647 lines
18 KiB
C++
647 lines
18 KiB
C++
/* fortran/dlaed2.f -- translated by f2c (version 20200916).
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You must link the resulting object file with libf2c:
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on Microsoft Windows system, link with libf2c.lib;
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on Linux or Unix systems, link with .../path/to/libf2c.a -lm
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or, if you install libf2c.a in a standard place, with -lf2c -lm
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-- in that order, at the end of the command line, as in
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cc *.o -lf2c -lm
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Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
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http://www.netlib.org/f2c/libf2c.zip
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*/
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#ifdef __cplusplus
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extern "C" {
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#endif
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#include "lmp_f2c.h"
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/* Table of constant values */
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static doublereal c_b3 = -1.;
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static integer c__1 = 1;
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/* > \brief \b DLAED2 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original
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matrix is tridiagonal. */
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/* =========== DOCUMENTATION =========== */
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/* Online html documentation available at */
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/* http://www.netlib.org/lapack/explore-html/ */
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/* > \htmlonly */
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/* > Download DLAED2 + dependencies */
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/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed2.
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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/dlaed2.
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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/dlaed2.
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f"> */
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/* > [TXT]</a> */
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/* > \endhtmlonly */
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/* Definition: */
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/* =========== */
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/* SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W, */
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/* Q2, INDX, INDXC, INDXP, COLTYP, INFO ) */
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/* .. Scalar Arguments .. */
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/* INTEGER INFO, K, LDQ, N, N1 */
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/* DOUBLE PRECISION RHO */
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/* .. */
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/* .. Array Arguments .. */
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/* INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ), */
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/* $ INDXQ( * ) */
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/* DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ), */
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/* $ W( * ), Z( * ) */
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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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/* > DLAED2 merges the two sets of eigenvalues together into a single */
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/* > sorted set. Then it tries to deflate the size of the problem. */
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/* > There are two ways in which deflation can occur: when two or more */
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/* > eigenvalues are close together or if there is a tiny entry in the */
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/* > Z vector. For each such occurrence the order of the related secular */
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/* > equation problem is reduced by one. */
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/* > \endverbatim */
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/* Arguments: */
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/* ========== */
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/* > \param[out] K */
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/* > \verbatim */
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/* > K is INTEGER */
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/* > The number of non-deflated eigenvalues, and the order of the */
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/* > related secular equation. 0 <= K <=N. */
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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 dimension of the symmetric tridiagonal matrix. N >= 0. */
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/* > \endverbatim */
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/* > */
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/* > \param[in] N1 */
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/* > \verbatim */
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/* > N1 is INTEGER */
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/* > The location of the last eigenvalue in the leading sub-matrix. */
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/* > min(1,N) <= N1 <= N/2. */
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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, D contains the eigenvalues of the two submatrices to */
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/* > be combined. */
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/* > On exit, D contains the trailing (N-K) updated eigenvalues */
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/* > (those which were deflated) sorted into increasing order. */
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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 DOUBLE PRECISION array, dimension (LDQ, N) */
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/* > On entry, Q contains the eigenvectors of two submatrices in */
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/* > the two square blocks with corners at (1,1), (N1,N1) */
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/* > and (N1+1, N1+1), (N,N). */
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/* > On exit, Q contains the trailing (N-K) updated eigenvectors */
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/* > (those which were deflated) in its last N-K columns. */
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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,out] INDXQ */
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/* > \verbatim */
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/* > INDXQ is INTEGER array, dimension (N) */
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/* > The permutation which separately sorts the two sub-problems */
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/* > in D into ascending order. Note that elements in the second */
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/* > half of this permutation must first have N1 added to their */
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/* > values. Destroyed on exit. */
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/* > \endverbatim */
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/* > */
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/* > \param[in,out] RHO */
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/* > \verbatim */
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/* > RHO is DOUBLE PRECISION */
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/* > On entry, the off-diagonal element associated with the rank-1 */
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/* > cut which originally split the two submatrices which are now */
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/* > being recombined. */
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/* > On exit, RHO has been modified to the value required by */
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/* > DLAED3. */
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/* > \endverbatim */
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/* > */
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/* > \param[in] Z */
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/* > \verbatim */
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/* > Z is DOUBLE PRECISION array, dimension (N) */
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/* > On entry, Z contains the updating vector (the last */
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/* > row of the first sub-eigenvector matrix and the first row of */
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/* > the second sub-eigenvector matrix). */
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/* > On exit, the contents of Z have been destroyed by the updating */
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/* > process. */
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/* > \endverbatim */
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/* > */
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/* > \param[out] DLAMDA */
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/* > \verbatim */
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/* > DLAMDA is DOUBLE PRECISION array, dimension (N) */
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/* > A copy of the first K eigenvalues which will be used by */
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/* > DLAED3 to form the secular equation. */
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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 DOUBLE PRECISION array, dimension (N) */
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/* > The first k values of the final deflation-altered z-vector */
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/* > which will be passed to DLAED3. */
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/* > \endverbatim */
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/* > */
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/* > \param[out] Q2 */
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/* > \verbatim */
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/* > Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2) */
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/* > A copy of the first K eigenvectors which will be used by */
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/* > DLAED3 in a matrix multiply (DGEMM) to solve for the new */
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/* > eigenvectors. */
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/* > \endverbatim */
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/* > */
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/* > \param[out] INDX */
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/* > \verbatim */
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/* > INDX is INTEGER array, dimension (N) */
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/* > The permutation used to sort the contents of DLAMDA into */
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/* > ascending order. */
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/* > \endverbatim */
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/* > */
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/* > \param[out] INDXC */
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/* > \verbatim */
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/* > INDXC is INTEGER array, dimension (N) */
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/* > The permutation used to arrange the columns of the deflated */
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/* > Q matrix into three groups: the first group contains non-zero */
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/* > elements only at and above N1, the second contains */
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/* > non-zero elements only below N1, and the third is dense. */
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/* > \endverbatim */
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/* > */
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/* > \param[out] INDXP */
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/* > \verbatim */
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/* > INDXP is INTEGER array, dimension (N) */
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/* > The permutation used to place deflated values of D at the end */
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/* > of the array. INDXP(1:K) points to the nondeflated D-values */
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/* > and INDXP(K+1:N) points to the deflated eigenvalues. */
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/* > \endverbatim */
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/* > */
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/* > \param[out] COLTYP */
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/* > \verbatim */
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/* > COLTYP is INTEGER array, dimension (N) */
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/* > During execution, a label which will indicate which of the */
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/* > following types a column in the Q2 matrix is: */
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/* > 1 : non-zero in the upper half only; */
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/* > 2 : dense; */
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/* > 3 : non-zero in the lower half only; */
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/* > 4 : deflated. */
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/* > On exit, COLTYP(i) is the number of columns of type i, */
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/* > for i=1 to 4 only. */
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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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/* > \endverbatim */
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/* Authors: */
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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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/* > \ingroup auxOTHERcomputational */
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/* > \par Contributors: */
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/* ================== */
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/* > */
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/* > Jeff Rutter, Computer Science Division, University of California */
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/* > at Berkeley, USA \n */
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/* > Modified by Francoise Tisseur, University of Tennessee */
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/* > */
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/* ===================================================================== */
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/* Subroutine */ int dlaed2_(integer *k, integer *n, integer *n1, doublereal *
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d__, doublereal *q, integer *ldq, integer *indxq, doublereal *rho,
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doublereal *z__, doublereal *dlamda, doublereal *w, doublereal *q2,
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integer *indx, integer *indxc, integer *indxp, integer *coltyp,
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integer *info)
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{
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/* System generated locals */
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integer q_dim1, q_offset, i__1, i__2;
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doublereal d__1, d__2, d__3, d__4;
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/* Builtin functions */
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double sqrt(doublereal);
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/* Local variables */
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doublereal c__;
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integer i__, j;
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doublereal s, t;
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integer k2, n2, ct, nj, pj, js, iq1, iq2, n1p1;
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doublereal eps, tau, tol;
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integer psm[4], imax, jmax;
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extern /* Subroutine */ int drot_(integer *, doublereal *, integer *,
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doublereal *, integer *, doublereal *, doublereal *);
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integer ctot[4];
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extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
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integer *), dcopy_(integer *, doublereal *, integer *, doublereal
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*, integer *);
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extern doublereal dlapy2_(doublereal *, doublereal *), dlamch_(char *,
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ftnlen);
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extern integer idamax_(integer *, doublereal *, integer *);
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extern /* Subroutine */ int dlamrg_(integer *, integer *, doublereal *,
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integer *, integer *, integer *), dlacpy_(char *, integer *,
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integer *, doublereal *, integer *, doublereal *, integer *,
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ftnlen), xerbla_(char *, integer *, ftnlen);
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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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/* .. Scalar Arguments .. */
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/* .. */
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/* .. Array Arguments .. */
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/* .. */
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/* ===================================================================== */
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/* .. Parameters .. */
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/* .. */
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/* .. Local Arrays .. */
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/* .. */
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/* .. Local Scalars .. */
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/* .. */
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/* .. External Functions .. */
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/* .. */
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/* .. External Subroutines .. */
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/* .. */
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/* .. Intrinsic Functions .. */
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/* .. */
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/* .. Executable Statements .. */
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/* Test the input parameters. */
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/* Parameter adjustments */
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--d__;
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q_dim1 = *ldq;
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q_offset = 1 + q_dim1;
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q -= q_offset;
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--indxq;
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--z__;
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--dlamda;
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--w;
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--q2;
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--indx;
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--indxc;
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--indxp;
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--coltyp;
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/* Function Body */
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*info = 0;
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if (*n < 0) {
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*info = -2;
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} else if (*ldq < max(1,*n)) {
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*info = -6;
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} else /* if(complicated condition) */ {
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/* Computing MIN */
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i__1 = 1, i__2 = *n / 2;
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if (min(i__1,i__2) > *n1 || *n / 2 < *n1) {
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*info = -3;
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}
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}
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if (*info != 0) {
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i__1 = -(*info);
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xerbla_((char *)"DLAED2", &i__1, (ftnlen)6);
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return 0;
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}
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/* Quick return if possible */
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if (*n == 0) {
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return 0;
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}
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n2 = *n - *n1;
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n1p1 = *n1 + 1;
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if (*rho < 0.) {
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dscal_(&n2, &c_b3, &z__[n1p1], &c__1);
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}
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/* Normalize z so that norm(z) = 1. Since z is the concatenation of */
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/* two normalized vectors, norm2(z) = sqrt(2). */
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t = 1. / sqrt(2.);
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dscal_(n, &t, &z__[1], &c__1);
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/* RHO = ABS( norm(z)**2 * RHO ) */
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*rho = (d__1 = *rho * 2., abs(d__1));
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/* Sort the eigenvalues into increasing order */
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i__1 = *n;
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for (i__ = n1p1; i__ <= i__1; ++i__) {
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indxq[i__] += *n1;
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/* L10: */
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}
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/* re-integrate the deflated parts from the last pass */
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i__1 = *n;
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for (i__ = 1; i__ <= i__1; ++i__) {
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dlamda[i__] = d__[indxq[i__]];
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/* L20: */
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}
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dlamrg_(n1, &n2, &dlamda[1], &c__1, &c__1, &indxc[1]);
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i__1 = *n;
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for (i__ = 1; i__ <= i__1; ++i__) {
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indx[i__] = indxq[indxc[i__]];
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/* L30: */
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}
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/* Calculate the allowable deflation tolerance */
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imax = idamax_(n, &z__[1], &c__1);
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jmax = idamax_(n, &d__[1], &c__1);
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eps = dlamch_((char *)"Epsilon", (ftnlen)7);
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/* Computing MAX */
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d__3 = (d__1 = d__[jmax], abs(d__1)), d__4 = (d__2 = z__[imax], abs(d__2))
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;
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tol = eps * 8. * max(d__3,d__4);
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/* If the rank-1 modifier is small enough, no more needs to be done */
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/* except to reorganize Q so that its columns correspond with the */
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/* elements in D. */
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if (*rho * (d__1 = z__[imax], abs(d__1)) <= tol) {
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*k = 0;
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iq2 = 1;
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i__1 = *n;
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for (j = 1; j <= i__1; ++j) {
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i__ = indx[j];
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dcopy_(n, &q[i__ * q_dim1 + 1], &c__1, &q2[iq2], &c__1);
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dlamda[j] = d__[i__];
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iq2 += *n;
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/* L40: */
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}
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dlacpy_((char *)"A", n, n, &q2[1], n, &q[q_offset], ldq, (ftnlen)1);
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dcopy_(n, &dlamda[1], &c__1, &d__[1], &c__1);
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goto L190;
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}
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/* If there are multiple eigenvalues then the problem deflates. Here */
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/* the number of equal eigenvalues are found. As each equal */
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/* eigenvalue is found, an elementary reflector is computed to rotate */
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/* the corresponding eigensubspace so that the corresponding */
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/* components of Z are zero in this new basis. */
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i__1 = *n1;
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for (i__ = 1; i__ <= i__1; ++i__) {
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coltyp[i__] = 1;
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/* L50: */
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}
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i__1 = *n;
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for (i__ = n1p1; i__ <= i__1; ++i__) {
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coltyp[i__] = 3;
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/* L60: */
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}
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*k = 0;
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k2 = *n + 1;
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i__1 = *n;
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for (j = 1; j <= i__1; ++j) {
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nj = indx[j];
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if (*rho * (d__1 = z__[nj], abs(d__1)) <= tol) {
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/* Deflate due to small z component. */
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--k2;
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coltyp[nj] = 4;
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indxp[k2] = nj;
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if (j == *n) {
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goto L100;
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}
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} else {
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pj = nj;
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goto L80;
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}
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/* L70: */
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}
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L80:
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++j;
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nj = indx[j];
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if (j > *n) {
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goto L100;
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}
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if (*rho * (d__1 = z__[nj], abs(d__1)) <= tol) {
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/* Deflate due to small z component. */
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--k2;
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coltyp[nj] = 4;
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indxp[k2] = nj;
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} else {
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/* Check if eigenvalues are close enough to allow deflation. */
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s = z__[pj];
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c__ = z__[nj];
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/* Find sqrt(a**2+b**2) without overflow or */
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/* destructive underflow. */
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tau = dlapy2_(&c__, &s);
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t = d__[nj] - d__[pj];
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c__ /= tau;
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s = -s / tau;
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if ((d__1 = t * c__ * s, abs(d__1)) <= tol) {
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/* Deflation is possible. */
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z__[nj] = tau;
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z__[pj] = 0.;
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if (coltyp[nj] != coltyp[pj]) {
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coltyp[nj] = 2;
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|
}
|
|
coltyp[pj] = 4;
|
|
drot_(n, &q[pj * q_dim1 + 1], &c__1, &q[nj * q_dim1 + 1], &c__1, &
|
|
c__, &s);
|
|
/* Computing 2nd power */
|
|
d__1 = c__;
|
|
/* Computing 2nd power */
|
|
d__2 = s;
|
|
t = d__[pj] * (d__1 * d__1) + d__[nj] * (d__2 * d__2);
|
|
/* Computing 2nd power */
|
|
d__1 = s;
|
|
/* Computing 2nd power */
|
|
d__2 = c__;
|
|
d__[nj] = d__[pj] * (d__1 * d__1) + d__[nj] * (d__2 * d__2);
|
|
d__[pj] = t;
|
|
--k2;
|
|
i__ = 1;
|
|
L90:
|
|
if (k2 + i__ <= *n) {
|
|
if (d__[pj] < d__[indxp[k2 + i__]]) {
|
|
indxp[k2 + i__ - 1] = indxp[k2 + i__];
|
|
indxp[k2 + i__] = pj;
|
|
++i__;
|
|
goto L90;
|
|
} else {
|
|
indxp[k2 + i__ - 1] = pj;
|
|
}
|
|
} else {
|
|
indxp[k2 + i__ - 1] = pj;
|
|
}
|
|
pj = nj;
|
|
} else {
|
|
++(*k);
|
|
dlamda[*k] = d__[pj];
|
|
w[*k] = z__[pj];
|
|
indxp[*k] = pj;
|
|
pj = nj;
|
|
}
|
|
}
|
|
goto L80;
|
|
L100:
|
|
|
|
/* Record the last eigenvalue. */
|
|
|
|
++(*k);
|
|
dlamda[*k] = d__[pj];
|
|
w[*k] = z__[pj];
|
|
indxp[*k] = pj;
|
|
|
|
/* Count up the total number of the various types of columns, then */
|
|
/* form a permutation which positions the four column types into */
|
|
/* four uniform groups (although one or more of these groups may be */
|
|
/* empty). */
|
|
|
|
for (j = 1; j <= 4; ++j) {
|
|
ctot[j - 1] = 0;
|
|
/* L110: */
|
|
}
|
|
i__1 = *n;
|
|
for (j = 1; j <= i__1; ++j) {
|
|
ct = coltyp[j];
|
|
++ctot[ct - 1];
|
|
/* L120: */
|
|
}
|
|
|
|
/* PSM(*) = Position in SubMatrix (of types 1 through 4) */
|
|
|
|
psm[0] = 1;
|
|
psm[1] = ctot[0] + 1;
|
|
psm[2] = psm[1] + ctot[1];
|
|
psm[3] = psm[2] + ctot[2];
|
|
*k = *n - ctot[3];
|
|
|
|
/* Fill out the INDXC array so that the permutation which it induces */
|
|
/* will place all type-1 columns first, all type-2 columns next, */
|
|
/* then all type-3's, and finally all type-4's. */
|
|
|
|
i__1 = *n;
|
|
for (j = 1; j <= i__1; ++j) {
|
|
js = indxp[j];
|
|
ct = coltyp[js];
|
|
indx[psm[ct - 1]] = js;
|
|
indxc[psm[ct - 1]] = j;
|
|
++psm[ct - 1];
|
|
/* L130: */
|
|
}
|
|
|
|
/* Sort the eigenvalues and corresponding eigenvectors into DLAMDA */
|
|
/* and Q2 respectively. The eigenvalues/vectors which were not */
|
|
/* deflated go into the first K slots of DLAMDA and Q2 respectively, */
|
|
/* while those which were deflated go into the last N - K slots. */
|
|
|
|
i__ = 1;
|
|
iq1 = 1;
|
|
iq2 = (ctot[0] + ctot[1]) * *n1 + 1;
|
|
i__1 = ctot[0];
|
|
for (j = 1; j <= i__1; ++j) {
|
|
js = indx[i__];
|
|
dcopy_(n1, &q[js * q_dim1 + 1], &c__1, &q2[iq1], &c__1);
|
|
z__[i__] = d__[js];
|
|
++i__;
|
|
iq1 += *n1;
|
|
/* L140: */
|
|
}
|
|
|
|
i__1 = ctot[1];
|
|
for (j = 1; j <= i__1; ++j) {
|
|
js = indx[i__];
|
|
dcopy_(n1, &q[js * q_dim1 + 1], &c__1, &q2[iq1], &c__1);
|
|
dcopy_(&n2, &q[*n1 + 1 + js * q_dim1], &c__1, &q2[iq2], &c__1);
|
|
z__[i__] = d__[js];
|
|
++i__;
|
|
iq1 += *n1;
|
|
iq2 += n2;
|
|
/* L150: */
|
|
}
|
|
|
|
i__1 = ctot[2];
|
|
for (j = 1; j <= i__1; ++j) {
|
|
js = indx[i__];
|
|
dcopy_(&n2, &q[*n1 + 1 + js * q_dim1], &c__1, &q2[iq2], &c__1);
|
|
z__[i__] = d__[js];
|
|
++i__;
|
|
iq2 += n2;
|
|
/* L160: */
|
|
}
|
|
|
|
iq1 = iq2;
|
|
i__1 = ctot[3];
|
|
for (j = 1; j <= i__1; ++j) {
|
|
js = indx[i__];
|
|
dcopy_(n, &q[js * q_dim1 + 1], &c__1, &q2[iq2], &c__1);
|
|
iq2 += *n;
|
|
z__[i__] = d__[js];
|
|
++i__;
|
|
/* L170: */
|
|
}
|
|
|
|
/* The deflated eigenvalues and their corresponding vectors go back */
|
|
/* into the last N - K slots of D and Q respectively. */
|
|
|
|
if (*k < *n) {
|
|
dlacpy_((char *)"A", n, &ctot[3], &q2[iq1], n, &q[(*k + 1) * q_dim1 + 1], ldq,
|
|
(ftnlen)1);
|
|
i__1 = *n - *k;
|
|
dcopy_(&i__1, &z__[*k + 1], &c__1, &d__[*k + 1], &c__1);
|
|
}
|
|
|
|
/* Copy CTOT into COLTYP for referencing in DLAED3. */
|
|
|
|
for (j = 1; j <= 4; ++j) {
|
|
coltyp[j] = ctot[j - 1];
|
|
/* L180: */
|
|
}
|
|
|
|
L190:
|
|
return 0;
|
|
|
|
/* End of DLAED2 */
|
|
|
|
} /* dlaed2_ */
|
|
|
|
#ifdef __cplusplus
|
|
}
|
|
#endif
|