612 lines
18 KiB
Fortran
612 lines
18 KiB
Fortran
*> \brief \b CSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download CSYTF2 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/csytf2.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/csytf2.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/csytf2.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE CSYTF2( UPLO, N, A, LDA, IPIV, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER UPLO
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* INTEGER INFO, LDA, N
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* ..
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* .. Array Arguments ..
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* INTEGER IPIV( * )
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* COMPLEX A( LDA, * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> CSYTF2 computes the factorization of a complex symmetric matrix A
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*> using the Bunch-Kaufman diagonal pivoting method:
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*>
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*> A = U*D*U**T or A = L*D*L**T
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*>
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*> where U (or L) is a product of permutation and unit upper (lower)
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*> triangular matrices, U**T is the transpose of U, and D is symmetric and
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*> block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
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*>
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*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] UPLO
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*> \verbatim
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*> UPLO is CHARACTER*1
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*> Specifies whether the upper or lower triangular part of the
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*> symmetric matrix A is stored:
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*> = 'U': Upper triangular
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*> = 'L': Lower triangular
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is COMPLEX array, dimension (LDA,N)
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*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
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*> n-by-n upper triangular part of A contains the upper
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*> triangular part of the matrix A, and the strictly lower
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*> triangular part of A is not referenced. If UPLO = 'L', the
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*> leading n-by-n lower triangular part of A contains the lower
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*> triangular part of the matrix A, and the strictly upper
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*> triangular part of A is not referenced.
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*>
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*> On exit, the block diagonal matrix D and the multipliers used
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*> to obtain the factor U or L (see below for further details).
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max(1,N).
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*> \endverbatim
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*>
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*> \param[out] IPIV
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*> \verbatim
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*> IPIV is INTEGER array, dimension (N)
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*> Details of the interchanges and the block structure of D.
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*>
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*> If UPLO = 'U':
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*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
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*> interchanged and D(k,k) is a 1-by-1 diagonal block.
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*>
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*> If IPIV(k) = IPIV(k-1) < 0, then rows and columns
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*> k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
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*> is a 2-by-2 diagonal block.
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*>
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*> If UPLO = 'L':
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*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
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*> interchanged and D(k,k) is a 1-by-1 diagonal block.
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*>
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*> If IPIV(k) = IPIV(k+1) < 0, then rows and columns
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*> k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
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*> is a 2-by-2 diagonal block.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -k, the k-th argument had an illegal value
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*> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
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*> has been completed, but the block diagonal matrix D is
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*> exactly singular, and division by zero will occur if it
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*> is used to solve a system of equations.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \date November 2013
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*
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*> \ingroup complexSYcomputational
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*
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> If UPLO = 'U', then A = U*D*U**T, where
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*> U = P(n)*U(n)* ... *P(k)U(k)* ...,
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*> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
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*> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
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*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
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*> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
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*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
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*>
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*> ( I v 0 ) k-s
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*> U(k) = ( 0 I 0 ) s
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*> ( 0 0 I ) n-k
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*> k-s s n-k
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*>
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*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
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*> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
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*> and A(k,k), and v overwrites A(1:k-2,k-1:k).
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*>
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*> If UPLO = 'L', then A = L*D*L**T, where
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*> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
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*> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
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*> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
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*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
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*> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
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*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
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*>
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*> ( I 0 0 ) k-1
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*> L(k) = ( 0 I 0 ) s
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*> ( 0 v I ) n-k-s+1
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*> k-1 s n-k-s+1
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*>
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*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
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*> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
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*> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
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*> \endverbatim
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*
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*> \par Contributors:
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* ==================
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*>
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*> \verbatim
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*>
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*> 09-29-06 - patch from
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*> Bobby Cheng, MathWorks
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*>
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*> Replace l.209 and l.377
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*> IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
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*> by
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*> IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN
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*>
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*> 1-96 - Based on modifications by J. Lewis, Boeing Computer Services
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*> Company
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*> \endverbatim
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*
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* =====================================================================
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SUBROUTINE CSYTF2( UPLO, N, A, LDA, IPIV, INFO )
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*
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* -- LAPACK computational routine (version 3.5.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* November 2013
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*
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* .. Scalar Arguments ..
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CHARACTER UPLO
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INTEGER INFO, LDA, N
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* ..
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* .. Array Arguments ..
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INTEGER IPIV( * )
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COMPLEX A( LDA, * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
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REAL EIGHT, SEVTEN
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PARAMETER ( EIGHT = 8.0E+0, SEVTEN = 17.0E+0 )
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COMPLEX CONE
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PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ) )
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* ..
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* .. Local Scalars ..
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LOGICAL UPPER
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INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP
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REAL ABSAKK, ALPHA, COLMAX, ROWMAX
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COMPLEX D11, D12, D21, D22, R1, T, WK, WKM1, WKP1, Z
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* ..
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* .. External Functions ..
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LOGICAL LSAME, SISNAN
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INTEGER ICAMAX
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EXTERNAL LSAME, ICAMAX, SISNAN
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* ..
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* .. External Subroutines ..
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EXTERNAL CSCAL, CSWAP, CSYR, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, AIMAG, MAX, REAL, SQRT
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* ..
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* .. Statement Functions ..
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REAL CABS1
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* ..
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* .. Statement Function definitions ..
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CABS1( Z ) = ABS( REAL( Z ) ) + ABS( AIMAG( Z ) )
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters.
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*
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INFO = 0
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UPPER = LSAME( UPLO, 'U' )
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IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
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INFO = -1
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ELSE IF( N.LT.0 ) THEN
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INFO = -2
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -4
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'CSYTF2', -INFO )
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RETURN
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END IF
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*
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* Initialize ALPHA for use in choosing pivot block size.
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*
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ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
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*
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IF( UPPER ) THEN
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*
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* Factorize A as U*D*U**T using the upper triangle of A
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*
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* K is the main loop index, decreasing from N to 1 in steps of
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* 1 or 2
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*
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K = N
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10 CONTINUE
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*
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* If K < 1, exit from loop
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*
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IF( K.LT.1 )
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$ GO TO 70
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KSTEP = 1
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*
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* Determine rows and columns to be interchanged and whether
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* a 1-by-1 or 2-by-2 pivot block will be used
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*
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ABSAKK = CABS1( A( K, K ) )
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*
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* IMAX is the row-index of the largest off-diagonal element in
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* column K, and COLMAX is its absolute value.
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* Determine both COLMAX and IMAX.
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*
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IF( K.GT.1 ) THEN
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IMAX = ICAMAX( K-1, A( 1, K ), 1 )
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COLMAX = CABS1( A( IMAX, K ) )
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ELSE
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COLMAX = ZERO
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END IF
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*
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IF( MAX( ABSAKK, COLMAX ).EQ.ZERO .OR. SISNAN(ABSAKK) ) THEN
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*
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* Column K is zero or underflow, or contains a NaN:
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* set INFO and continue
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*
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IF( INFO.EQ.0 )
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$ INFO = K
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KP = K
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ELSE
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IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
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*
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* no interchange, use 1-by-1 pivot block
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*
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KP = K
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ELSE
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*
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* JMAX is the column-index of the largest off-diagonal
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* element in row IMAX, and ROWMAX is its absolute value
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*
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JMAX = IMAX + ICAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA )
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ROWMAX = CABS1( A( IMAX, JMAX ) )
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IF( IMAX.GT.1 ) THEN
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JMAX = ICAMAX( IMAX-1, A( 1, IMAX ), 1 )
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ROWMAX = MAX( ROWMAX, CABS1( A( JMAX, IMAX ) ) )
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END IF
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*
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IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
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*
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* no interchange, use 1-by-1 pivot block
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*
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KP = K
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ELSE IF( CABS1( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
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*
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* interchange rows and columns K and IMAX, use 1-by-1
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* pivot block
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*
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KP = IMAX
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ELSE
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*
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* interchange rows and columns K-1 and IMAX, use 2-by-2
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* pivot block
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*
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KP = IMAX
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KSTEP = 2
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END IF
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END IF
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*
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KK = K - KSTEP + 1
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IF( KP.NE.KK ) THEN
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*
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* Interchange rows and columns KK and KP in the leading
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* submatrix A(1:k,1:k)
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*
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CALL CSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
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CALL CSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
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$ LDA )
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T = A( KK, KK )
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A( KK, KK ) = A( KP, KP )
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A( KP, KP ) = T
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IF( KSTEP.EQ.2 ) THEN
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T = A( K-1, K )
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A( K-1, K ) = A( KP, K )
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A( KP, K ) = T
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END IF
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END IF
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*
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* Update the leading submatrix
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*
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IF( KSTEP.EQ.1 ) THEN
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*
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* 1-by-1 pivot block D(k): column k now holds
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*
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* W(k) = U(k)*D(k)
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*
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* where U(k) is the k-th column of U
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*
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* Perform a rank-1 update of A(1:k-1,1:k-1) as
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*
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* A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
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*
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R1 = CONE / A( K, K )
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CALL CSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA )
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*
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* Store U(k) in column k
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*
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CALL CSCAL( K-1, R1, A( 1, K ), 1 )
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ELSE
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*
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* 2-by-2 pivot block D(k): columns k and k-1 now hold
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*
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* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
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*
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* where U(k) and U(k-1) are the k-th and (k-1)-th columns
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* of U
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*
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* Perform a rank-2 update of A(1:k-2,1:k-2) as
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*
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* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
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* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
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*
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IF( K.GT.2 ) THEN
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*
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D12 = A( K-1, K )
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D22 = A( K-1, K-1 ) / D12
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D11 = A( K, K ) / D12
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T = CONE / ( D11*D22-CONE )
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D12 = T / D12
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*
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DO 30 J = K - 2, 1, -1
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WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) )
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WK = D12*( D22*A( J, K )-A( J, K-1 ) )
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DO 20 I = J, 1, -1
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A( I, J ) = A( I, J ) - A( I, K )*WK -
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$ A( I, K-1 )*WKM1
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20 CONTINUE
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A( J, K ) = WK
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A( J, K-1 ) = WKM1
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30 CONTINUE
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*
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END IF
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*
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END IF
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END IF
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*
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* Store details of the interchanges in IPIV
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*
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IF( KSTEP.EQ.1 ) THEN
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IPIV( K ) = KP
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ELSE
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IPIV( K ) = -KP
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IPIV( K-1 ) = -KP
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END IF
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*
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* Decrease K and return to the start of the main loop
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*
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K = K - KSTEP
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GO TO 10
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*
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ELSE
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*
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* Factorize A as L*D*L**T using the lower triangle of A
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*
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* K is the main loop index, increasing from 1 to N in steps of
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* 1 or 2
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*
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K = 1
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40 CONTINUE
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*
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* If K > N, exit from loop
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*
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IF( K.GT.N )
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$ GO TO 70
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KSTEP = 1
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*
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* Determine rows and columns to be interchanged and whether
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* a 1-by-1 or 2-by-2 pivot block will be used
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*
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ABSAKK = CABS1( A( K, K ) )
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*
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* IMAX is the row-index of the largest off-diagonal element in
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* column K, and COLMAX is its absolute value.
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* Determine both COLMAX and IMAX.
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*
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IF( K.LT.N ) THEN
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IMAX = K + ICAMAX( N-K, A( K+1, K ), 1 )
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COLMAX = CABS1( A( IMAX, K ) )
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ELSE
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COLMAX = ZERO
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END IF
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*
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IF( MAX( ABSAKK, COLMAX ).EQ.ZERO .OR. SISNAN(ABSAKK) ) THEN
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*
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* Column K is zero or underflow, or contains a NaN:
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* set INFO and continue
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*
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IF( INFO.EQ.0 )
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$ INFO = K
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KP = K
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ELSE
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IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
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*
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* no interchange, use 1-by-1 pivot block
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*
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KP = K
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ELSE
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*
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* JMAX is the column-index of the largest off-diagonal
|
|
* element in row IMAX, and ROWMAX is its absolute value
|
|
*
|
|
JMAX = K - 1 + ICAMAX( IMAX-K, A( IMAX, K ), LDA )
|
|
ROWMAX = CABS1( A( IMAX, JMAX ) )
|
|
IF( IMAX.LT.N ) THEN
|
|
JMAX = IMAX + ICAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 )
|
|
ROWMAX = MAX( ROWMAX, CABS1( A( JMAX, IMAX ) ) )
|
|
END IF
|
|
*
|
|
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
|
|
*
|
|
* no interchange, use 1-by-1 pivot block
|
|
*
|
|
KP = K
|
|
ELSE IF( CABS1( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
|
|
*
|
|
* interchange rows and columns K and IMAX, use 1-by-1
|
|
* pivot block
|
|
*
|
|
KP = IMAX
|
|
ELSE
|
|
*
|
|
* interchange rows and columns K+1 and IMAX, use 2-by-2
|
|
* pivot block
|
|
*
|
|
KP = IMAX
|
|
KSTEP = 2
|
|
END IF
|
|
END IF
|
|
*
|
|
KK = K + KSTEP - 1
|
|
IF( KP.NE.KK ) THEN
|
|
*
|
|
* Interchange rows and columns KK and KP in the trailing
|
|
* submatrix A(k:n,k:n)
|
|
*
|
|
IF( KP.LT.N )
|
|
$ CALL CSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
|
|
CALL CSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
|
|
$ LDA )
|
|
T = A( KK, KK )
|
|
A( KK, KK ) = A( KP, KP )
|
|
A( KP, KP ) = T
|
|
IF( KSTEP.EQ.2 ) THEN
|
|
T = A( K+1, K )
|
|
A( K+1, K ) = A( KP, K )
|
|
A( KP, K ) = T
|
|
END IF
|
|
END IF
|
|
*
|
|
* Update the trailing submatrix
|
|
*
|
|
IF( KSTEP.EQ.1 ) THEN
|
|
*
|
|
* 1-by-1 pivot block D(k): column k now holds
|
|
*
|
|
* W(k) = L(k)*D(k)
|
|
*
|
|
* where L(k) is the k-th column of L
|
|
*
|
|
IF( K.LT.N ) THEN
|
|
*
|
|
* Perform a rank-1 update of A(k+1:n,k+1:n) as
|
|
*
|
|
* A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
|
|
*
|
|
R1 = CONE / A( K, K )
|
|
CALL CSYR( UPLO, N-K, -R1, A( K+1, K ), 1,
|
|
$ A( K+1, K+1 ), LDA )
|
|
*
|
|
* Store L(k) in column K
|
|
*
|
|
CALL CSCAL( N-K, R1, A( K+1, K ), 1 )
|
|
END IF
|
|
ELSE
|
|
*
|
|
* 2-by-2 pivot block D(k)
|
|
*
|
|
IF( K.LT.N-1 ) THEN
|
|
*
|
|
* Perform a rank-2 update of A(k+2:n,k+2:n) as
|
|
*
|
|
* A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T
|
|
* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T
|
|
*
|
|
* where L(k) and L(k+1) are the k-th and (k+1)-th
|
|
* columns of L
|
|
*
|
|
D21 = A( K+1, K )
|
|
D11 = A( K+1, K+1 ) / D21
|
|
D22 = A( K, K ) / D21
|
|
T = CONE / ( D11*D22-CONE )
|
|
D21 = T / D21
|
|
*
|
|
DO 60 J = K + 2, N
|
|
WK = D21*( D11*A( J, K )-A( J, K+1 ) )
|
|
WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) )
|
|
DO 50 I = J, N
|
|
A( I, J ) = A( I, J ) - A( I, K )*WK -
|
|
$ A( I, K+1 )*WKP1
|
|
50 CONTINUE
|
|
A( J, K ) = WK
|
|
A( J, K+1 ) = WKP1
|
|
60 CONTINUE
|
|
END IF
|
|
END IF
|
|
END IF
|
|
*
|
|
* Store details of the interchanges in IPIV
|
|
*
|
|
IF( KSTEP.EQ.1 ) THEN
|
|
IPIV( K ) = KP
|
|
ELSE
|
|
IPIV( K ) = -KP
|
|
IPIV( K+1 ) = -KP
|
|
END IF
|
|
*
|
|
* Increase K and return to the start of the main loop
|
|
*
|
|
K = K + KSTEP
|
|
GO TO 40
|
|
*
|
|
END IF
|
|
*
|
|
70 CONTINUE
|
|
RETURN
|
|
*
|
|
* End of CSYTF2
|
|
*
|
|
END
|