972 lines
30 KiB
Fortran
972 lines
30 KiB
Fortran
*> \brief \b ZLASYF_RK computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.
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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 ZLASYF_RK + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlasyf_rk.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/zlasyf_rk.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/zlasyf_rk.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 ZLASYF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
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* INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER UPLO
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* INTEGER INFO, KB, LDA, LDW, N, NB
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* ..
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* .. Array Arguments ..
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* INTEGER IPIV( * )
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* COMPLEX*16 A( LDA, * ), E( * ), W( LDW, * )
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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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*> ZLASYF_RK computes a partial factorization of a complex symmetric
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*> matrix A using the bounded Bunch-Kaufman (rook) diagonal
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*> pivoting method. The partial factorization has the form:
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*>
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*> A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
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*> ( 0 U22 ) ( 0 D ) ( U12**T U22**T )
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*>
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*> A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L',
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*> ( L21 I ) ( 0 A22 ) ( 0 I )
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*>
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*> where the order of D is at most NB. The actual order is returned in
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*> the argument KB, and is either NB or NB-1, or N if N <= NB.
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*>
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*> ZLASYF_RK is an auxiliary routine called by ZSYTRF_RK. It uses
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*> blocked code (calling Level 3 BLAS) to update the submatrix
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*> A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
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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] NB
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*> \verbatim
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*> NB is INTEGER
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*> The maximum number of columns of the matrix A that should be
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*> factored. NB should be at least 2 to allow for 2-by-2 pivot
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*> blocks.
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*> \endverbatim
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*>
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*> \param[out] KB
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*> \verbatim
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*> KB is INTEGER
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*> The number of columns of A that were actually factored.
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*> KB is either NB-1 or NB, or N if N <= NB.
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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*16 array, dimension (LDA,N)
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*> On entry, the symmetric matrix A.
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*> If UPLO = 'U': the leading N-by-N upper triangular part
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*> of A contains the upper triangular part of the matrix A,
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*> and the strictly lower triangular part of A is not
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*> referenced.
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*>
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*> If UPLO = 'L': the leading N-by-N lower triangular part
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*> of A contains the lower triangular part of the matrix A,
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*> and the strictly upper triangular part of A is not
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*> referenced.
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*>
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*> On exit, contains:
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*> a) ONLY diagonal elements of the symmetric block diagonal
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*> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
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*> (superdiagonal (or subdiagonal) elements of D
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*> are stored on exit in array E), and
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*> b) If UPLO = 'U': factor U in the superdiagonal part of A.
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*> If UPLO = 'L': factor L in the subdiagonal part of A.
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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] E
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*> \verbatim
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*> E is COMPLEX*16 array, dimension (N)
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*> On exit, contains the superdiagonal (or subdiagonal)
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*> elements of the symmetric block diagonal matrix D
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*> with 1-by-1 or 2-by-2 diagonal blocks, where
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*> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
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*> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
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*>
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*> NOTE: For 1-by-1 diagonal block D(k), where
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*> 1 <= k <= N, the element E(k) is set to 0 in both
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*> UPLO = 'U' or UPLO = 'L' cases.
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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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*> IPIV describes the permutation matrix P in the factorization
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*> of matrix A as follows. The absolute value of IPIV(k)
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*> represents the index of row and column that were
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*> interchanged with the k-th row and column. The value of UPLO
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*> describes the order in which the interchanges were applied.
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*> Also, the sign of IPIV represents the block structure of
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*> the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
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*> diagonal blocks which correspond to 1 or 2 interchanges
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*> at each factorization step.
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*>
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*> If UPLO = 'U',
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*> ( in factorization order, k decreases from N to 1 ):
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*> a) A single positive entry IPIV(k) > 0 means:
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*> D(k,k) is a 1-by-1 diagonal block.
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*> If IPIV(k) != k, rows and columns k and IPIV(k) were
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*> interchanged in the submatrix A(1:N,N-KB+1:N);
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*> If IPIV(k) = k, no interchange occurred.
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*>
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*>
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*> b) A pair of consecutive negative entries
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*> IPIV(k) < 0 and IPIV(k-1) < 0 means:
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*> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
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*> (NOTE: negative entries in IPIV appear ONLY in pairs).
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*> 1) If -IPIV(k) != k, rows and columns
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*> k and -IPIV(k) were interchanged
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*> in the matrix A(1:N,N-KB+1:N).
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*> If -IPIV(k) = k, no interchange occurred.
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*> 2) If -IPIV(k-1) != k-1, rows and columns
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*> k-1 and -IPIV(k-1) were interchanged
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*> in the submatrix A(1:N,N-KB+1:N).
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*> If -IPIV(k-1) = k-1, no interchange occurred.
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*>
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*> c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
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*>
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*> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
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*>
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*> If UPLO = 'L',
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*> ( in factorization order, k increases from 1 to N ):
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*> a) A single positive entry IPIV(k) > 0 means:
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*> D(k,k) is a 1-by-1 diagonal block.
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*> If IPIV(k) != k, rows and columns k and IPIV(k) were
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*> interchanged in the submatrix A(1:N,1:KB).
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*> If IPIV(k) = k, no interchange occurred.
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*>
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*> b) A pair of consecutive negative entries
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*> IPIV(k) < 0 and IPIV(k+1) < 0 means:
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*> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
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*> (NOTE: negative entries in IPIV appear ONLY in pairs).
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*> 1) If -IPIV(k) != k, rows and columns
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*> k and -IPIV(k) were interchanged
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*> in the submatrix A(1:N,1:KB).
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*> If -IPIV(k) = k, no interchange occurred.
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*> 2) If -IPIV(k+1) != k+1, rows and columns
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*> k-1 and -IPIV(k-1) were interchanged
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*> in the submatrix A(1:N,1:KB).
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*> If -IPIV(k+1) = k+1, no interchange occurred.
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*>
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*> c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
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*>
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*> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
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*> \endverbatim
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*>
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*> \param[out] W
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*> \verbatim
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*> W is COMPLEX*16 array, dimension (LDW,NB)
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*> \endverbatim
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*>
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*> \param[in] LDW
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*> \verbatim
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*> LDW is INTEGER
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*> The leading dimension of the array W. LDW >= max(1,N).
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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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*>
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*> < 0: If INFO = -k, the k-th argument had an illegal value
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*>
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*> > 0: If INFO = k, the matrix A is singular, because:
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*> If UPLO = 'U': column k in the upper
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*> triangular part of A contains all zeros.
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*> If UPLO = 'L': column k in the lower
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*> triangular part of A contains all zeros.
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*>
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*> Therefore D(k,k) is exactly zero, and superdiagonal
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*> elements of column k of U (or subdiagonal elements of
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*> column k of L ) are all zeros. The factorization has
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*> been completed, but the block diagonal matrix D is
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*> exactly singular, and division by zero will occur if
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*> it is used to solve a system of equations.
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*>
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*> NOTE: INFO only stores the first occurrence of
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*> a singularity, any subsequent occurrence of singularity
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*> is not stored in INFO even though the factorization
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*> always completes.
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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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*> \ingroup complex16SYcomputational
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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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*> December 2016, Igor Kozachenko,
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*> Computer Science Division,
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*> University of California, Berkeley
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*>
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*> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
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*> School of Mathematics,
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*> University of Manchester
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*>
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*> \endverbatim
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*
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* =====================================================================
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SUBROUTINE ZLASYF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
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$ INFO )
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*
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* -- LAPACK computational routine --
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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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*
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* .. Scalar Arguments ..
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CHARACTER UPLO
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INTEGER INFO, KB, LDA, LDW, N, NB
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* ..
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* .. Array Arguments ..
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INTEGER IPIV( * )
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COMPLEX*16 A( LDA, * ), E( * ), W( LDW, * )
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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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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
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DOUBLE PRECISION EIGHT, SEVTEN
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PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
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COMPLEX*16 CONE, CZERO
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PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ),
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$ CZERO = ( 0.0D+0, 0.0D+0 ) )
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* ..
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* .. Local Scalars ..
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LOGICAL DONE
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INTEGER IMAX, ITEMP, J, JB, JJ, JMAX, K, KK, KW, KKW,
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$ KP, KSTEP, P, II
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DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX, SFMIN, DTEMP
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COMPLEX*16 D11, D12, D21, D22, R1, T, Z
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER IZAMAX
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DOUBLE PRECISION DLAMCH
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EXTERNAL LSAME, IZAMAX, DLAMCH
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* ..
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* .. External Subroutines ..
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EXTERNAL ZCOPY, ZGEMM, ZGEMV, ZSCAL, ZSWAP
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, DBLE, DIMAG, MAX, MIN, SQRT
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* ..
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* .. Statement Functions ..
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DOUBLE PRECISION CABS1
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* ..
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* .. Statement Function definitions ..
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CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
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* ..
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* .. Executable Statements ..
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*
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INFO = 0
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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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* Compute machine safe minimum
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*
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SFMIN = DLAMCH( 'S' )
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*
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IF( LSAME( UPLO, 'U' ) ) THEN
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*
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* Factorize the trailing columns of A using the upper triangle
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* of A and working backwards, and compute the matrix W = U12*D
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* for use in updating A11
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*
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* Initialize the first entry of array E, where superdiagonal
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* elements of D are stored
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*
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E( 1 ) = CZERO
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*
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* K is the main loop index, decreasing from N in steps of 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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* KW is the column of W which corresponds to column K of A
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*
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KW = NB + K - N
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*
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* Exit from loop
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*
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IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
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$ GO TO 30
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*
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KSTEP = 1
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P = K
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*
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* Copy column K of A to column KW of W and update it
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*
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CALL ZCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
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IF( K.LT.N )
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$ CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ),
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$ LDA, W( K, KW+1 ), LDW, CONE, W( 1, KW ), 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( W( K, KW ) )
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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 = IZAMAX( K-1, W( 1, KW ), 1 )
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COLMAX = CABS1( W( IMAX, KW ) )
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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 ) THEN
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*
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* Column K is zero or underflow: 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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CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
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*
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* Set E( K ) to zero
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*
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IF( K.GT.1 )
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$ E( K ) = CZERO
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*
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ELSE
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*
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* ============================================================
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*
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* Test for interchange
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*
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* Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
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* (used to handle NaN and Inf)
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*
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IF( .NOT.( ABSAKK.LT.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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*
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ELSE
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*
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DONE = .FALSE.
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*
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* Loop until pivot found
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*
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12 CONTINUE
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*
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* Begin pivot search loop body
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*
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*
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* Copy column IMAX to column KW-1 of W and update it
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*
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CALL ZCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
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CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
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$ W( IMAX+1, KW-1 ), 1 )
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*
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IF( K.LT.N )
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$ CALL ZGEMV( 'No transpose', K, N-K, -CONE,
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$ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
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$ CONE, W( 1, KW-1 ), 1 )
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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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* Determine both ROWMAX and JMAX.
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*
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IF( IMAX.NE.K ) THEN
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JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ),
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$ 1 )
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ROWMAX = CABS1( W( JMAX, KW-1 ) )
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ELSE
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ROWMAX = ZERO
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END IF
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*
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IF( IMAX.GT.1 ) THEN
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ITEMP = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
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DTEMP = CABS1( W( ITEMP, KW-1 ) )
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IF( DTEMP.GT.ROWMAX ) THEN
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ROWMAX = DTEMP
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JMAX = ITEMP
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END IF
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END IF
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*
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* Equivalent to testing for
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* CABS1( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX
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* (used to handle NaN and Inf)
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*
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IF( .NOT.(CABS1( W( IMAX, KW-1 ) ).LT.ALPHA*ROWMAX ) )
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$ THEN
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*
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* interchange rows and columns K and IMAX,
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* use 1-by-1 pivot block
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*
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KP = IMAX
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*
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* copy column KW-1 of W to column KW of W
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*
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CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
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*
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DONE = .TRUE.
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*
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* Equivalent to testing for ROWMAX.EQ.COLMAX,
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* (used to handle NaN and Inf)
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*
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ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
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$ THEN
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*
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* interchange rows and columns K-1 and IMAX,
|
|
* use 2-by-2 pivot block
|
|
*
|
|
KP = IMAX
|
|
KSTEP = 2
|
|
DONE = .TRUE.
|
|
ELSE
|
|
*
|
|
* Pivot not found: set params and repeat
|
|
*
|
|
P = IMAX
|
|
COLMAX = ROWMAX
|
|
IMAX = JMAX
|
|
*
|
|
* Copy updated JMAXth (next IMAXth) column to Kth of W
|
|
*
|
|
CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
|
|
*
|
|
END IF
|
|
*
|
|
* End pivot search loop body
|
|
*
|
|
IF( .NOT. DONE ) GOTO 12
|
|
*
|
|
END IF
|
|
*
|
|
* ============================================================
|
|
*
|
|
KK = K - KSTEP + 1
|
|
*
|
|
* KKW is the column of W which corresponds to column KK of A
|
|
*
|
|
KKW = NB + KK - N
|
|
*
|
|
IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
|
|
*
|
|
* Copy non-updated column K to column P
|
|
*
|
|
CALL ZCOPY( K-P, A( P+1, K ), 1, A( P, P+1 ), LDA )
|
|
CALL ZCOPY( P, A( 1, K ), 1, A( 1, P ), 1 )
|
|
*
|
|
* Interchange rows K and P in last N-K+1 columns of A
|
|
* and last N-K+2 columns of W
|
|
*
|
|
CALL ZSWAP( N-K+1, A( K, K ), LDA, A( P, K ), LDA )
|
|
CALL ZSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), LDW )
|
|
END IF
|
|
*
|
|
* Updated column KP is already stored in column KKW of W
|
|
*
|
|
IF( KP.NE.KK ) THEN
|
|
*
|
|
* Copy non-updated column KK to column KP
|
|
*
|
|
A( KP, K ) = A( KK, K )
|
|
CALL ZCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
|
|
$ LDA )
|
|
CALL ZCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
|
|
*
|
|
* Interchange rows KK and KP in last N-KK+1 columns
|
|
* of A and W
|
|
*
|
|
CALL ZSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
|
|
CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
|
|
$ LDW )
|
|
END IF
|
|
*
|
|
IF( KSTEP.EQ.1 ) THEN
|
|
*
|
|
* 1-by-1 pivot block D(k): column KW of W now holds
|
|
*
|
|
* W(k) = U(k)*D(k)
|
|
*
|
|
* where U(k) is the k-th column of U
|
|
*
|
|
* Store U(k) in column k of A
|
|
*
|
|
CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
|
|
IF( K.GT.1 ) THEN
|
|
IF( CABS1( A( K, K ) ).GE.SFMIN ) THEN
|
|
R1 = CONE / A( K, K )
|
|
CALL ZSCAL( K-1, R1, A( 1, K ), 1 )
|
|
ELSE IF( A( K, K ).NE.CZERO ) THEN
|
|
DO 14 II = 1, K - 1
|
|
A( II, K ) = A( II, K ) / A( K, K )
|
|
14 CONTINUE
|
|
END IF
|
|
*
|
|
* Store the superdiagonal element of D in array E
|
|
*
|
|
E( K ) = CZERO
|
|
*
|
|
END IF
|
|
*
|
|
ELSE
|
|
*
|
|
* 2-by-2 pivot block D(k): columns KW and KW-1 of W now
|
|
* hold
|
|
*
|
|
* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
|
|
*
|
|
* where U(k) and U(k-1) are the k-th and (k-1)-th columns
|
|
* of U
|
|
*
|
|
IF( K.GT.2 ) THEN
|
|
*
|
|
* Store U(k) and U(k-1) in columns k and k-1 of A
|
|
*
|
|
D12 = W( K-1, KW )
|
|
D11 = W( K, KW ) / D12
|
|
D22 = W( K-1, KW-1 ) / D12
|
|
T = CONE / ( D11*D22-CONE )
|
|
DO 20 J = 1, K - 2
|
|
A( J, K-1 ) = T*( (D11*W( J, KW-1 )-W( J, KW ) ) /
|
|
$ D12 )
|
|
A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
|
|
$ D12 )
|
|
20 CONTINUE
|
|
END IF
|
|
*
|
|
* Copy diagonal elements of D(K) to A,
|
|
* copy superdiagonal element of D(K) to E(K) and
|
|
* ZERO out superdiagonal entry of A
|
|
*
|
|
A( K-1, K-1 ) = W( K-1, KW-1 )
|
|
A( K-1, K ) = CZERO
|
|
A( K, K ) = W( K, KW )
|
|
E( K ) = W( K-1, KW )
|
|
E( K-1 ) = CZERO
|
|
*
|
|
END IF
|
|
*
|
|
* End column K is nonsingular
|
|
*
|
|
END IF
|
|
*
|
|
* Store details of the interchanges in IPIV
|
|
*
|
|
IF( KSTEP.EQ.1 ) THEN
|
|
IPIV( K ) = KP
|
|
ELSE
|
|
IPIV( K ) = -P
|
|
IPIV( K-1 ) = -KP
|
|
END IF
|
|
*
|
|
* Decrease K and return to the start of the main loop
|
|
*
|
|
K = K - KSTEP
|
|
GO TO 10
|
|
*
|
|
30 CONTINUE
|
|
*
|
|
* Update the upper triangle of A11 (= A(1:k,1:k)) as
|
|
*
|
|
* A11 := A11 - U12*D*U12**T = A11 - U12*W**T
|
|
*
|
|
* computing blocks of NB columns at a time
|
|
*
|
|
DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
|
|
JB = MIN( NB, K-J+1 )
|
|
*
|
|
* Update the upper triangle of the diagonal block
|
|
*
|
|
DO 40 JJ = J, J + JB - 1
|
|
CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
|
|
$ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
|
|
$ A( J, JJ ), 1 )
|
|
40 CONTINUE
|
|
*
|
|
* Update the rectangular superdiagonal block
|
|
*
|
|
IF( J.GE.2 )
|
|
$ CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB,
|
|
$ N-K, -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ),
|
|
$ LDW, CONE, A( 1, J ), LDA )
|
|
50 CONTINUE
|
|
*
|
|
* Set KB to the number of columns factorized
|
|
*
|
|
KB = N - K
|
|
*
|
|
ELSE
|
|
*
|
|
* Factorize the leading columns of A using the lower triangle
|
|
* of A and working forwards, and compute the matrix W = L21*D
|
|
* for use in updating A22
|
|
*
|
|
* Initialize the unused last entry of the subdiagonal array E.
|
|
*
|
|
E( N ) = CZERO
|
|
*
|
|
* K is the main loop index, increasing from 1 in steps of 1 or 2
|
|
*
|
|
K = 1
|
|
70 CONTINUE
|
|
*
|
|
* Exit from loop
|
|
*
|
|
IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
|
|
$ GO TO 90
|
|
*
|
|
KSTEP = 1
|
|
P = K
|
|
*
|
|
* Copy column K of A to column K of W and update it
|
|
*
|
|
CALL ZCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
|
|
IF( K.GT.1 )
|
|
$ CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
|
|
$ LDA, W( K, 1 ), LDW, CONE, W( K, K ), 1 )
|
|
*
|
|
* Determine rows and columns to be interchanged and whether
|
|
* a 1-by-1 or 2-by-2 pivot block will be used
|
|
*
|
|
ABSAKK = CABS1( W( K, K ) )
|
|
*
|
|
* IMAX is the row-index of the largest off-diagonal element in
|
|
* column K, and COLMAX is its absolute value.
|
|
* Determine both COLMAX and IMAX.
|
|
*
|
|
IF( K.LT.N ) THEN
|
|
IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
|
|
COLMAX = CABS1( W( IMAX, K ) )
|
|
ELSE
|
|
COLMAX = ZERO
|
|
END IF
|
|
*
|
|
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
|
|
*
|
|
* Column K is zero or underflow: set INFO and continue
|
|
*
|
|
IF( INFO.EQ.0 )
|
|
$ INFO = K
|
|
KP = K
|
|
CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
|
|
*
|
|
* Set E( K ) to zero
|
|
*
|
|
IF( K.LT.N )
|
|
$ E( K ) = CZERO
|
|
*
|
|
ELSE
|
|
*
|
|
* ============================================================
|
|
*
|
|
* Test for interchange
|
|
*
|
|
* Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
|
|
* (used to handle NaN and Inf)
|
|
*
|
|
IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
|
|
*
|
|
* no interchange, use 1-by-1 pivot block
|
|
*
|
|
KP = K
|
|
*
|
|
ELSE
|
|
*
|
|
DONE = .FALSE.
|
|
*
|
|
* Loop until pivot found
|
|
*
|
|
72 CONTINUE
|
|
*
|
|
* Begin pivot search loop body
|
|
*
|
|
*
|
|
* Copy column IMAX to column K+1 of W and update it
|
|
*
|
|
CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
|
|
CALL ZCOPY( N-IMAX+1, A( IMAX, IMAX ), 1,
|
|
$ W( IMAX, K+1 ), 1 )
|
|
IF( K.GT.1 )
|
|
$ CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE,
|
|
$ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
|
|
$ CONE, W( K, K+1 ), 1 )
|
|
*
|
|
* JMAX is the column-index of the largest off-diagonal
|
|
* element in row IMAX, and ROWMAX is its absolute value.
|
|
* Determine both ROWMAX and JMAX.
|
|
*
|
|
IF( IMAX.NE.K ) THEN
|
|
JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
|
|
ROWMAX = CABS1( W( JMAX, K+1 ) )
|
|
ELSE
|
|
ROWMAX = ZERO
|
|
END IF
|
|
*
|
|
IF( IMAX.LT.N ) THEN
|
|
ITEMP = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
|
|
DTEMP = CABS1( W( ITEMP, K+1 ) )
|
|
IF( DTEMP.GT.ROWMAX ) THEN
|
|
ROWMAX = DTEMP
|
|
JMAX = ITEMP
|
|
END IF
|
|
END IF
|
|
*
|
|
* Equivalent to testing for
|
|
* CABS1( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX
|
|
* (used to handle NaN and Inf)
|
|
*
|
|
IF( .NOT.( CABS1( W( IMAX, K+1 ) ).LT.ALPHA*ROWMAX ) )
|
|
$ THEN
|
|
*
|
|
* interchange rows and columns K and IMAX,
|
|
* use 1-by-1 pivot block
|
|
*
|
|
KP = IMAX
|
|
*
|
|
* copy column K+1 of W to column K of W
|
|
*
|
|
CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
|
|
*
|
|
DONE = .TRUE.
|
|
*
|
|
* Equivalent to testing for ROWMAX.EQ.COLMAX,
|
|
* (used to handle NaN and Inf)
|
|
*
|
|
ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
|
|
$ THEN
|
|
*
|
|
* interchange rows and columns K+1 and IMAX,
|
|
* use 2-by-2 pivot block
|
|
*
|
|
KP = IMAX
|
|
KSTEP = 2
|
|
DONE = .TRUE.
|
|
ELSE
|
|
*
|
|
* Pivot not found: set params and repeat
|
|
*
|
|
P = IMAX
|
|
COLMAX = ROWMAX
|
|
IMAX = JMAX
|
|
*
|
|
* Copy updated JMAXth (next IMAXth) column to Kth of W
|
|
*
|
|
CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
|
|
*
|
|
END IF
|
|
*
|
|
* End pivot search loop body
|
|
*
|
|
IF( .NOT. DONE ) GOTO 72
|
|
*
|
|
END IF
|
|
*
|
|
* ============================================================
|
|
*
|
|
KK = K + KSTEP - 1
|
|
*
|
|
IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
|
|
*
|
|
* Copy non-updated column K to column P
|
|
*
|
|
CALL ZCOPY( P-K, A( K, K ), 1, A( P, K ), LDA )
|
|
CALL ZCOPY( N-P+1, A( P, K ), 1, A( P, P ), 1 )
|
|
*
|
|
* Interchange rows K and P in first K columns of A
|
|
* and first K+1 columns of W
|
|
*
|
|
CALL ZSWAP( K, A( K, 1 ), LDA, A( P, 1 ), LDA )
|
|
CALL ZSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
|
|
END IF
|
|
*
|
|
* Updated column KP is already stored in column KK of W
|
|
*
|
|
IF( KP.NE.KK ) THEN
|
|
*
|
|
* Copy non-updated column KK to column KP
|
|
*
|
|
A( KP, K ) = A( KK, K )
|
|
CALL ZCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
|
|
CALL ZCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
|
|
*
|
|
* Interchange rows KK and KP in first KK columns of A and W
|
|
*
|
|
CALL ZSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
|
|
CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
|
|
END IF
|
|
*
|
|
IF( KSTEP.EQ.1 ) THEN
|
|
*
|
|
* 1-by-1 pivot block D(k): column k of W now holds
|
|
*
|
|
* W(k) = L(k)*D(k)
|
|
*
|
|
* where L(k) is the k-th column of L
|
|
*
|
|
* Store L(k) in column k of A
|
|
*
|
|
CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
|
|
IF( K.LT.N ) THEN
|
|
IF( CABS1( A( K, K ) ).GE.SFMIN ) THEN
|
|
R1 = CONE / A( K, K )
|
|
CALL ZSCAL( N-K, R1, A( K+1, K ), 1 )
|
|
ELSE IF( A( K, K ).NE.CZERO ) THEN
|
|
DO 74 II = K + 1, N
|
|
A( II, K ) = A( II, K ) / A( K, K )
|
|
74 CONTINUE
|
|
END IF
|
|
*
|
|
* Store the subdiagonal element of D in array E
|
|
*
|
|
E( K ) = CZERO
|
|
*
|
|
END IF
|
|
*
|
|
ELSE
|
|
*
|
|
* 2-by-2 pivot block D(k): columns k and k+1 of W now hold
|
|
*
|
|
* ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
|
|
*
|
|
* where L(k) and L(k+1) are the k-th and (k+1)-th columns
|
|
* of L
|
|
*
|
|
IF( K.LT.N-1 ) THEN
|
|
*
|
|
* Store L(k) and L(k+1) in columns k and k+1 of A
|
|
*
|
|
D21 = W( K+1, K )
|
|
D11 = W( K+1, K+1 ) / D21
|
|
D22 = W( K, K ) / D21
|
|
T = CONE / ( D11*D22-CONE )
|
|
DO 80 J = K + 2, N
|
|
A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
|
|
$ D21 )
|
|
A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
|
|
$ D21 )
|
|
80 CONTINUE
|
|
END IF
|
|
*
|
|
* Copy diagonal elements of D(K) to A,
|
|
* copy subdiagonal element of D(K) to E(K) and
|
|
* ZERO out subdiagonal entry of A
|
|
*
|
|
A( K, K ) = W( K, K )
|
|
A( K+1, K ) = CZERO
|
|
A( K+1, K+1 ) = W( K+1, K+1 )
|
|
E( K ) = W( K+1, K )
|
|
E( K+1 ) = CZERO
|
|
*
|
|
END IF
|
|
*
|
|
* End column K is nonsingular
|
|
*
|
|
END IF
|
|
*
|
|
* Store details of the interchanges in IPIV
|
|
*
|
|
IF( KSTEP.EQ.1 ) THEN
|
|
IPIV( K ) = KP
|
|
ELSE
|
|
IPIV( K ) = -P
|
|
IPIV( K+1 ) = -KP
|
|
END IF
|
|
*
|
|
* Increase K and return to the start of the main loop
|
|
*
|
|
K = K + KSTEP
|
|
GO TO 70
|
|
*
|
|
90 CONTINUE
|
|
*
|
|
* Update the lower triangle of A22 (= A(k:n,k:n)) as
|
|
*
|
|
* A22 := A22 - L21*D*L21**T = A22 - L21*W**T
|
|
*
|
|
* computing blocks of NB columns at a time
|
|
*
|
|
DO 110 J = K, N, NB
|
|
JB = MIN( NB, N-J+1 )
|
|
*
|
|
* Update the lower triangle of the diagonal block
|
|
*
|
|
DO 100 JJ = J, J + JB - 1
|
|
CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
|
|
$ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
|
|
$ A( JJ, JJ ), 1 )
|
|
100 CONTINUE
|
|
*
|
|
* Update the rectangular subdiagonal block
|
|
*
|
|
IF( J+JB.LE.N )
|
|
$ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
|
|
$ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
|
|
$ LDW, CONE, A( J+JB, J ), LDA )
|
|
110 CONTINUE
|
|
*
|
|
* Set KB to the number of columns factorized
|
|
*
|
|
KB = K - 1
|
|
*
|
|
END IF
|
|
*
|
|
RETURN
|
|
*
|
|
* End of ZLASYF_RK
|
|
*
|
|
END
|