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*> \brief \b DSYCON_3
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYCON_3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsycon_3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsycon_3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsycon_3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYCON_3( UPLO, N, A, LDA, E, IPIV, ANORM, RCOND,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION A( LDA, * ), E ( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*> DSYCON_3 estimates the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric matrix A using the factorization
*> computed by DSYTRF_RK or DSYTRF_BK:
*>
*> A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
*>
*> where U (or L) is unit upper (or lower) triangular matrix,
*> U**T (or L**T) is the transpose of U (or L), P is a permutation
*> matrix, P**T is the transpose of P, and D is symmetric and block
*> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*> This routine uses BLAS3 solver DSYTRS_3.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the details of the factorization are
*> stored as an upper or lower triangular matrix:
*> = 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T);
*> = 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> Diagonal of the block diagonal matrix D and factors U or L
*> as computed by DSYTRF_RK and DSYTRF_BK:
*> a) ONLY diagonal elements of the symmetric block diagonal
*> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
*> (superdiagonal (or subdiagonal) elements of D
*> should be provided on entry in array E), and
*> b) If UPLO = 'U': factor U in the superdiagonal part of A.
*> If UPLO = 'L': factor L in the subdiagonal part of A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N)
*> On entry, contains the superdiagonal (or subdiagonal)
*> elements of the symmetric block diagonal matrix D
*> with 1-by-1 or 2-by-2 diagonal blocks, where
*> If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
*> If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
*>
*> NOTE: For 1-by-1 diagonal block D(k), where
*> 1 <= k <= N, the element E(k) is not referenced in both
*> UPLO = 'U' or UPLO = 'L' cases.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D
*> as determined by DSYTRF_RK or DSYTRF_BK.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is DOUBLE PRECISION
*> The 1-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*> estimate of the 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup doubleSYcomputational
*
*> \par Contributors:
* ==================
*> \verbatim
*>
*> December 2016, Igor Kozachenko,
*> Computer Science Division,
*> University of California, Berkeley
*>
*> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
*> School of Mathematics,
*> University of Manchester
*>
*> \endverbatim
*
* =====================================================================
SUBROUTINE DSYCON_3( UPLO, N, A, LDA, E, IPIV, ANORM, RCOND,
$ WORK, IWORK, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), E( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, KASE
DOUBLE PRECISION AINVNM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLACN2, DSYTRS_3, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYCON_3', -INFO )
RETURN
END IF
*
* Quick return if possible
*
RCOND = ZERO
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
ELSE IF( ANORM.LE.ZERO ) THEN
RETURN
END IF
*
* Check that the diagonal matrix D is nonsingular.
*
IF( UPPER ) THEN
*
* Upper triangular storage: examine D from bottom to top
*
DO I = N, 1, -1
IF( IPIV( I ).GT.0 .AND. A( I, I ).EQ.ZERO )
$ RETURN
END DO
ELSE
*
* Lower triangular storage: examine D from top to bottom.
*
DO I = 1, N
IF( IPIV( I ).GT.0 .AND. A( I, I ).EQ.ZERO )
$ RETURN
END DO
END IF
*
* Estimate the 1-norm of the inverse.
*
KASE = 0
30 CONTINUE
CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
*
* Multiply by inv(L*D*L**T) or inv(U*D*U**T).
*
CALL DSYTRS_3( UPLO, N, 1, A, LDA, E, IPIV, WORK, N, INFO )
GO TO 30
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / AINVNM ) / ANORM
*
RETURN
*
* End of DSYCON_3
*
END
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