944 lines
29 KiB
Fortran
944 lines
29 KiB
Fortran
*> \brief \b DSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 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 DSYTF2_RK + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytf2_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/dsytf2_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/dsytf2_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 DSYTF2_RK( UPLO, N, A, LDA, E, 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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* DOUBLE PRECISION A( LDA, * ), E ( * )
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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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*> DSYTF2_RK computes the factorization of a real symmetric matrix A
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*> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
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*>
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*> A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
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*>
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*> where U (or L) is unit upper (or lower) triangular matrix,
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*> U**T (or L**T) is the transpose of U (or L), P is a permutation
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*> matrix, P**T is the transpose of P, and D is symmetric and block
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*> 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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*> For more information see Further Details section.
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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 DOUBLE PRECISION 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 DOUBLE PRECISION 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. For more info see Further
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*> Details section.
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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 matrix A(1:N,1:N);
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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-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,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 matrix A(1:N,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), 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 matrix A(1:N,1:N).
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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 matrix A(1:N,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 matrix A(1:N,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), 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] 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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*> \date December 2016
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*
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*> \ingroup doubleSYcomputational
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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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*> TODO: put further details
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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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*> 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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*> 01-01-96 - Based on modifications by
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*> J. Lewis, Boeing Computer Services Company
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*> A. Petitet, Computer Science Dept.,
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*> Univ. of Tenn., Knoxville abd , USA
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*> \endverbatim
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*
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* =====================================================================
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SUBROUTINE DSYTF2_RK( UPLO, N, A, LDA, E, IPIV, INFO )
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*
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* -- LAPACK computational routine (version 3.7.0) --
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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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* December 2016
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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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DOUBLE PRECISION A( LDA, * ), E( * )
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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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* ..
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* .. Local Scalars ..
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LOGICAL UPPER, DONE
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INTEGER I, IMAX, J, JMAX, ITEMP, K, KK, KP, KSTEP,
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$ P, II
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DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
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$ ROWMAX, DTEMP, T, WK, WKM1, WKP1, SFMIN
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER IDAMAX
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DOUBLE PRECISION DLAMCH
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EXTERNAL LSAME, IDAMAX, DLAMCH
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* ..
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* .. External Subroutines ..
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EXTERNAL DSCAL, DSWAP, DSYR, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, SQRT
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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( 'DSYTF2_RK', -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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* Compute machine safe minimum
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*
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SFMIN = DLAMCH( 'S' )
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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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* 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 ) = ZERO
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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 34
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KSTEP = 1
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P = K
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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 = ABS( 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 = IDAMAX( K-1, A( 1, K ), 1 )
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COLMAX = ABS( 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) ) 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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*
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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 ) = ZERO
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*
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ELSE
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*
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* Test for interchange
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*
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* Equivalent to testing for (used to handle NaN and Inf)
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* ABSAKK.GE.ALPHA*COLMAX
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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,
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* 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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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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* 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 + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ),
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$ LDA )
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ROWMAX = ABS( A( IMAX, JMAX ) )
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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 = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
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DTEMP = ABS( A( ITEMP, IMAX ) )
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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 (used to handle NaN and Inf)
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* ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
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*
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IF( .NOT.( ABS( A( IMAX, IMAX ) ).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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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 ) ) THEN
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*
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* interchange rows and columns K+1 and IMAX,
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* use 2-by-2 pivot block
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*
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KP = IMAX
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KSTEP = 2
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DONE = .TRUE.
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ELSE
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*
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* Pivot NOT found, set variables and repeat
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*
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P = IMAX
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COLMAX = ROWMAX
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IMAX = JMAX
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END IF
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*
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* End pivot search loop body
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*
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IF( .NOT. DONE ) GOTO 12
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*
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END IF
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*
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* Swap TWO rows and TWO columns
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*
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* First swap
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*
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IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
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*
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* Interchange rows and column K and P in the leading
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* submatrix A(1:k,1:k) if we have a 2-by-2 pivot
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*
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IF( P.GT.1 )
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$ CALL DSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 )
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IF( P.LT.(K-1) )
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$ CALL DSWAP( K-P-1, A( P+1, K ), 1, A( P, P+1 ),
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$ LDA )
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T = A( K, K )
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A( K, K ) = A( P, P )
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A( P, P ) = T
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*
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* Convert upper triangle of A into U form by applying
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* the interchanges in columns k+1:N.
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*
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IF( K.LT.N )
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$ CALL DSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ), LDA )
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*
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END IF
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*
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* Second swap
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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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IF( KP.GT.1 )
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$ CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
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IF( ( KK.GT.1 ) .AND. ( KP.LT.(KK-1) ) )
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$ CALL DSWAP( 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 )
|
|
A( K-1, K ) = A( KP, K )
|
|
A( KP, K ) = T
|
|
END IF
|
|
*
|
|
* Convert upper triangle of A into U form by applying
|
|
* the interchanges in columns k+1:N.
|
|
*
|
|
IF( K.LT.N )
|
|
$ CALL DSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
|
|
$ LDA )
|
|
*
|
|
END IF
|
|
*
|
|
* Update the leading submatrix
|
|
*
|
|
IF( KSTEP.EQ.1 ) THEN
|
|
*
|
|
* 1-by-1 pivot block D(k): column k now holds
|
|
*
|
|
* W(k) = U(k)*D(k)
|
|
*
|
|
* where U(k) is the k-th column of U
|
|
*
|
|
IF( K.GT.1 ) THEN
|
|
*
|
|
* Perform a rank-1 update of A(1:k-1,1:k-1) and
|
|
* store U(k) in column k
|
|
*
|
|
IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
|
|
*
|
|
* Perform a rank-1 update of A(1:k-1,1:k-1) as
|
|
* A := A - U(k)*D(k)*U(k)**T
|
|
* = A - W(k)*1/D(k)*W(k)**T
|
|
*
|
|
D11 = ONE / A( K, K )
|
|
CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
|
|
*
|
|
* Store U(k) in column k
|
|
*
|
|
CALL DSCAL( K-1, D11, A( 1, K ), 1 )
|
|
ELSE
|
|
*
|
|
* Store L(k) in column K
|
|
*
|
|
D11 = A( K, K )
|
|
DO 16 II = 1, K - 1
|
|
A( II, K ) = A( II, K ) / D11
|
|
16 CONTINUE
|
|
*
|
|
* Perform a rank-1 update of A(k+1:n,k+1:n) as
|
|
* A := A - U(k)*D(k)*U(k)**T
|
|
* = A - W(k)*(1/D(k))*W(k)**T
|
|
* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
|
|
*
|
|
CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
|
|
END IF
|
|
*
|
|
* Store the superdiagonal element of D in array E
|
|
*
|
|
E( K ) = ZERO
|
|
*
|
|
END IF
|
|
*
|
|
ELSE
|
|
*
|
|
* 2-by-2 pivot block D(k): columns k and k-1 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
|
|
*
|
|
* Perform a rank-2 update of A(1:k-2,1:k-2) as
|
|
*
|
|
* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
|
|
* = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T
|
|
*
|
|
* and store L(k) and L(k+1) in columns k and k+1
|
|
*
|
|
IF( K.GT.2 ) THEN
|
|
*
|
|
D12 = A( K-1, K )
|
|
D22 = A( K-1, K-1 ) / D12
|
|
D11 = A( K, K ) / D12
|
|
T = ONE / ( D11*D22-ONE )
|
|
*
|
|
DO 30 J = K - 2, 1, -1
|
|
*
|
|
WKM1 = T*( D11*A( J, K-1 )-A( J, K ) )
|
|
WK = T*( D22*A( J, K )-A( J, K-1 ) )
|
|
*
|
|
DO 20 I = J, 1, -1
|
|
A( I, J ) = A( I, J ) - (A( I, K ) / D12 )*WK -
|
|
$ ( A( I, K-1 ) / D12 )*WKM1
|
|
20 CONTINUE
|
|
*
|
|
* Store U(k) and U(k-1) in cols k and k-1 for row J
|
|
*
|
|
A( J, K ) = WK / D12
|
|
A( J, K-1 ) = WKM1 / D12
|
|
*
|
|
30 CONTINUE
|
|
*
|
|
END IF
|
|
*
|
|
* Copy superdiagonal elements of D(K) to E(K) and
|
|
* ZERO out superdiagonal entry of A
|
|
*
|
|
E( K ) = A( K-1, K )
|
|
E( K-1 ) = ZERO
|
|
A( K-1, K ) = ZERO
|
|
*
|
|
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
|
|
*
|
|
34 CONTINUE
|
|
*
|
|
ELSE
|
|
*
|
|
* Factorize A as L*D*L**T using the lower triangle of A
|
|
*
|
|
* Initialize the unused last entry of the subdiagonal array E.
|
|
*
|
|
E( N ) = ZERO
|
|
*
|
|
* K is the main loop index, increasing from 1 to N in steps of
|
|
* 1 or 2
|
|
*
|
|
K = 1
|
|
40 CONTINUE
|
|
*
|
|
* If K > N, exit from loop
|
|
*
|
|
IF( K.GT.N )
|
|
$ GO TO 64
|
|
KSTEP = 1
|
|
P = K
|
|
*
|
|
* Determine rows and columns to be interchanged and whether
|
|
* a 1-by-1 or 2-by-2 pivot block will be used
|
|
*
|
|
ABSAKK = ABS( A( 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 + IDAMAX( N-K, A( K+1, K ), 1 )
|
|
COLMAX = ABS( A( 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
|
|
*
|
|
* Set E( K ) to zero
|
|
*
|
|
IF( K.LT.N )
|
|
$ E( K ) = ZERO
|
|
*
|
|
ELSE
|
|
*
|
|
* Test for interchange
|
|
*
|
|
* Equivalent to testing for (used to handle NaN and Inf)
|
|
* ABSAKK.GE.ALPHA*COLMAX
|
|
*
|
|
IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
|
|
*
|
|
* no interchange, use 1-by-1 pivot block
|
|
*
|
|
KP = K
|
|
*
|
|
ELSE
|
|
*
|
|
DONE = .FALSE.
|
|
*
|
|
* Loop until pivot found
|
|
*
|
|
42 CONTINUE
|
|
*
|
|
* Begin pivot search loop body
|
|
*
|
|
* 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 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
|
|
ROWMAX = ABS( A( IMAX, JMAX ) )
|
|
ELSE
|
|
ROWMAX = ZERO
|
|
END IF
|
|
*
|
|
IF( IMAX.LT.N ) THEN
|
|
ITEMP = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ),
|
|
$ 1 )
|
|
DTEMP = ABS( A( ITEMP, IMAX ) )
|
|
IF( DTEMP.GT.ROWMAX ) THEN
|
|
ROWMAX = DTEMP
|
|
JMAX = ITEMP
|
|
END IF
|
|
END IF
|
|
*
|
|
* Equivalent to testing for (used to handle NaN and Inf)
|
|
* ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
|
|
*
|
|
IF( .NOT.( ABS( A( IMAX, IMAX ) ).LT.ALPHA*ROWMAX ) )
|
|
$ THEN
|
|
*
|
|
* interchange rows and columns K and IMAX,
|
|
* use 1-by-1 pivot block
|
|
*
|
|
KP = IMAX
|
|
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 variables and repeat
|
|
*
|
|
P = IMAX
|
|
COLMAX = ROWMAX
|
|
IMAX = JMAX
|
|
END IF
|
|
*
|
|
* End pivot search loop body
|
|
*
|
|
IF( .NOT. DONE ) GOTO 42
|
|
*
|
|
END IF
|
|
*
|
|
* Swap TWO rows and TWO columns
|
|
*
|
|
* First swap
|
|
*
|
|
IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
|
|
*
|
|
* Interchange rows and column K and P in the trailing
|
|
* submatrix A(k:n,k:n) if we have a 2-by-2 pivot
|
|
*
|
|
IF( P.LT.N )
|
|
$ CALL DSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
|
|
IF( P.GT.(K+1) )
|
|
$ CALL DSWAP( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA )
|
|
T = A( K, K )
|
|
A( K, K ) = A( P, P )
|
|
A( P, P ) = T
|
|
*
|
|
* Convert lower triangle of A into L form by applying
|
|
* the interchanges in columns 1:k-1.
|
|
*
|
|
IF ( K.GT.1 )
|
|
$ CALL DSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA )
|
|
*
|
|
END IF
|
|
*
|
|
* Second swap
|
|
*
|
|
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 DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
|
|
IF( ( KK.LT.N ) .AND. ( KP.GT.(KK+1) ) )
|
|
$ CALL DSWAP( 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
|
|
*
|
|
* Convert lower triangle of A into L form by applying
|
|
* the interchanges in columns 1:k-1.
|
|
*
|
|
IF ( K.GT.1 )
|
|
$ CALL DSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
|
|
*
|
|
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) and
|
|
* store L(k) in column k
|
|
*
|
|
IF( ABS( A( K, K ) ).GE.SFMIN ) 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
|
|
*
|
|
D11 = ONE / A( K, K )
|
|
CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
|
|
$ A( K+1, K+1 ), LDA )
|
|
*
|
|
* Store L(k) in column k
|
|
*
|
|
CALL DSCAL( N-K, D11, A( K+1, K ), 1 )
|
|
ELSE
|
|
*
|
|
* Store L(k) in column k
|
|
*
|
|
D11 = A( K, K )
|
|
DO 46 II = K + 1, N
|
|
A( II, K ) = A( II, K ) / D11
|
|
46 CONTINUE
|
|
*
|
|
* 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
|
|
* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
|
|
*
|
|
CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
|
|
$ A( K+1, K+1 ), LDA )
|
|
END IF
|
|
*
|
|
* Store the subdiagonal element of D in array E
|
|
*
|
|
E( K ) = ZERO
|
|
*
|
|
END IF
|
|
*
|
|
ELSE
|
|
*
|
|
* 2-by-2 pivot block D(k): columns k and k+1 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
|
|
*
|
|
*
|
|
* 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 - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T
|
|
*
|
|
* and store L(k) and L(k+1) in columns k and k+1
|
|
*
|
|
IF( K.LT.N-1 ) THEN
|
|
*
|
|
D21 = A( K+1, K )
|
|
D11 = A( K+1, K+1 ) / D21
|
|
D22 = A( K, K ) / D21
|
|
T = ONE / ( D11*D22-ONE )
|
|
*
|
|
DO 60 J = K + 2, N
|
|
*
|
|
* Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
|
|
*
|
|
WK = T*( D11*A( J, K )-A( J, K+1 ) )
|
|
WKP1 = T*( D22*A( J, K+1 )-A( J, K ) )
|
|
*
|
|
* Perform a rank-2 update of A(k+2:n,k+2:n)
|
|
*
|
|
DO 50 I = J, N
|
|
A( I, J ) = A( I, J ) - ( A( I, K ) / D21 )*WK -
|
|
$ ( A( I, K+1 ) / D21 )*WKP1
|
|
50 CONTINUE
|
|
*
|
|
* Store L(k) and L(k+1) in cols k and k+1 for row J
|
|
*
|
|
A( J, K ) = WK / D21
|
|
A( J, K+1 ) = WKP1 / D21
|
|
*
|
|
60 CONTINUE
|
|
*
|
|
END IF
|
|
*
|
|
* Copy subdiagonal elements of D(K) to E(K) and
|
|
* ZERO out subdiagonal entry of A
|
|
*
|
|
E( K ) = A( K+1, K )
|
|
E( K+1 ) = ZERO
|
|
A( K+1, K ) = ZERO
|
|
*
|
|
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 40
|
|
*
|
|
64 CONTINUE
|
|
*
|
|
END IF
|
|
*
|
|
RETURN
|
|
*
|
|
* End of DSYTF2_RK
|
|
*
|
|
END
|