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

 subroutine zposvxx ( character fact, character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, 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 )

ZPOSVXX computes the solution to system of linear equations A * X = B for PO matrices

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

Purpose:
```    ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
to compute the solution to a complex*16 system of linear equations
A * X = B, where A is an N-by-N Hermitian positive definite matrix
and X and B are N-by-NRHS matrices.

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

ZPOSVXX 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
ZPOSVXX 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 ZPOSVXX 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 Cholesky decomposition is used to
factor the matrix A (after equilibration if FACT = 'E') as
A = U**T* U,  if UPLO = 'U', or
A = L * L**T,  if UPLO = 'L',
where U is an upper triangular matrix and L is a lower triangular
matrix.

3. If the leading principal minor of order i is not positive,
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(S) 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 contains the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by S. A and AF 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) On entry, the Hermitian matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). 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. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. 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 triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(S)*A*diag(S). If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A. If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T 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] 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 row 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 489 of file zposvxx.f.

493*
494* -- LAPACK driver routine --
495* -- LAPACK is a software package provided by Univ. of Tennessee, --
496* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
497*
498* .. Scalar Arguments ..
499 CHARACTER EQUED, FACT, UPLO
500 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
501 \$ N_ERR_BNDS
502 DOUBLE PRECISION RCOND, RPVGRW
503* ..
504* .. Array Arguments ..
505 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
506 \$ WORK( * ), X( LDX, * )
507 DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
508 \$ ERR_BNDS_NORM( NRHS, * ),
509 \$ ERR_BNDS_COMP( NRHS, * )
510* ..
511*
512* ==================================================================
513*
514* .. Parameters ..
515 DOUBLE PRECISION ZERO, ONE
516 parameter( zero = 0.0d+0, one = 1.0d+0 )
517 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
518 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
519 INTEGER CMP_ERR_I, PIV_GROWTH_I
520 parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
521 \$ berr_i = 3 )
522 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
523 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
524 \$ piv_growth_i = 9 )
525* ..
526* .. Local Scalars ..
527 LOGICAL EQUIL, NOFACT, RCEQU
528 INTEGER INFEQU, J
529 DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
530* ..
531* .. External Functions ..
532 EXTERNAL lsame, dlamch, zla_porpvgrw
533 LOGICAL LSAME
534 DOUBLE PRECISION DLAMCH, ZLA_PORPVGRW
535* ..
536* .. External Subroutines ..
537 EXTERNAL zpoequb, zpotrf, zpotrs, zlacpy,
539* ..
540* .. Intrinsic Functions ..
541 INTRINSIC max, min
542* ..
543* .. Executable Statements ..
544*
545 info = 0
546 nofact = lsame( fact, 'N' )
547 equil = lsame( fact, 'E' )
548 smlnum = dlamch( 'Safe minimum' )
549 bignum = one / smlnum
550 IF( nofact .OR. equil ) THEN
551 equed = 'N'
552 rcequ = .false.
553 ELSE
554 rcequ = lsame( equed, 'Y' )
555 ENDIF
556*
557* Default is failure. If an input parameter is wrong or
558* factorization fails, make everything look horrible. Only the
559* pivot growth is set here, the rest is initialized in ZPORFSX.
560*
561 rpvgrw = zero
562*
563* Test the input parameters. PARAMS is not tested until ZPORFSX.
564*
565 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
566 \$ lsame( fact, 'F' ) ) THEN
567 info = -1
568 ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
569 \$ .NOT.lsame( uplo, 'L' ) ) THEN
570 info = -2
571 ELSE IF( n.LT.0 ) THEN
572 info = -3
573 ELSE IF( nrhs.LT.0 ) THEN
574 info = -4
575 ELSE IF( lda.LT.max( 1, n ) ) THEN
576 info = -6
577 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
578 info = -8
579 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
580 \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
581 info = -9
582 ELSE
583 IF ( rcequ ) THEN
584 smin = bignum
585 smax = zero
586 DO 10 j = 1, n
587 smin = min( smin, s( j ) )
588 smax = max( smax, s( j ) )
589 10 CONTINUE
590 IF( smin.LE.zero ) THEN
591 info = -10
592 ELSE IF( n.GT.0 ) THEN
593 scond = max( smin, smlnum ) / min( smax, bignum )
594 ELSE
595 scond = one
596 END IF
597 END IF
598 IF( info.EQ.0 ) THEN
599 IF( ldb.LT.max( 1, n ) ) THEN
600 info = -12
601 ELSE IF( ldx.LT.max( 1, n ) ) THEN
602 info = -14
603 END IF
604 END IF
605 END IF
606*
607 IF( info.NE.0 ) THEN
608 CALL xerbla( 'ZPOSVXX', -info )
609 RETURN
610 END IF
611*
612 IF( equil ) THEN
613*
614* Compute row and column scalings to equilibrate the matrix A.
615*
616 CALL zpoequb( n, a, lda, s, scond, amax, infequ )
617 IF( infequ.EQ.0 ) THEN
618*
619* Equilibrate the matrix.
620*
621 CALL zlaqhe( uplo, n, a, lda, s, scond, amax, equed )
622 rcequ = lsame( equed, 'Y' )
623 END IF
624 END IF
625*
626* Scale the right-hand side.
627*
628 IF( rcequ ) CALL zlascl2( n, nrhs, s, b, ldb )
629*
630 IF( nofact .OR. equil ) THEN
631*
632* Compute the Cholesky factorization of A.
633*
634 CALL zlacpy( uplo, n, n, a, lda, af, ldaf )
635 CALL zpotrf( uplo, n, af, ldaf, info )
636*
637* Return if INFO is non-zero.
638*
639 IF( info.GT.0 ) THEN
640*
641* Pivot in column INFO is exactly 0
642* Compute the reciprocal pivot growth factor of the
643* leading rank-deficient INFO columns of A.
644*
645 rpvgrw = zla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
646 RETURN
647 END IF
648 END IF
649*
650* Compute the reciprocal pivot growth factor RPVGRW.
651*
652 rpvgrw = zla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
653*
654* Compute the solution matrix X.
655*
656 CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
657 CALL zpotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
658*
659* Use iterative refinement to improve the computed solution and
660* compute error bounds and backward error estimates for it.
661*
662 CALL zporfsx( uplo, equed, n, nrhs, a, lda, af, ldaf,
663 \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
664 \$ err_bnds_comp, nparams, params, work, rwork, info )
665
666*
667* Scale solutions.
668*
669 IF ( rcequ ) THEN
670 CALL zlascl2( n, nrhs, s, x, ldx )
671 END IF
672*
673 RETURN
674*
675* End of ZPOSVXX
676*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
double precision function zla_porpvgrw(uplo, ncols, a, lda, af, ldaf, work)
ZLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian...
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
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
subroutine zlaqhe(uplo, n, a, lda, s, scond, amax, equed)
ZLAQHE scales a Hermitian matrix.
Definition zlaqhe.f:134
subroutine zlascl2(m, n, d, x, ldx)
ZLASCL2 performs diagonal scaling on a matrix.
Definition zlascl2.f:91
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine zpoequb(n, a, lda, s, scond, amax, info)
ZPOEQUB
Definition zpoequb.f:119
subroutine zporfsx(uplo, equed, n, nrhs, a, lda, af, ldaf, s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
ZPORFSX
Definition zporfsx.f:393
subroutine zpotrf(uplo, n, a, lda, info)
ZPOTRF
Definition zpotrf.f:107
subroutine zpotrs(uplo, n, nrhs, a, lda, b, ldb, info)
ZPOTRS
Definition zpotrs.f:110
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