LAPACK  3.9.0 LAPACK: Linear Algebra PACKage

◆ zsysvxx()

 subroutine zsysvxx ( character FACT, character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, 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 )

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

Purpose:
```    ZSYSVXX uses the diagonal pivoting factorization to compute the
solution to a complex*16 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. ZSYSVXX 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.

ZSYSVXX 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
ZSYSVXX 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 ZSYSVXX 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 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*16 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*16 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 DSYTRF. 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 DSYTRF. 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 DSYTRF.``` [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 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 508 of file zsysvxx.f.

508 *
509 * -- LAPACK driver routine (version 3.7.0) --
510 * -- LAPACK is a software package provided by Univ. of Tennessee, --
511 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
512 * April 2012
513 *
514 * .. Scalar Arguments ..
515  CHARACTER EQUED, FACT, UPLO
516  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
517  \$ N_ERR_BNDS
518  DOUBLE PRECISION RCOND, RPVGRW
519 * ..
520 * .. Array Arguments ..
521  INTEGER IPIV( * )
522  COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
523  \$ X( LDX, * ), WORK( * )
524  DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
525  \$ ERR_BNDS_NORM( NRHS, * ),
526  \$ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
527 * ..
528 *
529 * ==================================================================
530 *
531 * .. Parameters ..
532  DOUBLE PRECISION ZERO, ONE
533  parameter( zero = 0.0d+0, one = 1.0d+0 )
534  INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
535  INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
536  INTEGER CMP_ERR_I, PIV_GROWTH_I
537  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
538  \$ berr_i = 3 )
539  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
540  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
541  \$ piv_growth_i = 9 )
542 * ..
543 * .. Local Scalars ..
544  LOGICAL EQUIL, NOFACT, RCEQU
545  INTEGER INFEQU, J
546  DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
547 * ..
548 * .. External Functions ..
549  EXTERNAL lsame, dlamch, zla_syrpvgrw
550  LOGICAL LSAME
551  DOUBLE PRECISION DLAMCH, ZLA_SYRPVGRW
552 * ..
553 * .. External Subroutines ..
554  EXTERNAL zsyequb, zsytrf, zsytrs, zlacpy,
556 * ..
557 * .. Intrinsic Functions ..
558  INTRINSIC max, min
559 * ..
560 * .. Executable Statements ..
561 *
562  info = 0
563  nofact = lsame( fact, 'N' )
564  equil = lsame( fact, 'E' )
565  smlnum = dlamch( 'Safe minimum' )
566  bignum = one / smlnum
567  IF( nofact .OR. equil ) THEN
568  equed = 'N'
569  rcequ = .false.
570  ELSE
571  rcequ = lsame( equed, 'Y' )
572  ENDIF
573 *
574 * Default is failure. If an input parameter is wrong or
575 * factorization fails, make everything look horrible. Only the
576 * pivot growth is set here, the rest is initialized in ZSYRFSX.
577 *
578  rpvgrw = zero
579 *
580 * Test the input parameters. PARAMS is not tested until ZSYRFSX.
581 *
582  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
583  \$ lsame( fact, 'F' ) ) THEN
584  info = -1
585  ELSE IF( .NOT.lsame(uplo, 'U') .AND.
586  \$ .NOT.lsame(uplo, 'L') ) THEN
587  info = -2
588  ELSE IF( n.LT.0 ) THEN
589  info = -3
590  ELSE IF( nrhs.LT.0 ) THEN
591  info = -4
592  ELSE IF( lda.LT.max( 1, n ) ) THEN
593  info = -6
594  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
595  info = -8
596  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
597  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
598  info = -10
599  ELSE
600  IF ( rcequ ) THEN
601  smin = bignum
602  smax = zero
603  DO 10 j = 1, n
604  smin = min( smin, s( j ) )
605  smax = max( smax, s( j ) )
606  10 CONTINUE
607  IF( smin.LE.zero ) THEN
608  info = -11
609  ELSE IF( n.GT.0 ) THEN
610  scond = max( smin, smlnum ) / min( smax, bignum )
611  ELSE
612  scond = one
613  END IF
614  END IF
615  IF( info.EQ.0 ) THEN
616  IF( ldb.LT.max( 1, n ) ) THEN
617  info = -13
618  ELSE IF( ldx.LT.max( 1, n ) ) THEN
619  info = -15
620  END IF
621  END IF
622  END IF
623 *
624  IF( info.NE.0 ) THEN
625  CALL xerbla( 'ZSYSVXX', -info )
626  RETURN
627  END IF
628 *
629  IF( equil ) THEN
630 *
631 * Compute row and column scalings to equilibrate the matrix A.
632 *
633  CALL zsyequb( uplo, n, a, lda, s, scond, amax, work, infequ )
634  IF( infequ.EQ.0 ) THEN
635 *
636 * Equilibrate the matrix.
637 *
638  CALL zlaqsy( uplo, n, a, lda, s, scond, amax, equed )
639  rcequ = lsame( equed, 'Y' )
640  END IF
641
642  END IF
643 *
644 * Scale the right hand-side.
645 *
646  IF( rcequ ) CALL zlascl2( n, nrhs, s, b, ldb )
647 *
648  IF( nofact .OR. equil ) THEN
649 *
650 * Compute the LDL^T or UDU^T factorization of A.
651 *
652  CALL zlacpy( uplo, n, n, a, lda, af, ldaf )
653  CALL zsytrf( uplo, n, af, ldaf, ipiv, work, 5*max(1,n), info )
654 *
655 * Return if INFO is non-zero.
656 *
657  IF( info.GT.0 ) THEN
658 *
659 * Pivot in column INFO is exactly 0
660 * Compute the reciprocal pivot growth factor of the
661 * leading rank-deficient INFO columns of A.
662 *
663  IF ( n.GT.0 )
664  \$ rpvgrw = zla_syrpvgrw( uplo, n, info, a, lda, af,
665  \$ ldaf, ipiv, rwork )
666  RETURN
667  END IF
668  END IF
669 *
670 * Compute the reciprocal pivot growth factor RPVGRW.
671 *
672  IF ( n.GT.0 )
673  \$ rpvgrw = zla_syrpvgrw( uplo, n, info, a, lda, af, ldaf,
674  \$ ipiv, rwork )
675 *
676 * Compute the solution matrix X.
677 *
678  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
679  CALL zsytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
680 *
681 * Use iterative refinement to improve the computed solution and
682 * compute error bounds and backward error estimates for it.
683 *
684  CALL zsyrfsx( uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv,
685  \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
686  \$ err_bnds_comp, nparams, params, work, rwork, info )
687 *
688 * Scale solutions.
689 *
690  IF ( rcequ ) THEN
691  CALL zlascl2 (n, nrhs, s, x, ldx )
692  END IF
693 *
694  RETURN
695 *
696 * End of ZSYSVXX
697 *
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zlaqsy
subroutine zlaqsy(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
ZLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
Definition: zlaqsy.f:136
zla_syrpvgrw
double precision function zla_syrpvgrw(UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK)
ZLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite m...
Definition: zla_syrpvgrw.f:125
zsytrs
subroutine zsytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZSYTRS
Definition: zsytrs.f:122
zsyequb
subroutine zsyequb(UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO)
ZSYEQUB
Definition: zsyequb.f:134
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
zsyrfsx
subroutine zsyrfsx(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)
ZSYRFSX
Definition: zsyrfsx.f:404
zsytrf
subroutine zsytrf(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
ZSYTRF
Definition: zsytrf.f:184
dlamch
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:70