claqhe( uplo, n, a, lda, s, scond, amax, equed )
639 rcequ = lsame( equed, 'Y' )
640 END IF
641 END IF
642*
643* Scale the right-hand side.
644*
645 IF( rcequ ) CALL clascl2( n, nrhs, s, b, ldb )
646*
647 IF( nofact .OR. equil ) THEN
648*
649* Compute the LDL^H or UDU^H factorization of A.
650*
651 CALL clacpy( uplo, n, n, a, lda, af, ldaf )
652 CALL chetrf( uplo, n, af, ldaf, ipiv, work, 5*max(1,n), info )
653*
654* Return if INFO is non-zero.
655*
656 IF( info.GT.0 ) THEN
657*
658* Pivot in column INFO is exactly 0
659* Compute the reciprocal pivot growth factor of the
660* leading rank-deficient INFO columns of A.
661*
662 IF( n.GT.0 )
663 \$ rpvgrw = cla_herpvgrw( uplo, n, info, a, lda, af, ldaf,
664 \$ ipiv, rwork )
665 RETURN
666 END IF
667 END IF
668*
669* Compute the reciprocal pivot growth factor RPVGRW.
670*
671 IF( n.GT.0 )
672 \$ rpvgrw = cla_herpvgrw( uplo, n, info, a, lda, af, ldaf, ipiv,
673 \$ rwork )
674*
675* Compute the solution matrix X.
676*
677 CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
678 CALL chetrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
679*
680* Use iterative refinement to improve the computed solution and
681* compute error bounds and backward error estimates for it.
682*
683 CALL cherfsx( uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv,
684 \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
685 \$ err_bnds_comp, nparams, params, work, rwork, info )
686*
687* Scale solutions.
688*
689 IF ( rcequ ) THEN
690 CALL clascl2 ( n, nrhs, s, x, ldx )
691 END IF
692*
693 RETURN
694*
695* End of CHESVXX
696*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine cheequb(uplo, n, a, lda, s, scond, amax, work, info)
CHEEQUB
Definition cheequb.f:132
subroutine cherfsx(uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv, s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
CHERFSX
Definition cherfsx.f:401
subroutine chetrf(uplo, n, a, lda, ipiv, work, lwork, info)
CHETRF
Definition chetrf.f:177
subroutine chetrs(uplo, n, nrhs, a, lda, ipiv, b, ldb, info)
CHETRS
Definition chetrs.f:120
real function cla_herpvgrw(uplo, n, info, a, lda, af, ldaf, ipiv, work)
CLA_HERPVGRW
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
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
subroutine claqhe(uplo, n, a, lda, s, scond, amax, equed)
CLAQHE scales a Hermitian matrix.
Definition claqhe.f:134
subroutine clascl2(m, n, d, x, ldx)
CLASCL2 performs diagonal scaling on a matrix.
Definition clascl2.f:91
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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