LAPACK  3.10.1 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 ZSYTRF. 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 ZSYTRF. 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 ZSYTRF.``` [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.```

Definition at line 502 of file zsysvxx.f.

506 *
507 * -- LAPACK driver routine --
508 * -- LAPACK is a software package provided by Univ. of Tennessee, --
509 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
510 *
511 * .. Scalar Arguments ..
512  CHARACTER EQUED, FACT, UPLO
513  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
514  \$ N_ERR_BNDS
515  DOUBLE PRECISION RCOND, RPVGRW
516 * ..
517 * .. Array Arguments ..
518  INTEGER IPIV( * )
519  COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
520  \$ X( LDX, * ), WORK( * )
521  DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
522  \$ ERR_BNDS_NORM( NRHS, * ),
523  \$ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
524 * ..
525 *
526 * ==================================================================
527 *
528 * .. Parameters ..
529  DOUBLE PRECISION ZERO, ONE
530  parameter( zero = 0.0d+0, one = 1.0d+0 )
531  INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
532  INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
533  INTEGER CMP_ERR_I, PIV_GROWTH_I
534  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
535  \$ berr_i = 3 )
536  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
537  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
538  \$ piv_growth_i = 9 )
539 * ..
540 * .. Local Scalars ..
541  LOGICAL EQUIL, NOFACT, RCEQU
542  INTEGER INFEQU, J
543  DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
544 * ..
545 * .. External Functions ..
546  EXTERNAL lsame, dlamch, zla_syrpvgrw
547  LOGICAL LSAME
548  DOUBLE PRECISION DLAMCH, ZLA_SYRPVGRW
549 * ..
550 * .. External Subroutines ..
551  EXTERNAL zsyequb, zsytrf, zsytrs, zlacpy,
553 * ..
554 * .. Intrinsic Functions ..
555  INTRINSIC max, min
556 * ..
557 * .. Executable Statements ..
558 *
559  info = 0
560  nofact = lsame( fact, 'N' )
561  equil = lsame( fact, 'E' )
562  smlnum = dlamch( 'Safe minimum' )
563  bignum = one / smlnum
564  IF( nofact .OR. equil ) THEN
565  equed = 'N'
566  rcequ = .false.
567  ELSE
568  rcequ = lsame( equed, 'Y' )
569  ENDIF
570 *
571 * Default is failure. If an input parameter is wrong or
572 * factorization fails, make everything look horrible. Only the
573 * pivot growth is set here, the rest is initialized in ZSYRFSX.
574 *
575  rpvgrw = zero
576 *
577 * Test the input parameters. PARAMS is not tested until ZSYRFSX.
578 *
579  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
580  \$ lsame( fact, 'F' ) ) THEN
581  info = -1
582  ELSE IF( .NOT.lsame(uplo, 'U') .AND.
583  \$ .NOT.lsame(uplo, 'L') ) THEN
584  info = -2
585  ELSE IF( n.LT.0 ) THEN
586  info = -3
587  ELSE IF( nrhs.LT.0 ) THEN
588  info = -4
589  ELSE IF( lda.LT.max( 1, n ) ) THEN
590  info = -6
591  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
592  info = -8
593  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
594  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
595  info = -10
596  ELSE
597  IF ( rcequ ) THEN
598  smin = bignum
599  smax = zero
600  DO 10 j = 1, n
601  smin = min( smin, s( j ) )
602  smax = max( smax, s( j ) )
603  10 CONTINUE
604  IF( smin.LE.zero ) THEN
605  info = -11
606  ELSE IF( n.GT.0 ) THEN
607  scond = max( smin, smlnum ) / min( smax, bignum )
608  ELSE
609  scond = one
610  END IF
611  END IF
612  IF( info.EQ.0 ) THEN
613  IF( ldb.LT.max( 1, n ) ) THEN
614  info = -13
615  ELSE IF( ldx.LT.max( 1, n ) ) THEN
616  info = -15
617  END IF
618  END IF
619  END IF
620 *
621  IF( info.NE.0 ) THEN
622  CALL xerbla( 'ZSYSVXX', -info )
623  RETURN
624  END IF
625 *
626  IF( equil ) THEN
627 *
628 * Compute row and column scalings to equilibrate the matrix A.
629 *
630  CALL zsyequb( uplo, n, a, lda, s, scond, amax, work, infequ )
631  IF( infequ.EQ.0 ) THEN
632 *
633 * Equilibrate the matrix.
634 *
635  CALL zlaqsy( uplo, n, a, lda, s, scond, amax, equed )
636  rcequ = lsame( equed, 'Y' )
637  END IF
638
639  END IF
640 *
641 * Scale the right hand-side.
642 *
643  IF( rcequ ) CALL zlascl2( n, nrhs, s, b, ldb )
644 *
645  IF( nofact .OR. equil ) THEN
646 *
647 * Compute the LDL^T or UDU^T factorization of A.
648 *
649  CALL zlacpy( uplo, n, n, a, lda, af, ldaf )
650  CALL zsytrf( uplo, n, af, ldaf, ipiv, work, 5*max(1,n), info )
651 *
652 * Return if INFO is non-zero.
653 *
654  IF( info.GT.0 ) THEN
655 *
656 * Pivot in column INFO is exactly 0
657 * Compute the reciprocal pivot growth factor of the
658 * leading rank-deficient INFO columns of A.
659 *
660  IF ( n.GT.0 )
661  \$ rpvgrw = zla_syrpvgrw( uplo, n, info, a, lda, af,
662  \$ ldaf, ipiv, rwork )
663  RETURN
664  END IF
665  END IF
666 *
667 * Compute the reciprocal pivot growth factor RPVGRW.
668 *
669  IF ( n.GT.0 )
670  \$ rpvgrw = zla_syrpvgrw( uplo, n, info, a, lda, af, ldaf,
671  \$ ipiv, rwork )
672 *
673 * Compute the solution matrix X.
674 *
675  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
676  CALL zsytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
677 *
678 * Use iterative refinement to improve the computed solution and
679 * compute error bounds and backward error estimates for it.
680 *
681  CALL zsyrfsx( uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv,
682  \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
683  \$ err_bnds_comp, nparams, params, work, rwork, info )
684 *
685 * Scale solutions.
686 *
687  IF ( rcequ ) THEN
688  CALL zlascl2 (n, nrhs, s, x, ldx )
689  END IF
690 *
691  RETURN
692 *
693 * End of ZSYSVXX
694 *
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 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 zlaqsy(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
ZLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
Definition: zlaqsy.f:134
subroutine zsyequb(UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO)
ZSYEQUB
Definition: zsyequb.f:132
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:123
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:402
subroutine zsytrf(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
ZSYTRF
Definition: zsytrf.f:182
subroutine zsytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZSYTRS
Definition: zsytrs.f:120
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