LAPACK  3.10.0 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.```

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 *
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 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 vector.
Definition: zlascl2.f:91
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
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:107
subroutine zpoequb(N, A, LDA, S, SCOND, AMAX, INFO)
ZPOEQUB
Definition: zpoequb.f:119
subroutine zpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
ZPOTRS
Definition: zpotrs.f:110
subroutine zpotrf(UPLO, N, A, LDA, INFO)
ZPOTRF VARIANT: right looking block version of the algorithm, calling Level 3 BLAS.
Definition: zpotrf.f:102
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