 LAPACK  3.9.0 LAPACK: Linear Algebra PACKage

## ◆ 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

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 i-by-i principal minor is not positive definite,
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.```
Date
April 2012

Definition at line 495 of file zposvxx.f.

495 *
496 * -- LAPACK driver routine (version 3.7.0) --
497 * -- LAPACK is a software package provided by Univ. of Tennessee, --
498 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
499 * April 2012
500 *
501 * .. Scalar Arguments ..
502  CHARACTER EQUED, FACT, UPLO
503  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
504  \$ N_ERR_BNDS
505  DOUBLE PRECISION RCOND, RPVGRW
506 * ..
507 * .. Array Arguments ..
508  COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
509  \$ WORK( * ), X( LDX, * )
510  DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
511  \$ ERR_BNDS_NORM( NRHS, * ),
512  \$ ERR_BNDS_COMP( NRHS, * )
513 * ..
514 *
515 * ==================================================================
516 *
517 * .. Parameters ..
518  DOUBLE PRECISION ZERO, ONE
519  parameter( zero = 0.0d+0, one = 1.0d+0 )
520  INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
521  INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
522  INTEGER CMP_ERR_I, PIV_GROWTH_I
523  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
524  \$ berr_i = 3 )
525  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
526  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
527  \$ piv_growth_i = 9 )
528 * ..
529 * .. Local Scalars ..
530  LOGICAL EQUIL, NOFACT, RCEQU
531  INTEGER INFEQU, J
532  DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
533 * ..
534 * .. External Functions ..
535  EXTERNAL lsame, dlamch, zla_porpvgrw
536  LOGICAL LSAME
537  DOUBLE PRECISION DLAMCH, ZLA_PORPVGRW
538 * ..
539 * .. External Subroutines ..
540  EXTERNAL zpoequb, zpotrf, zpotrs, zlacpy,
542 * ..
543 * .. Intrinsic Functions ..
544  INTRINSIC max, min
545 * ..
546 * .. Executable Statements ..
547 *
548  info = 0
549  nofact = lsame( fact, 'N' )
550  equil = lsame( fact, 'E' )
551  smlnum = dlamch( 'Safe minimum' )
552  bignum = one / smlnum
553  IF( nofact .OR. equil ) THEN
554  equed = 'N'
555  rcequ = .false.
556  ELSE
557  rcequ = lsame( equed, 'Y' )
558  ENDIF
559 *
560 * Default is failure. If an input parameter is wrong or
561 * factorization fails, make everything look horrible. Only the
562 * pivot growth is set here, the rest is initialized in ZPORFSX.
563 *
564  rpvgrw = zero
565 *
566 * Test the input parameters. PARAMS is not tested until ZPORFSX.
567 *
568  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
569  \$ lsame( fact, 'F' ) ) THEN
570  info = -1
571  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
572  \$ .NOT.lsame( uplo, 'L' ) ) THEN
573  info = -2
574  ELSE IF( n.LT.0 ) THEN
575  info = -3
576  ELSE IF( nrhs.LT.0 ) THEN
577  info = -4
578  ELSE IF( lda.LT.max( 1, n ) ) THEN
579  info = -6
580  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
581  info = -8
582  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
583  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
584  info = -9
585  ELSE
586  IF ( rcequ ) THEN
587  smin = bignum
588  smax = zero
589  DO 10 j = 1, n
590  smin = min( smin, s( j ) )
591  smax = max( smax, s( j ) )
592  10 CONTINUE
593  IF( smin.LE.zero ) THEN
594  info = -10
595  ELSE IF( n.GT.0 ) THEN
596  scond = max( smin, smlnum ) / min( smax, bignum )
597  ELSE
598  scond = one
599  END IF
600  END IF
601  IF( info.EQ.0 ) THEN
602  IF( ldb.LT.max( 1, n ) ) THEN
603  info = -12
604  ELSE IF( ldx.LT.max( 1, n ) ) THEN
605  info = -14
606  END IF
607  END IF
608  END IF
609 *
610  IF( info.NE.0 ) THEN
611  CALL xerbla( 'ZPOSVXX', -info )
612  RETURN
613  END IF
614 *
615  IF( equil ) THEN
616 *
617 * Compute row and column scalings to equilibrate the matrix A.
618 *
619  CALL zpoequb( n, a, lda, s, scond, amax, infequ )
620  IF( infequ.EQ.0 ) THEN
621 *
622 * Equilibrate the matrix.
623 *
624  CALL zlaqhe( uplo, n, a, lda, s, scond, amax, equed )
625  rcequ = lsame( equed, 'Y' )
626  END IF
627  END IF
628 *
629 * Scale the right-hand side.
630 *
631  IF( rcequ ) CALL zlascl2( n, nrhs, s, b, ldb )
632 *
633  IF( nofact .OR. equil ) THEN
634 *
635 * Compute the Cholesky factorization of A.
636 *
637  CALL zlacpy( uplo, n, n, a, lda, af, ldaf )
638  CALL zpotrf( uplo, n, af, ldaf, info )
639 *
640 * Return if INFO is non-zero.
641 *
642  IF( info.GT.0 ) THEN
643 *
644 * Pivot in column INFO is exactly 0
645 * Compute the reciprocal pivot growth factor of the
646 * leading rank-deficient INFO columns of A.
647 *
648  rpvgrw = zla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
649  RETURN
650  END IF
651  END IF
652 *
653 * Compute the reciprocal pivot growth factor RPVGRW.
654 *
655  rpvgrw = zla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
656 *
657 * Compute the solution matrix X.
658 *
659  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
660  CALL zpotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
661 *
662 * Use iterative refinement to improve the computed solution and
663 * compute error bounds and backward error estimates for it.
664 *
665  CALL zporfsx( uplo, equed, n, nrhs, a, lda, af, ldaf,
666  \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
667  \$ err_bnds_comp, nparams, params, work, rwork, info )
668
669 *
670 * Scale solutions.
671 *
672  IF ( rcequ ) THEN
673  CALL zlascl2( n, nrhs, s, x, ldx )
674  END IF
675 *
676  RETURN
677 *
678 * End of ZPOSVXX
679 *
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zpotrs
subroutine zpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
ZPOTRS
Definition: zpotrs.f:112
zpoequb
subroutine zpoequb(N, A, LDA, S, SCOND, AMAX, INFO)
ZPOEQUB
Definition: zpoequb.f:121
zlaqhe
subroutine zlaqhe(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
ZLAQHE scales a Hermitian matrix.
Definition: zlaqhe.f:136
zporfsx
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:395
zlascl2
subroutine zlascl2(M, N, D, X, LDX)
ZLASCL2 performs diagonal scaling on a vector.
Definition: zlascl2.f:93
zlacpy
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
lsame
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
zpotrf
subroutine zpotrf(UPLO, N, A, LDA, INFO)
ZPOTRF VARIANT: right looking block version of the algorithm, calling Level 3 BLAS.
Definition: zpotrf.f:102
dlamch
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:70
zla_porpvgrw
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...
Definition: zla_porpvgrw.f:109