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383 lines
12 KiB
383 lines
12 KiB
2 years ago
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*> \brief \b ZLAED7 used by ZSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
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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 ZLAED7 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaed7.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/zlaed7.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/zlaed7.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 ZLAED7( N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
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* LDQ, RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM,
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* GIVPTR, GIVCOL, GIVNUM, WORK, RWORK, IWORK,
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* INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER CURLVL, CURPBM, CUTPNT, INFO, LDQ, N, QSIZ,
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* $ TLVLS
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* DOUBLE PRECISION RHO
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* ..
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* .. Array Arguments ..
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* INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
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* $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
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* DOUBLE PRECISION D( * ), GIVNUM( 2, * ), QSTORE( * ), RWORK( * )
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* COMPLEX*16 Q( LDQ, * ), WORK( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> ZLAED7 computes the updated eigensystem of a diagonal
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*> matrix after modification by a rank-one symmetric matrix. This
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*> routine is used only for the eigenproblem which requires all
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*> eigenvalues and optionally eigenvectors of a dense or banded
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*> Hermitian matrix that has been reduced to tridiagonal form.
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*>
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*> T = Q(in) ( D(in) + RHO * Z*Z**H ) Q**H(in) = Q(out) * D(out) * Q**H(out)
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*>
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*> where Z = Q**Hu, u is a vector of length N with ones in the
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*> CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
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*>
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*> The eigenvectors of the original matrix are stored in Q, and the
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*> eigenvalues are in D. The algorithm consists of three stages:
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*>
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*> The first stage consists of deflating the size of the problem
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*> when there are multiple eigenvalues or if there is a zero in
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*> the Z vector. For each such occurrence the dimension of the
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*> secular equation problem is reduced by one. This stage is
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*> performed by the routine DLAED2.
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*>
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*> The second stage consists of calculating the updated
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*> eigenvalues. This is done by finding the roots of the secular
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*> equation via the routine DLAED4 (as called by SLAED3).
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*> This routine also calculates the eigenvectors of the current
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*> problem.
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*>
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*> The final stage consists of computing the updated eigenvectors
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*> directly using the updated eigenvalues. The eigenvectors for
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*> the current problem are multiplied with the eigenvectors from
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*> the overall problem.
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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] N
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*> \verbatim
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*> N is INTEGER
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*> The dimension of the symmetric tridiagonal matrix. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] CUTPNT
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*> \verbatim
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*> CUTPNT is INTEGER
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*> Contains the location of the last eigenvalue in the leading
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*> sub-matrix. min(1,N) <= CUTPNT <= N.
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*> \endverbatim
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*>
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*> \param[in] QSIZ
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*> \verbatim
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*> QSIZ is INTEGER
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*> The dimension of the unitary matrix used to reduce
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*> the full matrix to tridiagonal form. QSIZ >= N.
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*> \endverbatim
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*>
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*> \param[in] TLVLS
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*> \verbatim
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*> TLVLS is INTEGER
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*> The total number of merging levels in the overall divide and
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*> conquer tree.
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*> \endverbatim
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*>
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*> \param[in] CURLVL
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*> \verbatim
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*> CURLVL is INTEGER
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*> The current level in the overall merge routine,
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*> 0 <= curlvl <= tlvls.
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*> \endverbatim
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*>
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*> \param[in] CURPBM
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*> \verbatim
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*> CURPBM is INTEGER
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*> The current problem in the current level in the overall
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*> merge routine (counting from upper left to lower right).
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*> \endverbatim
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*>
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*> \param[in,out] D
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*> \verbatim
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*> D is DOUBLE PRECISION array, dimension (N)
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*> On entry, the eigenvalues of the rank-1-perturbed matrix.
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*> On exit, the eigenvalues of the repaired matrix.
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*> \endverbatim
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*>
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*> \param[in,out] Q
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*> \verbatim
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*> Q is COMPLEX*16 array, dimension (LDQ,N)
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*> On entry, the eigenvectors of the rank-1-perturbed matrix.
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*> On exit, the eigenvectors of the repaired tridiagonal matrix.
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*> \endverbatim
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*>
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*> \param[in] LDQ
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*> \verbatim
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*> LDQ is INTEGER
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*> The leading dimension of the array Q. LDQ >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in] RHO
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*> \verbatim
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*> RHO is DOUBLE PRECISION
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*> Contains the subdiagonal element used to create the rank-1
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*> modification.
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*> \endverbatim
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*>
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*> \param[out] INDXQ
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*> \verbatim
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*> INDXQ is INTEGER array, dimension (N)
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*> This contains the permutation which will reintegrate the
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*> subproblem just solved back into sorted order,
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*> ie. D( INDXQ( I = 1, N ) ) will be in ascending order.
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*> \endverbatim
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*>
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*> \param[out] IWORK
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*> \verbatim
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*> IWORK is INTEGER array, dimension (4*N)
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*> \endverbatim
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*>
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*> \param[out] RWORK
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*> \verbatim
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*> RWORK is DOUBLE PRECISION array,
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*> dimension (3*N+2*QSIZ*N)
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is COMPLEX*16 array, dimension (QSIZ*N)
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*> \endverbatim
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*>
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*> \param[in,out] QSTORE
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*> \verbatim
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*> QSTORE is DOUBLE PRECISION array, dimension (N**2+1)
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*> Stores eigenvectors of submatrices encountered during
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*> divide and conquer, packed together. QPTR points to
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*> beginning of the submatrices.
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*> \endverbatim
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*>
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*> \param[in,out] QPTR
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*> \verbatim
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*> QPTR is INTEGER array, dimension (N+2)
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*> List of indices pointing to beginning of submatrices stored
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*> in QSTORE. The submatrices are numbered starting at the
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*> bottom left of the divide and conquer tree, from left to
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*> right and bottom to top.
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*> \endverbatim
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*>
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*> \param[in] PRMPTR
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*> \verbatim
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*> PRMPTR is INTEGER array, dimension (N lg N)
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*> Contains a list of pointers which indicate where in PERM a
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*> level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
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*> indicates the size of the permutation and also the size of
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*> the full, non-deflated problem.
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*> \endverbatim
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*>
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*> \param[in] PERM
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*> \verbatim
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*> PERM is INTEGER array, dimension (N lg N)
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*> Contains the permutations (from deflation and sorting) to be
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*> applied to each eigenblock.
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*> \endverbatim
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*>
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*> \param[in] GIVPTR
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*> \verbatim
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*> GIVPTR is INTEGER array, dimension (N lg N)
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*> Contains a list of pointers which indicate where in GIVCOL a
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*> level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
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*> indicates the number of Givens rotations.
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*> \endverbatim
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*>
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*> \param[in] GIVCOL
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*> \verbatim
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*> GIVCOL is INTEGER array, dimension (2, N lg N)
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*> Each pair of numbers indicates a pair of columns to take place
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*> in a Givens rotation.
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*> \endverbatim
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*>
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*> \param[in] GIVNUM
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*> \verbatim
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*> GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
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*> Each number indicates the S value to be used in the
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*> corresponding Givens rotation.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit.
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*> < 0: if INFO = -i, the i-th argument had an illegal value.
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*> > 0: if INFO = 1, an eigenvalue did not converge
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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 complex16OTHERcomputational
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*
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* =====================================================================
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SUBROUTINE ZLAED7( N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
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$ LDQ, RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM,
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$ GIVPTR, GIVCOL, GIVNUM, WORK, RWORK, IWORK,
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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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INTEGER CURLVL, CURPBM, CUTPNT, INFO, LDQ, N, QSIZ,
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$ TLVLS
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DOUBLE PRECISION RHO
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* ..
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* .. Array Arguments ..
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INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
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$ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
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DOUBLE PRECISION D( * ), GIVNUM( 2, * ), QSTORE( * ), RWORK( * )
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COMPLEX*16 Q( LDQ, * ), WORK( * )
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* ..
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*
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* =====================================================================
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*
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* .. Local Scalars ..
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INTEGER COLTYP, CURR, I, IDLMDA, INDX,
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$ INDXC, INDXP, IQ, IW, IZ, K, N1, N2, PTR
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* ..
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* .. External Subroutines ..
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EXTERNAL DLAED9, DLAEDA, DLAMRG, XERBLA, ZLACRM, ZLAED8
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN
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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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*
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* IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
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* INFO = -1
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* ELSE IF( N.LT.0 ) THEN
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IF( N.LT.0 ) THEN
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INFO = -1
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ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN
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INFO = -2
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ELSE IF( QSIZ.LT.N ) THEN
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INFO = -3
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ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
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INFO = -9
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZLAED7', -INFO )
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( N.EQ.0 )
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$ RETURN
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*
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* The following values are for bookkeeping purposes only. They are
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* integer pointers which indicate the portion of the workspace
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* used by a particular array in DLAED2 and SLAED3.
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*
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IZ = 1
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IDLMDA = IZ + N
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IW = IDLMDA + N
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IQ = IW + N
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*
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INDX = 1
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INDXC = INDX + N
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COLTYP = INDXC + N
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INDXP = COLTYP + N
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*
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* Form the z-vector which consists of the last row of Q_1 and the
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* first row of Q_2.
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*
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PTR = 1 + 2**TLVLS
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DO 10 I = 1, CURLVL - 1
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PTR = PTR + 2**( TLVLS-I )
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10 CONTINUE
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CURR = PTR + CURPBM
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CALL DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR,
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$ GIVCOL, GIVNUM, QSTORE, QPTR, RWORK( IZ ),
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$ RWORK( IZ+N ), INFO )
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*
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* When solving the final problem, we no longer need the stored data,
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* so we will overwrite the data from this level onto the previously
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* used storage space.
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*
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IF( CURLVL.EQ.TLVLS ) THEN
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QPTR( CURR ) = 1
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PRMPTR( CURR ) = 1
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GIVPTR( CURR ) = 1
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END IF
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*
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* Sort and Deflate eigenvalues.
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*
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CALL ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, RWORK( IZ ),
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$ RWORK( IDLMDA ), WORK, QSIZ, RWORK( IW ),
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$ IWORK( INDXP ), IWORK( INDX ), INDXQ,
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$ PERM( PRMPTR( CURR ) ), GIVPTR( CURR+1 ),
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$ GIVCOL( 1, GIVPTR( CURR ) ),
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$ GIVNUM( 1, GIVPTR( CURR ) ), INFO )
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PRMPTR( CURR+1 ) = PRMPTR( CURR ) + N
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GIVPTR( CURR+1 ) = GIVPTR( CURR+1 ) + GIVPTR( CURR )
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*
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|
* Solve Secular Equation.
|
||
|
|
*
|
||
|
|
IF( K.NE.0 ) THEN
|
||
|
|
CALL DLAED9( K, 1, K, N, D, RWORK( IQ ), K, RHO,
|
||
|
|
$ RWORK( IDLMDA ), RWORK( IW ),
|
||
|
|
$ QSTORE( QPTR( CURR ) ), K, INFO )
|
||
|
|
CALL ZLACRM( QSIZ, K, WORK, QSIZ, QSTORE( QPTR( CURR ) ), K, Q,
|
||
|
|
$ LDQ, RWORK( IQ ) )
|
||
|
|
QPTR( CURR+1 ) = QPTR( CURR ) + K**2
|
||
|
|
IF( INFO.NE.0 ) THEN
|
||
|
|
RETURN
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
* Prepare the INDXQ sorting permutation.
|
||
|
|
*
|
||
|
|
N1 = K
|
||
|
|
N2 = N - K
|
||
|
|
CALL DLAMRG( N1, N2, D, 1, -1, INDXQ )
|
||
|
|
ELSE
|
||
|
|
QPTR( CURR+1 ) = QPTR( CURR )
|
||
|
|
DO 20 I = 1, N
|
||
|
|
INDXQ( I ) = I
|
||
|
|
20 CONTINUE
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
RETURN
|
||
|
|
*
|
||
|
|
* End of ZLAED7
|
||
|
|
*
|
||
|
|
END
|