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

CSYSVXX computes the solution to system of linear equations A * X = B for SY matrices

Purpose:
```    CSYSVXX uses the diagonal pivoting factorization to compute the
solution to a complex system of linear equations A * X = B, where
A is an N-by-N symmetric matrix and X and B are N-by-NRHS
matrices.

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

CSYSVXX 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
CSYSVXX 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 CSYSVXX 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 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 symmetric 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 array, dimension (LDA,N) The symmetric 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 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**T or A = L*D*L**T as computed by SSYTRF. 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**T or A = L*D*L**T.``` [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 SSYTRF. 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 SSYTRF.``` [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 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 511 of file csysvxx.f.

511 *
512 * -- LAPACK driver routine (version 3.6.0) --
513 * -- LAPACK is a software package provided by Univ. of Tennessee, --
514 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
515 * April 2012
516 *
517 * .. Scalar Arguments ..
518  CHARACTER equed, fact, uplo
519  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs, nparams,
520  \$ n_err_bnds
521  REAL rcond, rpvgrw
522 * ..
523 * .. Array Arguments ..
524  INTEGER ipiv( * )
525  COMPLEX a( lda, * ), af( ldaf, * ), b( ldb, * ),
526  \$ x( ldx, * ), work( * )
527  REAL s( * ), params( * ), berr( * ),
528  \$ err_bnds_norm( nrhs, * ),
529  \$ err_bnds_comp( nrhs, * ), rwork( * )
530 * ..
531 *
532 * ==================================================================
533 *
534 * .. Parameters ..
535  REAL zero, one
536  parameter ( zero = 0.0e+0, one = 1.0e+0 )
537  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
538  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
539  INTEGER cmp_err_i, piv_growth_i
540  parameter ( final_nrm_err_i = 1, final_cmp_err_i = 2,
541  \$ berr_i = 3 )
542  parameter ( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
543  parameter ( cmp_rcond_i = 7, cmp_err_i = 8,
544  \$ piv_growth_i = 9 )
545 * ..
546 * .. Local Scalars ..
547  LOGICAL equil, nofact, rcequ
548  INTEGER infequ, j
549  REAL amax, bignum, smin, smax, scond, smlnum
550 * ..
551 * .. External Functions ..
552  EXTERNAL lsame, slamch, cla_syrpvgrw
553  LOGICAL lsame
554  REAL slamch, cla_syrpvgrw
555 * ..
556 * .. External Subroutines ..
557  EXTERNAL csycon, csyequb, csytrf, csytrs, clacpy,
559 * ..
560 * .. Intrinsic Functions ..
561  INTRINSIC max, min
562 * ..
563 * .. Executable Statements ..
564 *
565  info = 0
566  nofact = lsame( fact, 'N' )
567  equil = lsame( fact, 'E' )
568  smlnum = slamch( 'Safe minimum' )
569  bignum = one / smlnum
570  IF( nofact .OR. equil ) THEN
571  equed = 'N'
572  rcequ = .false.
573  ELSE
574  rcequ = lsame( equed, 'Y' )
575  ENDIF
576 *
577 * Default is failure. If an input parameter is wrong or
578 * factorization fails, make everything look horrible. Only the
579 * pivot growth is set here, the rest is initialized in CSYRFSX.
580 *
581  rpvgrw = zero
582 *
583 * Test the input parameters. PARAMS is not tested until CSYRFSX.
584 *
585  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
586  \$ lsame( fact, 'F' ) ) THEN
587  info = -1
588  ELSE IF( .NOT.lsame(uplo, 'U') .AND.
589  \$ .NOT.lsame(uplo, 'L') ) THEN
590  info = -2
591  ELSE IF( n.LT.0 ) THEN
592  info = -3
593  ELSE IF( nrhs.LT.0 ) THEN
594  info = -4
595  ELSE IF( lda.LT.max( 1, n ) ) THEN
596  info = -6
597  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
598  info = -8
599  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
600  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
601  info = -10
602  ELSE
603  IF ( rcequ ) THEN
604  smin = bignum
605  smax = zero
606  DO 10 j = 1, n
607  smin = min( smin, s( j ) )
608  smax = max( smax, s( j ) )
609  10 CONTINUE
610  IF( smin.LE.zero ) THEN
611  info = -11
612  ELSE IF( n.GT.0 ) THEN
613  scond = max( smin, smlnum ) / min( smax, bignum )
614  ELSE
615  scond = one
616  END IF
617  END IF
618  IF( info.EQ.0 ) THEN
619  IF( ldb.LT.max( 1, n ) ) THEN
620  info = -13
621  ELSE IF( ldx.LT.max( 1, n ) ) THEN
622  info = -15
623  END IF
624  END IF
625  END IF
626 *
627  IF( info.NE.0 ) THEN
628  CALL xerbla( 'CSYSVXX', -info )
629  RETURN
630  END IF
631 *
632  IF( equil ) THEN
633 *
634 * Compute row and column scalings to equilibrate the matrix A.
635 *
636  CALL csyequb( uplo, n, a, lda, s, scond, amax, work, infequ )
637  IF( infequ.EQ.0 ) THEN
638 *
639 * Equilibrate the matrix.
640 *
641  CALL claqsy( uplo, n, a, lda, s, scond, amax, equed )
642  rcequ = lsame( equed, 'Y' )
643  END IF
644
645  END IF
646 *
647 * Scale the right hand-side.
648 *
649  IF( rcequ ) CALL clascl2( n, nrhs, s, b, ldb )
650 *
651  IF( nofact .OR. equil ) THEN
652 *
653 * Compute the LDL^T or UDU^T factorization of A.
654 *
655  CALL clacpy( uplo, n, n, a, lda, af, ldaf )
656  CALL csytrf( uplo, n, af, ldaf, ipiv, work, 5*max(1,n), info )
657 *
658 * Return if INFO is non-zero.
659 *
660  IF( info.GT.0 ) THEN
661 *
662 * Pivot in column INFO is exactly 0
663 * Compute the reciprocal pivot growth factor of the
664 * leading rank-deficient INFO columns of A.
665 *
666  IF ( n.GT.0 )
667  \$ rpvgrw = cla_syrpvgrw( uplo, n, info, a, lda, af,
668  \$ ldaf, ipiv, rwork )
669  RETURN
670  END IF
671  END IF
672 *
673 * Compute the reciprocal pivot growth factor RPVGRW.
674 *
675  IF ( n.GT.0 )
676  \$ rpvgrw = cla_syrpvgrw( uplo, n, info, a, lda, af, ldaf,
677  \$ ipiv, rwork )
678 *
679 * Compute the solution matrix X.
680 *
681  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
682  CALL csytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
683 *
684 * Use iterative refinement to improve the computed solution and
685 * compute error bounds and backward error estimates for it.
686 *
687  CALL csyrfsx( uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv,
688  \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
689  \$ err_bnds_comp, nparams, params, work, rwork, info )
690 *
691 * Scale solutions.
692 *
693  IF ( rcequ ) THEN
694  CALL clascl2 (n, nrhs, s, x, ldx )
695  END IF
696 *
697  RETURN
698 *
699 * End of CSYSVXX
700 *
subroutine claqsy(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
CLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
Definition: claqsy.f:136
subroutine csytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
CSYTRS
Definition: csytrs.f:122
subroutine csyequb(UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO)
CSYEQUB
Definition: csyequb.f:138
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine csyrfsx(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)
CSYRFSX
Definition: csyrfsx.f:404
subroutine clascl2(M, N, D, X, LDX)
CLASCL2 performs diagonal scaling on a vector.
Definition: clascl2.f:93
subroutine csytrf(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
CSYTRF
Definition: csytrf.f:184
subroutine csycon(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
CSYCON
Definition: csycon.f:127
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 cla_syrpvgrw(UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK)
CLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite m...
Definition: cla_syrpvgrw.f:125
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

Here is the call graph for this function:

Here is the caller graph for this function: