LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ cposvxx()

 subroutine cposvxx ( character fact, character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, 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 )

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

Purpose:
```    CPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
to compute the solution to a complex system of linear equations
A * X = B, where A is an N-by-N Hermitian positive definite matrix
and X and B are N-by-NRHS matrices.

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

CPOSVXX 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
CPOSVXX 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 CPOSVXX 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 principal minor of order i is not positive,
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 COMPLEX array, dimension (LDA,N) On entry, the Hermitian 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 COMPLEX 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 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.```

Definition at line 492 of file cposvxx.f.

496*
497* -- LAPACK driver routine --
498* -- LAPACK is a software package provided by Univ. of Tennessee, --
499* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
500*
501* .. Scalar Arguments ..
502 CHARACTER EQUED, FACT, UPLO
503 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
504 \$ N_ERR_BNDS
505 REAL RCOND, RPVGRW
506* ..
507* .. Array Arguments ..
508 COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
509 \$ WORK( * ), X( LDX, * )
510 REAL S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
511 \$ ERR_BNDS_NORM( NRHS, * ),
512 \$ ERR_BNDS_COMP( NRHS, * )
513* ..
514*
515* ==================================================================
516*
517* .. Parameters ..
518 REAL ZERO, ONE
519 parameter( zero = 0.0e+0, one = 1.0e+0 )
520 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
521 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
522 INTEGER CMP_ERR_I, PIV_GROWTH_I
523 parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
524 \$ berr_i = 3 )
525 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
526 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
527 \$ piv_growth_i = 9 )
528* ..
529* .. Local Scalars ..
530 LOGICAL EQUIL, NOFACT, RCEQU
531 INTEGER INFEQU, J
532 REAL AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
533* ..
534* .. External Functions ..
535 EXTERNAL lsame, slamch, cla_porpvgrw
536 LOGICAL LSAME
537 REAL SLAMCH, CLA_PORPVGRW
538* ..
539* .. External Subroutines ..
540 EXTERNAL cpoequb, cpotrf, cpotrs, clacpy,
542* ..
543* .. Intrinsic Functions ..
544 INTRINSIC max, min
545* ..
546* .. Executable Statements ..
547*
548 info = 0
549 nofact = lsame( fact, 'N' )
550 equil = lsame( fact, 'E' )
551 smlnum = slamch( 'Safe minimum' )
552 bignum = one / smlnum
553 IF( nofact .OR. equil ) THEN
554 equed = 'N'
555 rcequ = .false.
556 ELSE
557 rcequ = lsame( equed, 'Y' )
558 ENDIF
559*
560* Default is failure. If an input parameter is wrong or
561* factorization fails, make everything look horrible. Only the
562* pivot growth is set here, the rest is initialized in CPORFSX.
563*
564 rpvgrw = zero
565*
566* Test the input parameters. PARAMS is not tested until CPORFSX.
567*
568 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
569 \$ lsame( fact, 'F' ) ) THEN
570 info = -1
571 ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
572 \$ .NOT.lsame( uplo, 'L' ) ) THEN
573 info = -2
574 ELSE IF( n.LT.0 ) THEN
575 info = -3
576 ELSE IF( nrhs.LT.0 ) THEN
577 info = -4
578 ELSE IF( lda.LT.max( 1, n ) ) THEN
579 info = -6
580 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
581 info = -8
582 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
583 \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
584 info = -9
585 ELSE
586 IF ( rcequ ) THEN
587 smin = bignum
588 smax = zero
589 DO 10 j = 1, n
590 smin = min( smin, s( j ) )
591 smax = max( smax, s( j ) )
592 10 CONTINUE
593 IF( smin.LE.zero ) THEN
594 info = -10
595 ELSE IF( n.GT.0 ) THEN
596 scond = max( smin, smlnum ) / min( smax, bignum )
597 ELSE
598 scond = one
599 END IF
600 END IF
601 IF( info.EQ.0 ) THEN
602 IF( ldb.LT.max( 1, n ) ) THEN
603 info = -12
604 ELSE IF( ldx.LT.max( 1, n ) ) THEN
605 info = -14
606 END IF
607 END IF
608 END IF
609*
610 IF( info.NE.0 ) THEN
611 CALL xerbla( 'CPOSVXX', -info )
612 RETURN
613 END IF
614*
615 IF( equil ) THEN
616*
617* Compute row and column scalings to equilibrate the matrix A.
618*
619 CALL cpoequb( n, a, lda, s, scond, amax, infequ )
620 IF( infequ.EQ.0 ) THEN
621*
622* Equilibrate the matrix.
623*
624 CALL claqhe( uplo, n, a, lda, s, scond, amax, equed )
625 rcequ = lsame( equed, 'Y' )
626 END IF
627 END IF
628*
629* Scale the right-hand side.
630*
631 IF( rcequ ) CALL clascl2( n, nrhs, s, b, ldb )
632*
633 IF( nofact .OR. equil ) THEN
634*
635* Compute the Cholesky factorization of A.
636*
637 CALL clacpy( uplo, n, n, a, lda, af, ldaf )
638 CALL cpotrf( uplo, n, af, ldaf, info )
639*
640* Return if INFO is non-zero.
641*
642 IF( info.GT.0 ) THEN
643*
644* Pivot in column INFO is exactly 0
645* Compute the reciprocal pivot growth factor of the
646* leading rank-deficient INFO columns of A.
647*
648 rpvgrw = cla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
649 RETURN
650 END IF
651 END IF
652*
653* Compute the reciprocal pivot growth factor RPVGRW.
654*
655 rpvgrw = cla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
656*
657* Compute the solution matrix X.
658*
659 CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
660 CALL cpotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
661*
662* Use iterative refinement to improve the computed solution and
663* compute error bounds and backward error estimates for it.
664*
665 CALL cporfsx( uplo, equed, n, nrhs, a, lda, af, ldaf,
666 \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
667 \$ err_bnds_comp, nparams, params, work, rwork, info )
668
669*
670* Scale solutions.
671*
672 IF ( rcequ ) THEN
673 CALL clascl2( n, nrhs, s, x, ldx )
674 END IF
675*
676 RETURN
677*
678* End of CPOSVXX
679*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
real function cla_porpvgrw(uplo, ncols, a, lda, af, ldaf, work)
CLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian...
subroutine clacpy(uplo, m, n, a, lda, b, ldb)
CLACPY copies all or part of one two-dimensional array to another.
Definition clacpy.f:103
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
subroutine claqhe(uplo, n, a, lda, s, scond, amax, equed)
CLAQHE scales a Hermitian matrix.
Definition claqhe.f:134
subroutine clascl2(m, n, d, x, ldx)
CLASCL2 performs diagonal scaling on a matrix.
Definition clascl2.f:91
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine cpoequb(n, a, lda, s, scond, amax, info)
CPOEQUB
Definition cpoequb.f:119
subroutine cporfsx(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, rwork, info)
CPORFSX
Definition cporfsx.f:393
subroutine cpotrf(uplo, n, a, lda, info)
CPOTRF
Definition cpotrf.f:107
subroutine cpotrs(uplo, n, nrhs, a, lda, b, ldb, info)
CPOTRS
Definition cpotrs.f:110
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