 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

◆ sposvxx()

 subroutine sposvxx ( character FACT, character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, 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 )

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

Purpose:
```    SPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
to compute the solution to a real 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. SPOSVXX 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.

SPOSVXX 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
SPOSVXX 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 SPOSVXX 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 REAL 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 REAL 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 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
April 2012

Definition at line 499 of file sposvxx.f.

499 *
500 * -- LAPACK driver routine (version 3.7.0) --
501 * -- LAPACK is a software package provided by Univ. of Tennessee, --
502 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
503 * April 2012
504 *
505 * .. Scalar Arguments ..
506  CHARACTER equed, fact, uplo
507  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs, nparams,
508  \$ n_err_bnds
509  REAL rcond, rpvgrw
510 * ..
511 * .. Array Arguments ..
512  INTEGER iwork( * )
513  REAL a( lda, * ), af( ldaf, * ), b( ldb, * ),
514  \$ x( ldx, * ), work( * )
515  REAL s( * ), params( * ), berr( * ),
516  \$ err_bnds_norm( nrhs, * ),
517  \$ err_bnds_comp( nrhs, * )
518 * ..
519 *
520 * ==================================================================
521 *
522 * .. Parameters ..
523  REAL zero, one
524  parameter( zero = 0.0e+0, one = 1.0e+0 )
525  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
526  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
527  INTEGER cmp_err_i, piv_growth_i
528  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
529  \$ berr_i = 3 )
530  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
531  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
532  \$ piv_growth_i = 9 )
533 * ..
534 * .. Local Scalars ..
535  LOGICAL equil, nofact, rcequ
536  INTEGER infequ, j
537  REAL amax, bignum, smin, smax,
538  \$ scond, smlnum
539 * ..
540 * .. External Functions ..
541  EXTERNAL lsame, slamch, sla_porpvgrw
542  LOGICAL lsame
543  REAL slamch, sla_porpvgrw
544 * ..
545 * .. External Subroutines ..
546  EXTERNAL spoequb, spotrf, spotrs, slacpy, slaqsy,
548 * ..
549 * .. Intrinsic Functions ..
550  INTRINSIC max, min
551 * ..
552 * .. Executable Statements ..
553 *
554  info = 0
555  nofact = lsame( fact, 'N' )
556  equil = lsame( fact, 'E' )
557  smlnum = slamch( 'Safe minimum' )
558  bignum = one / smlnum
559  IF( nofact .OR. equil ) THEN
560  equed = 'N'
561  rcequ = .false.
562  ELSE
563  rcequ = lsame( equed, 'Y' )
564  ENDIF
565 *
566 * Default is failure. If an input parameter is wrong or
567 * factorization fails, make everything look horrible. Only the
568 * pivot growth is set here, the rest is initialized in SPORFSX.
569 *
570  rpvgrw = zero
571 *
572 * Test the input parameters. PARAMS is not tested until SPORFSX.
573 *
574  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
575  \$ lsame( fact, 'F' ) ) THEN
576  info = -1
577  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
578  \$ .NOT.lsame( uplo, 'L' ) ) THEN
579  info = -2
580  ELSE IF( n.LT.0 ) THEN
581  info = -3
582  ELSE IF( nrhs.LT.0 ) THEN
583  info = -4
584  ELSE IF( lda.LT.max( 1, n ) ) THEN
585  info = -6
586  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
587  info = -8
588  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
589  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
590  info = -9
591  ELSE
592  IF ( rcequ ) THEN
593  smin = bignum
594  smax = zero
595  DO 10 j = 1, n
596  smin = min( smin, s( j ) )
597  smax = max( smax, s( j ) )
598  10 CONTINUE
599  IF( smin.LE.zero ) THEN
600  info = -10
601  ELSE IF( n.GT.0 ) THEN
602  scond = max( smin, smlnum ) / min( smax, bignum )
603  ELSE
604  scond = one
605  END IF
606  END IF
607  IF( info.EQ.0 ) THEN
608  IF( ldb.LT.max( 1, n ) ) THEN
609  info = -12
610  ELSE IF( ldx.LT.max( 1, n ) ) THEN
611  info = -14
612  END IF
613  END IF
614  END IF
615 *
616  IF( info.NE.0 ) THEN
617  CALL xerbla( 'SPOSVXX', -info )
618  RETURN
619  END IF
620 *
621  IF( equil ) THEN
622 *
623 * Compute row and column scalings to equilibrate the matrix A.
624 *
625  CALL spoequb( n, a, lda, s, scond, amax, infequ )
626  IF( infequ.EQ.0 ) THEN
627 *
628 * Equilibrate the matrix.
629 *
630  CALL slaqsy( uplo, n, a, lda, s, scond, amax, equed )
631  rcequ = lsame( equed, 'Y' )
632  END IF
633  END IF
634 *
635 * Scale the right-hand side.
636 *
637  IF( rcequ ) CALL slascl2( n, nrhs, s, b, ldb )
638 *
639  IF( nofact .OR. equil ) THEN
640 *
641 * Compute the Cholesky factorization of A.
642 *
643  CALL slacpy( uplo, n, n, a, lda, af, ldaf )
644  CALL spotrf( uplo, n, af, ldaf, info )
645 *
646 * Return if INFO is non-zero.
647 *
648  IF( info.NE.0 ) THEN
649 *
650 * Pivot in column INFO is exactly 0
651 * Compute the reciprocal pivot growth factor of the
652 * leading rank-deficient INFO columns of A.
653 *
654  rpvgrw = sla_porpvgrw( uplo, info, a, lda, af, ldaf, work )
655  RETURN
656  ENDIF
657  END IF
658 *
659 * Compute the reciprocal growth factor RPVGRW.
660 *
661  rpvgrw = sla_porpvgrw( uplo, n, a, lda, af, ldaf, work )
662 *
663 * Compute the solution matrix X.
664 *
665  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
666  CALL spotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
667 *
668 * Use iterative refinement to improve the computed solution and
669 * compute error bounds and backward error estimates for it.
670 *
671  CALL sporfsx( uplo, equed, n, nrhs, a, lda, af, ldaf,
672  \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
673  \$ err_bnds_comp, nparams, params, work, iwork, info )
674
675 *
676 * Scale solutions.
677 *
678  IF ( rcequ ) THEN
679  CALL slascl2 ( n, nrhs, s, x, ldx )
680  END IF
681 *
682  RETURN
683 *
684 * End of SPOSVXX
685 *
subroutine spotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
SPOTRS
Definition: spotrs.f:112
subroutine spoequb(N, A, LDA, S, SCOND, AMAX, INFO)
SPOEQUB
Definition: spoequb.f:120
subroutine spotrf(UPLO, N, A, LDA, INFO)
SPOTRF
Definition: spotrf.f:109
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine sporfsx(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, IWORK, INFO)
SPORFSX
Definition: sporfsx.f:396
subroutine slascl2(M, N, D, X, LDX)
SLASCL2 performs diagonal scaling on a vector.
Definition: slascl2.f:92
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
real function sla_porpvgrw(UPLO, NCOLS, A, LDA, AF, LDAF, WORK)
SLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian...
Definition: sla_porpvgrw.f:106
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 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
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