LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine cposvxx ( character FACT, character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, 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 )

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

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

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

CPOSVXX 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
CPOSVXX 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 CPOSVXX 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 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 array, dimension (LDA,N) On entry, the symmetric 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 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 REAL 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 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.```
Date
April 2012

Definition at line 498 of file cposvxx.f.

498 *
499 * -- LAPACK driver routine (version 3.4.1) --
500 * -- LAPACK is a software package provided by Univ. of Tennessee, --
501 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
502 * April 2012
503 *
504 * .. Scalar Arguments ..
505  CHARACTER equed, fact, uplo
506  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs, nparams,
507  \$ n_err_bnds
508  REAL rcond, rpvgrw
509 * ..
510 * .. Array Arguments ..
511  COMPLEX a( lda, * ), af( ldaf, * ), b( ldb, * ),
512  \$ work( * ), x( ldx, * )
513  REAL s( * ), params( * ), berr( * ), rwork( * ),
514  \$ err_bnds_norm( nrhs, * ),
515  \$ err_bnds_comp( nrhs, * )
516 * ..
517 *
518 * ==================================================================
519 *
520 * .. Parameters ..
521  REAL zero, one
522  parameter ( zero = 0.0e+0, one = 1.0e+0 )
523  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
524  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
525  INTEGER cmp_err_i, piv_growth_i
526  parameter ( final_nrm_err_i = 1, final_cmp_err_i = 2,
527  \$ berr_i = 3 )
528  parameter ( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
529  parameter ( cmp_rcond_i = 7, cmp_err_i = 8,
530  \$ piv_growth_i = 9 )
531 * ..
532 * .. Local Scalars ..
533  LOGICAL equil, nofact, rcequ
534  INTEGER infequ, j
535  REAL amax, bignum, smin, smax, scond, smlnum
536 * ..
537 * .. External Functions ..
538  EXTERNAL lsame, slamch, cla_porpvgrw
539  LOGICAL lsame
540  REAL slamch, cla_porpvgrw
541 * ..
542 * .. External Subroutines ..
543  EXTERNAL cpocon, cpoequb, cpotrf, cpotrs, clacpy,
545 * ..
546 * .. Intrinsic Functions ..
547  INTRINSIC max, min
548 * ..
549 * .. Executable Statements ..
550 *
551  info = 0
552  nofact = lsame( fact, 'N' )
553  equil = lsame( fact, 'E' )
554  smlnum = slamch( 'Safe minimum' )
555  bignum = one / smlnum
556  IF( nofact .OR. equil ) THEN
557  equed = 'N'
558  rcequ = .false.
559  ELSE
560  rcequ = lsame( equed, 'Y' )
561  ENDIF
562 *
563 * Default is failure. If an input parameter is wrong or
564 * factorization fails, make everything look horrible. Only the
565 * pivot growth is set here, the rest is initialized in CPORFSX.
566 *
567  rpvgrw = zero
568 *
569 * Test the input parameters. PARAMS is not tested until CPORFSX.
570 *
571  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
572  \$ lsame( fact, 'F' ) ) THEN
573  info = -1
574  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
575  \$ .NOT.lsame( uplo, 'L' ) ) THEN
576  info = -2
577  ELSE IF( n.LT.0 ) THEN
578  info = -3
579  ELSE IF( nrhs.LT.0 ) THEN
580  info = -4
581  ELSE IF( lda.LT.max( 1, n ) ) THEN
582  info = -6
583  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
584  info = -8
585  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
586  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
587  info = -9
588  ELSE
589  IF ( rcequ ) THEN
590  smin = bignum
591  smax = zero
592  DO 10 j = 1, n
593  smin = min( smin, s( j ) )
594  smax = max( smax, s( j ) )
595  10 CONTINUE
596  IF( smin.LE.zero ) THEN
597  info = -10
598  ELSE IF( n.GT.0 ) THEN
599  scond = max( smin, smlnum ) / min( smax, bignum )
600  ELSE
601  scond = one
602  END IF
603  END IF
604  IF( info.EQ.0 ) THEN
605  IF( ldb.LT.max( 1, n ) ) THEN
606  info = -12
607  ELSE IF( ldx.LT.max( 1, n ) ) THEN
608  info = -14
609  END IF
610  END IF
611  END IF
612 *
613  IF( info.NE.0 ) THEN
614  CALL xerbla( 'CPOSVXX', -info )
615  RETURN
616  END IF
617 *
618  IF( equil ) THEN
619 *
620 * Compute row and column scalings to equilibrate the matrix A.
621 *
622  CALL cpoequb( n, a, lda, s, scond, amax, infequ )
623  IF( infequ.EQ.0 ) THEN
624 *
625 * Equilibrate the matrix.
626 *
627  CALL claqhe( uplo, n, a, lda, s, scond, amax, equed )
628  rcequ = lsame( equed, 'Y' )
629  END IF
630  END IF
631 *
632 * Scale the right-hand side.
633 *
634  IF( rcequ ) CALL clascl2( n, nrhs, s, b, ldb )
635 *
636  IF( nofact .OR. equil ) THEN
637 *
638 * Compute the Cholesky factorization of A.
639 *
640  CALL clacpy( uplo, n, n, a, lda, af, ldaf )
641  CALL cpotrf( uplo, n, af, ldaf, info )
642 *
643 * Return if INFO is non-zero.
644 *
645  IF( info.GT.0 ) THEN
646 *
647 * Pivot in column INFO is exactly 0
648 * Compute the reciprocal pivot growth factor of the
649 * leading rank-deficient INFO columns of A.
650 *
651  rpvgrw = cla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
652  RETURN
653  END IF
654  END IF
655 *
656 * Compute the reciprocal pivot growth factor RPVGRW.
657 *
658  rpvgrw = cla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
659 *
660 * Compute the solution matrix X.
661 *
662  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
663  CALL cpotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
664 *
665 * Use iterative refinement to improve the computed solution and
666 * compute error bounds and backward error estimates for it.
667 *
668  CALL cporfsx( uplo, equed, n, nrhs, a, lda, af, ldaf,
669  \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
670  \$ err_bnds_comp, nparams, params, work, rwork, info )
671
672 *
673 * Scale solutions.
674 *
675  IF ( rcequ ) THEN
676  CALL clascl2( n, nrhs, s, x, ldx )
677  END IF
678 *
679  RETURN
680 *
681 * End of CPOSVXX
682 *
subroutine claqhe(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
CLAQHE scales a Hermitian matrix.
Definition: claqhe.f:136
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine cpocon(UPLO, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO)
CPOCON
Definition: cpocon.f:123
subroutine cpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
CPOTRS
Definition: cpotrs.f:112
subroutine clascl2(M, N, D, X, LDX)
CLASCL2 performs diagonal scaling on a vector.
Definition: clascl2.f:93
subroutine cpoequb(N, A, LDA, S, SCOND, AMAX, INFO)
CPOEQUB
Definition: cpoequb.f:115
subroutine cpotrf(UPLO, N, A, LDA, INFO)
CPOTRF
Definition: cpotrf.f:109
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
real function cla_porpvgrw(UPLO, NCOLS, A, LDA, AF, LDAF, WORK)
CLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian...
Definition: cla_porpvgrw.f:107
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine cporfsx(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)
CPORFSX
Definition: cporfsx.f:395

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