LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine ssysvxx ( character FACT, character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, character EQUED, real, dimension( * ) S, real, dimension( ldb, * ) B, integer LDB, real, 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, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

SSYSVXX

Purpose:
```    SSYSVXX uses the diagonal pivoting factorization to compute the
solution to a real 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. SSYSVXX 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.

SSYSVXX 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
SSYSVXX 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 SSYSVXX 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 REAL 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 REAL 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 REAL 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 REAL 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
November 2015

Definition at line 510 of file ssysvxx.f.

510 *
511 * -- LAPACK driver routine (version 3.6.0) --
512 * -- LAPACK is a software package provided by Univ. of Tennessee, --
513 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
514 * November 2015
515 *
516 * .. Scalar Arguments ..
517  CHARACTER equed, fact, uplo
518  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs, nparams,
519  \$ n_err_bnds
520  REAL rcond, rpvgrw
521 * ..
522 * .. Array Arguments ..
523  INTEGER ipiv( * ), iwork( * )
524  REAL a( lda, * ), af( ldaf, * ), b( ldb, * ),
525  \$ x( ldx, * ), work( * )
526  REAL s( * ), params( * ), berr( * ),
527  \$ err_bnds_norm( nrhs, * ),
528  \$ err_bnds_comp( nrhs, * )
529 * ..
530 *
531 * ==================================================================
532 *
533 * .. Parameters ..
534  REAL zero, one
535  parameter ( zero = 0.0e+0, one = 1.0e+0 )
536  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
537  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
538  INTEGER cmp_err_i, piv_growth_i
539  parameter ( final_nrm_err_i = 1, final_cmp_err_i = 2,
540  \$ berr_i = 3 )
541  parameter ( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
542  parameter ( cmp_rcond_i = 7, cmp_err_i = 8,
543  \$ piv_growth_i = 9 )
544 * ..
545 * .. Local Scalars ..
546  LOGICAL equil, nofact, rcequ
547  INTEGER infequ, j
548  REAL amax, bignum, smin, smax, scond, smlnum
549 * ..
550 * .. External Functions ..
551  EXTERNAL lsame, slamch, sla_syrpvgrw
552  LOGICAL lsame
553  REAL slamch, sla_syrpvgrw
554 * ..
555 * .. External Subroutines ..
556  EXTERNAL ssycon, ssyequb, ssytrf, ssytrs,
558 * ..
559 * .. Intrinsic Functions ..
560  INTRINSIC max, min
561 * ..
562 * .. Executable Statements ..
563 *
564  info = 0
565  nofact = lsame( fact, 'N' )
566  equil = lsame( fact, 'E' )
567  smlnum = slamch( 'Safe minimum' )
568  bignum = one / smlnum
569  IF( nofact .OR. equil ) THEN
570  equed = 'N'
571  rcequ = .false.
572  ELSE
573  rcequ = lsame( equed, 'Y' )
574  ENDIF
575 *
576 * Default is failure. If an input parameter is wrong or
577 * factorization fails, make everything look horrible. Only the
578 * pivot growth is set here, the rest is initialized in SSYRFSX.
579 *
580  rpvgrw = zero
581 *
582 * Test the input parameters. PARAMS is not tested until SSYRFSX.
583 *
584  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
585  \$ lsame( fact, 'F' ) ) THEN
586  info = -1
587  ELSE IF( .NOT.lsame(uplo, 'U') .AND.
588  \$ .NOT.lsame(uplo, 'L') ) THEN
589  info = -2
590  ELSE IF( n.LT.0 ) THEN
591  info = -3
592  ELSE IF( nrhs.LT.0 ) THEN
593  info = -4
594  ELSE IF( lda.LT.max( 1, n ) ) THEN
595  info = -6
596  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
597  info = -8
598  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
599  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
600  info = -10
601  ELSE
602  IF ( rcequ ) THEN
603  smin = bignum
604  smax = zero
605  DO 10 j = 1, n
606  smin = min( smin, s( j ) )
607  smax = max( smax, s( j ) )
608  10 CONTINUE
609  IF( smin.LE.zero ) THEN
610  info = -11
611  ELSE IF( n.GT.0 ) THEN
612  scond = max( smin, smlnum ) / min( smax, bignum )
613  ELSE
614  scond = one
615  END IF
616  END IF
617  IF( info.EQ.0 ) THEN
618  IF( ldb.LT.max( 1, n ) ) THEN
619  info = -13
620  ELSE IF( ldx.LT.max( 1, n ) ) THEN
621  info = -15
622  END IF
623  END IF
624  END IF
625 *
626  IF( info.NE.0 ) THEN
627  CALL xerbla( 'SSYSVXX', -info )
628  RETURN
629  END IF
630 *
631  IF( equil ) THEN
632 *
633 * Compute row and column scalings to equilibrate the matrix A.
634 *
635  CALL ssyequb( uplo, n, a, lda, s, scond, amax, work, infequ )
636  IF( infequ.EQ.0 ) THEN
637 *
638 * Equilibrate the matrix.
639 *
640  CALL slaqsy( uplo, n, a, lda, s, scond, amax, equed )
641  rcequ = lsame( equed, 'Y' )
642  END IF
643  END IF
644 *
645 * Scale the right-hand side.
646 *
647  IF( rcequ ) CALL slascl2( n, nrhs, s, b, ldb )
648 *
649  IF( nofact .OR. equil ) THEN
650 *
651 * Compute the LDL^T or UDU^T factorization of A.
652 *
653  CALL slacpy( uplo, n, n, a, lda, af, ldaf )
654  CALL ssytrf( uplo, n, af, ldaf, ipiv, work, 5*max(1,n), info )
655 *
656 * Return if INFO is non-zero.
657 *
658  IF( info.GT.0 ) THEN
659 *
660 * Pivot in column INFO is exactly 0
661 * Compute the reciprocal pivot growth factor of the
662 * leading rank-deficient INFO columns of A.
663 *
664  IF ( n.GT.0 )
665  \$ rpvgrw = sla_syrpvgrw(uplo, n, info, a, lda, af,
666  \$ ldaf, ipiv, work )
667  RETURN
668  END IF
669  END IF
670 *
671 * Compute the reciprocal pivot growth factor RPVGRW.
672 *
673  IF ( n.GT.0 )
674  \$ rpvgrw = sla_syrpvgrw( uplo, n, info, a, lda, af, ldaf,
675  \$ ipiv, work )
676 *
677 * Compute the solution matrix X.
678 *
679  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
680  CALL ssytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
681 *
682 * Use iterative refinement to improve the computed solution and
683 * compute error bounds and backward error estimates for it.
684 *
685  CALL ssyrfsx( uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv,
686  \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
687  \$ err_bnds_comp, nparams, params, work, iwork, info )
688 *
689 * Scale solutions.
690 *
691  IF ( rcequ ) THEN
692  CALL slascl2 ( n, nrhs, s, x, ldx )
693  END IF
694 *
695  RETURN
696 *
697 * End of SSYSVXX
698 *
subroutine ssytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SSYTRS
Definition: ssytrs.f:122
subroutine ssytrf(UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
SSYTRF
Definition: ssytrf.f:184
subroutine slascl2(M, N, D, X, LDX)
SLASCL2 performs diagonal scaling on a vector.
Definition: slascl2.f:92
subroutine ssyrfsx(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, IWORK, INFO)
SSYRFSX
Definition: ssyrfsx.f:404
subroutine ssycon(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
SSYCON
Definition: ssycon.f:132
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine ssyequb(UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO)
SSYEQUB
Definition: ssyequb.f:137
real function sla_syrpvgrw(UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK)
SLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite m...
Definition: sla_syrpvgrw.f:124
subroutine slaqsy(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
SLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
Definition: slaqsy.f:135
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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