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

 subroutine zhesvxx ( character FACT, character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, character EQUED, double precision, dimension( * ) S, 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 )

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

Purpose:
```    ZHESVXX uses the diagonal pivoting factorization to compute the
solution to a complex*16 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. ZHESVXX 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.

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

1. If FACT = 'E', double precision 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*16 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*16 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 ZHETRF. 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 ZHETRF. 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 ZHETRF.``` [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 DOUBLE PRECISION 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*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 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*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(S))*X.``` [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 502 of file zhesvxx.f.

506*
507* -- LAPACK driver routine --
508* -- LAPACK is a software package provided by Univ. of Tennessee, --
509* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
510*
511* .. Scalar Arguments ..
512 CHARACTER EQUED, FACT, UPLO
513 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
514 \$ N_ERR_BNDS
515 DOUBLE PRECISION RCOND, RPVGRW
516* ..
517* .. Array Arguments ..
518 INTEGER IPIV( * )
519 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
520 \$ WORK( * ), X( LDX, * )
521 DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
522 \$ ERR_BNDS_NORM( NRHS, * ),
523 \$ ERR_BNDS_COMP( NRHS, * )
524* ..
525*
526* ==================================================================
527*
528* .. Parameters ..
529 DOUBLE PRECISION ZERO, ONE
530 parameter( zero = 0.0d+0, one = 1.0d+0 )
531 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
532 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
533 INTEGER CMP_ERR_I, PIV_GROWTH_I
534 parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
535 \$ berr_i = 3 )
536 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
537 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
538 \$ piv_growth_i = 9 )
539* ..
540* .. Local Scalars ..
541 LOGICAL EQUIL, NOFACT, RCEQU
542 INTEGER INFEQU, J
543 DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
544* ..
545* .. External Functions ..
546 EXTERNAL lsame, dlamch, zla_herpvgrw
547 LOGICAL LSAME
548 DOUBLE PRECISION DLAMCH, ZLA_HERPVGRW
549* ..
550* .. External Subroutines ..
551 EXTERNAL zheequb, zhetrf, zhetrs, zlacpy,
553* ..
554* .. Intrinsic Functions ..
555 INTRINSIC max, min
556* ..
557* .. Executable Statements ..
558*
559 info = 0
560 nofact = lsame( fact, 'N' )
561 equil = lsame( fact, 'E' )
562 smlnum = dlamch( 'Safe minimum' )
563 bignum = one / smlnum
564 IF( nofact .OR. equil ) THEN
565 equed = 'N'
566 rcequ = .false.
567 ELSE
568 rcequ = lsame( equed, 'Y' )
569 ENDIF
570*
571* Default is failure. If an input parameter is wrong or
572* factorization fails, make everything look horrible. Only the
573* pivot growth is set here, the rest is initialized in ZHERFSX.
574*
575 rpvgrw = zero
576*
577* Test the input parameters. PARAMS is not tested until ZHERFSX.
578*
579 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
580 \$ lsame( fact, 'F' ) ) THEN
581 info = -1
582 ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
583 \$ .NOT.lsame( uplo, 'L' ) ) THEN
584 info = -2
585 ELSE IF( n.LT.0 ) THEN
586 info = -3
587 ELSE IF( nrhs.LT.0 ) THEN
588 info = -4
589 ELSE IF( lda.LT.max( 1, n ) ) THEN
590 info = -6
591 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
592 info = -8
593 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
594 \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
595 info = -9
596 ELSE
597 IF ( rcequ ) THEN
598 smin = bignum
599 smax = zero
600 DO 10 j = 1, n
601 smin = min( smin, s( j ) )
602 smax = max( smax, s( j ) )
603 10 CONTINUE
604 IF( smin.LE.zero ) THEN
605 info = -10
606 ELSE IF( n.GT.0 ) THEN
607 scond = max( smin, smlnum ) / min( smax, bignum )
608 ELSE
609 scond = one
610 END IF
611 END IF
612 IF( info.EQ.0 ) THEN
613 IF( ldb.LT.max( 1, n ) ) THEN
614 info = -12
615 ELSE IF( ldx.LT.max( 1, n ) ) THEN
616 info = -14
617 END IF
618 END IF
619 END IF
620*
621 IF( info.NE.0 ) THEN
622 CALL xerbla( 'ZHESVXX', -info )
623 RETURN
624 END IF
625*
626 IF( equil ) THEN
627*
628* Compute row and column scalings to equilibrate the matrix A.
629*
630 CALL zheequb( uplo, n, a, lda, s, scond, amax, work, infequ )
631 IF( infequ.EQ.0 ) THEN
632*
633* Equilibrate the matrix.
634*
635 CALL zlaqhe( uplo, n, a, lda, s, scond, amax, equed )
636 rcequ = lsame( equed, 'Y' )
637 END IF
638 END IF
639*
640* Scale the right-hand side.
641*
642 IF( rcequ ) CALL zlascl2( n, nrhs, s, b, ldb )
643*
644 IF( nofact .OR. equil ) THEN
645*
646* Compute the LDL^H or UDU^H factorization of A.
647*
648 CALL zlacpy( uplo, n, n, a, lda, af, ldaf )
649 CALL zhetrf( uplo, n, af, ldaf, ipiv, work, 5*max(1,n), info )
650*
651* Return if INFO is non-zero.
652*
653 IF( info.GT.0 ) THEN
654*
655* Pivot in column INFO is exactly 0
656* Compute the reciprocal pivot growth factor of the
657* leading rank-deficient INFO columns of A.
658*
659 IF( n.GT.0 )
660 \$ rpvgrw = zla_herpvgrw( uplo, n, info, a, lda, af, ldaf,
661 \$ ipiv, rwork )
662 RETURN
663 END IF
664 END IF
665*
666* Compute the reciprocal pivot growth factor RPVGRW.
667*
668 IF( n.GT.0 )
669 \$ rpvgrw = zla_herpvgrw( uplo, n, info, a, lda, af, ldaf, ipiv,
670 \$ rwork )
671*
672* Compute the solution matrix X.
673*
674 CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
675 CALL zhetrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
676*
677* Use iterative refinement to improve the computed solution and
678* compute error bounds and backward error estimates for it.
679*
680 CALL zherfsx( uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv,
681 \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
682 \$ err_bnds_comp, nparams, params, work, rwork, info )
683*
684* Scale solutions.
685*
686 IF ( rcequ ) THEN
687 CALL zlascl2 ( n, nrhs, s, x, ldx )
688 END IF
689*
690 RETURN
691*
692* End of ZHESVXX
693*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zlaqhe(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
ZLAQHE scales a Hermitian matrix.
Definition: zlaqhe.f:134
subroutine zheequb(UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO)
ZHEEQUB
Definition: zheequb.f:132
double precision function zla_herpvgrw(UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK)
ZLA_HERPVGRW
Definition: zla_herpvgrw.f:123
subroutine zherfsx(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)
ZHERFSX
Definition: zherfsx.f:401
subroutine zhetrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZHETRS
Definition: zhetrs.f:120
subroutine zhetrf(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
ZHETRF
Definition: zhetrf.f:177
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zlascl2(M, N, D, X, LDX)
ZLASCL2 performs diagonal scaling on a matrix.
Definition: zlascl2.f:91
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