570 lines
18 KiB
C++
570 lines
18 KiB
C++
/* fortran/zlaed8.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 ZLAED8 used by ZSTEDC. Merges eigenvalues and deflates secular equation. Used when the original
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matrix is dense. */
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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 ZLAED8 + dependencies */
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/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaed8.
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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/zlaed8.
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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/zlaed8.
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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 ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA, */
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/* Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR, */
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/* GIVCOL, GIVNUM, INFO ) */
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/* .. Scalar Arguments .. */
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/* INTEGER CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N, QSIZ */
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/* DOUBLE PRECISION RHO */
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/* .. */
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/* .. Array Arguments .. */
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/* INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), */
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/* $ INDXQ( * ), PERM( * ) */
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/* DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ), */
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/* $ Z( * ) */
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/* COMPLEX*16 Q( LDQ, * ), Q2( LDQ2, * ) */
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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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/* > ZLAED8 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 element 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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/* > Contains the number of non-deflated eigenvalues. */
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/* > This is the order of the related secular equation. */
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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] 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 dense or band matrix to tridiagonal form. */
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/* > QSIZ >= N if ICOMPQ = 1. */
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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, Q contains the eigenvectors of the partially solved */
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/* > system which has been previously updated in matrix */
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/* > multiplies with other partially solved eigensystems. */
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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] 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. On exit, D contains the trailing (N-K) updated */
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/* > eigenvalues (those which were deflated) sorted into increasing */
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/* > order. */
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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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/* > Contains 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. RHO is modified during the computation to */
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/* > the value required by DLAED3. */
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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] Z */
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/* > \verbatim */
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/* > Z is DOUBLE PRECISION array, dimension (N) */
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/* > On input this vector 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). The contents of Z are */
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/* > destroyed during the updating 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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/* > Contains a copy of the first K eigenvalues which will be used */
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/* > by DLAED3 to form the secular equation. */
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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 COMPLEX*16 array, dimension (LDQ2,N) */
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/* > If ICOMPQ = 0, Q2 is not referenced. Otherwise, */
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/* > Contains a copy of the first K eigenvectors which will be used */
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/* > by DLAED7 in a matrix multiply (DGEMM) to update the new */
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/* > eigenvectors. */
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/* > \endverbatim */
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/* > */
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/* > \param[in] LDQ2 */
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/* > \verbatim */
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/* > LDQ2 is INTEGER */
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/* > The leading dimension of the array Q2. LDQ2 >= max( 1, N ). */
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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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/* > This will hold the first k values of the final */
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/* > deflation-altered z-vector and will be passed to DLAED3. */
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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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/* > This will contain the permutation used to place deflated */
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/* > values of D at the end of the array. On output INDXP(1:K) */
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/* > points to the nondeflated D-values and INDXP(K+1:N) */
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/* > points to the deflated eigenvalues. */
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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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/* > This will contain the permutation used to sort the contents of */
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/* > D into ascending order. */
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/* > \endverbatim */
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/* > */
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/* > \param[in] 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 separately sorts the two */
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/* > sub-problems in D into ascending order. Note that elements in */
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/* > the second half of this permutation must first have CUTPNT */
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/* > added to their values in order to be accurate. */
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/* > \endverbatim */
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/* > */
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/* > \param[out] PERM */
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/* > \verbatim */
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/* > PERM is INTEGER array, dimension (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[out] GIVPTR */
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/* > \verbatim */
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/* > GIVPTR is INTEGER */
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/* > Contains the number of Givens rotations which took place in */
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/* > this subproblem. */
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/* > \endverbatim */
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/* > */
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/* > \param[out] GIVCOL */
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/* > \verbatim */
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/* > GIVCOL is INTEGER array, dimension (2, 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[out] GIVNUM */
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/* > \verbatim */
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/* > GIVNUM is DOUBLE PRECISION array, dimension (2, 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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/* > \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 complex16OTHERcomputational */
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/* ===================================================================== */
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/* Subroutine */ int zlaed8_(integer *k, integer *n, integer *qsiz,
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doublecomplex *q, integer *ldq, doublereal *d__, doublereal *rho,
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integer *cutpnt, doublereal *z__, doublereal *dlamda, doublecomplex *
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q2, integer *ldq2, doublereal *w, integer *indxp, integer *indx,
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integer *indxq, integer *perm, integer *givptr, integer *givcol,
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doublereal *givnum, integer *info)
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{
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/* System generated locals */
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integer q_dim1, q_offset, q2_dim1, q2_offset, i__1;
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doublereal d__1;
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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, n1, n2, jp, n1p1;
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doublereal eps, tau, tol;
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integer jlam, imax, jmax;
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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 *), zdrot_(integer *, doublecomplex *, integer *,
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doublecomplex *, integer *, doublereal *, doublereal *), zcopy_(
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integer *, doublecomplex *, integer *, doublecomplex *, integer *)
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;
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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 *), xerbla_(char *, integer *,
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ftnlen), zlacpy_(char *, integer *, integer *, doublecomplex *,
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integer *, doublecomplex *, 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 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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q_dim1 = *ldq;
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q_offset = 1 + q_dim1;
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q -= q_offset;
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--d__;
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--z__;
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--dlamda;
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q2_dim1 = *ldq2;
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q2_offset = 1 + q2_dim1;
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q2 -= q2_offset;
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--w;
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--indxp;
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--indx;
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--indxq;
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--perm;
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givcol -= 3;
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givnum -= 3;
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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 (*qsiz < *n) {
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*info = -3;
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} else if (*ldq < max(1,*n)) {
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*info = -5;
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} else if (*cutpnt < min(1,*n) || *cutpnt > *n) {
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*info = -8;
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} else if (*ldq2 < max(1,*n)) {
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*info = -12;
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}
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if (*info != 0) {
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i__1 = -(*info);
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xerbla_((char *)"ZLAED8", &i__1, (ftnlen)6);
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return 0;
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}
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/* Need to initialize GIVPTR to O here in case of quick exit */
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/* to prevent an unspecified code behavior (usually sigfault) */
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/* when IWORK array on entry to *stedc is not zeroed */
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/* (or at least some IWORK entries which used in *laed7 for GIVPTR). */
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*givptr = 0;
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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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n1 = *cutpnt;
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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 */
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t = 1. / sqrt(2.);
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i__1 = *n;
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for (j = 1; j <= i__1; ++j) {
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indx[j] = j;
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/* L10: */
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}
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dscal_(n, &t, &z__[1], &c__1);
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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__ = *cutpnt + 1; i__ <= i__1; ++i__) {
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indxq[i__] += *cutpnt;
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/* L20: */
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}
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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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w[i__] = z__[indxq[i__]];
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/* L30: */
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}
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i__ = 1;
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j = *cutpnt + 1;
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dlamrg_(&n1, &n2, &dlamda[1], &c__1, &c__1, &indx[1]);
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i__1 = *n;
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for (i__ = 1; i__ <= i__1; ++i__) {
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d__[i__] = dlamda[indx[i__]];
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z__[i__] = w[indx[i__]];
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/* L40: */
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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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tol = eps * 8. * (d__1 = d__[jmax], abs(d__1));
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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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i__1 = *n;
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for (j = 1; j <= i__1; ++j) {
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perm[j] = indxq[indx[j]];
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zcopy_(qsiz, &q[perm[j] * q_dim1 + 1], &c__1, &q2[j * q2_dim1 + 1]
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, &c__1);
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/* L50: */
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}
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zlacpy_((char *)"A", qsiz, n, &q2[q2_dim1 + 1], ldq2, &q[q_dim1 + 1], ldq, (
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ftnlen)1);
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return 0;
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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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*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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if (*rho * (d__1 = z__[j], abs(d__1)) <= tol) {
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/* Deflate due to small z component. */
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--k2;
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indxp[k2] = j;
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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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jlam = j;
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goto L70;
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}
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/* L60: */
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}
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L70:
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++j;
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if (j > *n) {
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goto L90;
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}
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if (*rho * (d__1 = z__[j], abs(d__1)) <= tol) {
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/* Deflate due to small z component. */
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--k2;
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indxp[k2] = j;
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} else {
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/* Check if eigenvalues are close enough to allow deflation. */
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s = z__[jlam];
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c__ = z__[j];
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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__[j] - d__[jlam];
|
|
c__ /= tau;
|
|
s = -s / tau;
|
|
if ((d__1 = t * c__ * s, abs(d__1)) <= tol) {
|
|
|
|
/* Deflation is possible. */
|
|
|
|
z__[j] = tau;
|
|
z__[jlam] = 0.;
|
|
|
|
/* Record the appropriate Givens rotation */
|
|
|
|
++(*givptr);
|
|
givcol[(*givptr << 1) + 1] = indxq[indx[jlam]];
|
|
givcol[(*givptr << 1) + 2] = indxq[indx[j]];
|
|
givnum[(*givptr << 1) + 1] = c__;
|
|
givnum[(*givptr << 1) + 2] = s;
|
|
zdrot_(qsiz, &q[indxq[indx[jlam]] * q_dim1 + 1], &c__1, &q[indxq[
|
|
indx[j]] * q_dim1 + 1], &c__1, &c__, &s);
|
|
t = d__[jlam] * c__ * c__ + d__[j] * s * s;
|
|
d__[j] = d__[jlam] * s * s + d__[j] * c__ * c__;
|
|
d__[jlam] = t;
|
|
--k2;
|
|
i__ = 1;
|
|
L80:
|
|
if (k2 + i__ <= *n) {
|
|
if (d__[jlam] < d__[indxp[k2 + i__]]) {
|
|
indxp[k2 + i__ - 1] = indxp[k2 + i__];
|
|
indxp[k2 + i__] = jlam;
|
|
++i__;
|
|
goto L80;
|
|
} else {
|
|
indxp[k2 + i__ - 1] = jlam;
|
|
}
|
|
} else {
|
|
indxp[k2 + i__ - 1] = jlam;
|
|
}
|
|
jlam = j;
|
|
} else {
|
|
++(*k);
|
|
w[*k] = z__[jlam];
|
|
dlamda[*k] = d__[jlam];
|
|
indxp[*k] = jlam;
|
|
jlam = j;
|
|
}
|
|
}
|
|
goto L70;
|
|
L90:
|
|
|
|
/* Record the last eigenvalue. */
|
|
|
|
++(*k);
|
|
w[*k] = z__[jlam];
|
|
dlamda[*k] = d__[jlam];
|
|
indxp[*k] = jlam;
|
|
|
|
L100:
|
|
|
|
/* 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 = *n;
|
|
for (j = 1; j <= i__1; ++j) {
|
|
jp = indxp[j];
|
|
dlamda[j] = d__[jp];
|
|
perm[j] = indxq[indx[jp]];
|
|
zcopy_(qsiz, &q[perm[j] * q_dim1 + 1], &c__1, &q2[j * q2_dim1 + 1], &
|
|
c__1);
|
|
/* L110: */
|
|
}
|
|
|
|
/* The deflated eigenvalues and their corresponding vectors go back */
|
|
/* into the last N - K slots of D and Q respectively. */
|
|
|
|
if (*k < *n) {
|
|
i__1 = *n - *k;
|
|
dcopy_(&i__1, &dlamda[*k + 1], &c__1, &d__[*k + 1], &c__1);
|
|
i__1 = *n - *k;
|
|
zlacpy_((char *)"A", qsiz, &i__1, &q2[(*k + 1) * q2_dim1 + 1], ldq2, &q[(*k +
|
|
1) * q_dim1 + 1], ldq, (ftnlen)1);
|
|
}
|
|
|
|
return 0;
|
|
|
|
/* End of ZLAED8 */
|
|
|
|
} /* zlaed8_ */
|
|
|
|
#ifdef __cplusplus
|
|
}
|
|
#endif
|