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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b ZHEGST 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download ZHEGVX + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhegvx.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhegvx.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhegvx.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE ZHEGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, 00022 * VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, 00023 * LWORK, RWORK, IWORK, IFAIL, INFO ) 00024 * 00025 * .. Scalar Arguments .. 00026 * CHARACTER JOBZ, RANGE, UPLO 00027 * INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N 00028 * DOUBLE PRECISION ABSTOL, VL, VU 00029 * .. 00030 * .. Array Arguments .. 00031 * INTEGER IFAIL( * ), IWORK( * ) 00032 * DOUBLE PRECISION RWORK( * ), W( * ) 00033 * COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ), 00034 * $ Z( LDZ, * ) 00035 * .. 00036 * 00037 * 00038 *> \par Purpose: 00039 * ============= 00040 *> 00041 *> \verbatim 00042 *> 00043 *> ZHEGVX computes selected eigenvalues, and optionally, eigenvectors 00044 *> of a complex generalized Hermitian-definite eigenproblem, of the form 00045 *> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and 00046 *> B are assumed to be Hermitian and B is also positive definite. 00047 *> Eigenvalues and eigenvectors can be selected by specifying either a 00048 *> range of values or a range of indices for the desired eigenvalues. 00049 *> \endverbatim 00050 * 00051 * Arguments: 00052 * ========== 00053 * 00054 *> \param[in] ITYPE 00055 *> \verbatim 00056 *> ITYPE is INTEGER 00057 *> Specifies the problem type to be solved: 00058 *> = 1: A*x = (lambda)*B*x 00059 *> = 2: A*B*x = (lambda)*x 00060 *> = 3: B*A*x = (lambda)*x 00061 *> \endverbatim 00062 *> 00063 *> \param[in] JOBZ 00064 *> \verbatim 00065 *> JOBZ is CHARACTER*1 00066 *> = 'N': Compute eigenvalues only; 00067 *> = 'V': Compute eigenvalues and eigenvectors. 00068 *> \endverbatim 00069 *> 00070 *> \param[in] RANGE 00071 *> \verbatim 00072 *> RANGE is CHARACTER*1 00073 *> = 'A': all eigenvalues will be found. 00074 *> = 'V': all eigenvalues in the half-open interval (VL,VU] 00075 *> will be found. 00076 *> = 'I': the IL-th through IU-th eigenvalues will be found. 00077 *> \endverbatim 00078 *> 00079 *> \param[in] UPLO 00080 *> \verbatim 00081 *> UPLO is CHARACTER*1 00082 *> = 'U': Upper triangles of A and B are stored; 00083 *> = 'L': Lower triangles of A and B are stored. 00084 *> \endverbatim 00085 *> 00086 *> \param[in] N 00087 *> \verbatim 00088 *> N is INTEGER 00089 *> The order of the matrices A and B. N >= 0. 00090 *> \endverbatim 00091 *> 00092 *> \param[in,out] A 00093 *> \verbatim 00094 *> A is COMPLEX*16 array, dimension (LDA, N) 00095 *> On entry, the Hermitian matrix A. If UPLO = 'U', the 00096 *> leading N-by-N upper triangular part of A contains the 00097 *> upper triangular part of the matrix A. If UPLO = 'L', 00098 *> the leading N-by-N lower triangular part of A contains 00099 *> the lower triangular part of the matrix A. 00100 *> 00101 *> On exit, the lower triangle (if UPLO='L') or the upper 00102 *> triangle (if UPLO='U') of A, including the diagonal, is 00103 *> destroyed. 00104 *> \endverbatim 00105 *> 00106 *> \param[in] LDA 00107 *> \verbatim 00108 *> LDA is INTEGER 00109 *> The leading dimension of the array A. LDA >= max(1,N). 00110 *> \endverbatim 00111 *> 00112 *> \param[in,out] B 00113 *> \verbatim 00114 *> B is COMPLEX*16 array, dimension (LDB, N) 00115 *> On entry, the Hermitian matrix B. If UPLO = 'U', the 00116 *> leading N-by-N upper triangular part of B contains the 00117 *> upper triangular part of the matrix B. If UPLO = 'L', 00118 *> the leading N-by-N lower triangular part of B contains 00119 *> the lower triangular part of the matrix B. 00120 *> 00121 *> On exit, if INFO <= N, the part of B containing the matrix is 00122 *> overwritten by the triangular factor U or L from the Cholesky 00123 *> factorization B = U**H*U or B = L*L**H. 00124 *> \endverbatim 00125 *> 00126 *> \param[in] LDB 00127 *> \verbatim 00128 *> LDB is INTEGER 00129 *> The leading dimension of the array B. LDB >= max(1,N). 00130 *> \endverbatim 00131 *> 00132 *> \param[in] VL 00133 *> \verbatim 00134 *> VL is DOUBLE PRECISION 00135 *> \endverbatim 00136 *> 00137 *> \param[in] VU 00138 *> \verbatim 00139 *> VU is DOUBLE PRECISION 00140 *> 00141 *> If RANGE='V', the lower and upper bounds of the interval to 00142 *> be searched for eigenvalues. VL < VU. 00143 *> Not referenced if RANGE = 'A' or 'I'. 00144 *> \endverbatim 00145 *> 00146 *> \param[in] IL 00147 *> \verbatim 00148 *> IL is INTEGER 00149 *> \endverbatim 00150 *> 00151 *> \param[in] IU 00152 *> \verbatim 00153 *> IU is INTEGER 00154 *> 00155 *> If RANGE='I', the indices (in ascending order) of the 00156 *> smallest and largest eigenvalues to be returned. 00157 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 00158 *> Not referenced if RANGE = 'A' or 'V'. 00159 *> \endverbatim 00160 *> 00161 *> \param[in] ABSTOL 00162 *> \verbatim 00163 *> ABSTOL is DOUBLE PRECISION 00164 *> The absolute error tolerance for the eigenvalues. 00165 *> An approximate eigenvalue is accepted as converged 00166 *> when it is determined to lie in an interval [a,b] 00167 *> of width less than or equal to 00168 *> 00169 *> ABSTOL + EPS * max( |a|,|b| ) , 00170 *> 00171 *> where EPS is the machine precision. If ABSTOL is less than 00172 *> or equal to zero, then EPS*|T| will be used in its place, 00173 *> where |T| is the 1-norm of the tridiagonal matrix obtained 00174 *> by reducing C to tridiagonal form, where C is the symmetric 00175 *> matrix of the standard symmetric problem to which the 00176 *> generalized problem is transformed. 00177 *> 00178 *> Eigenvalues will be computed most accurately when ABSTOL is 00179 *> set to twice the underflow threshold 2*DLAMCH('S'), not zero. 00180 *> If this routine returns with INFO>0, indicating that some 00181 *> eigenvectors did not converge, try setting ABSTOL to 00182 *> 2*DLAMCH('S'). 00183 *> \endverbatim 00184 *> 00185 *> \param[out] M 00186 *> \verbatim 00187 *> M is INTEGER 00188 *> The total number of eigenvalues found. 0 <= M <= N. 00189 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 00190 *> \endverbatim 00191 *> 00192 *> \param[out] W 00193 *> \verbatim 00194 *> W is DOUBLE PRECISION array, dimension (N) 00195 *> The first M elements contain the selected 00196 *> eigenvalues in ascending order. 00197 *> \endverbatim 00198 *> 00199 *> \param[out] Z 00200 *> \verbatim 00201 *> Z is COMPLEX*16 array, dimension (LDZ, max(1,M)) 00202 *> If JOBZ = 'N', then Z is not referenced. 00203 *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z 00204 *> contain the orthonormal eigenvectors of the matrix A 00205 *> corresponding to the selected eigenvalues, with the i-th 00206 *> column of Z holding the eigenvector associated with W(i). 00207 *> The eigenvectors are normalized as follows: 00208 *> if ITYPE = 1 or 2, Z**T*B*Z = I; 00209 *> if ITYPE = 3, Z**T*inv(B)*Z = I. 00210 *> 00211 *> If an eigenvector fails to converge, then that column of Z 00212 *> contains the latest approximation to the eigenvector, and the 00213 *> index of the eigenvector is returned in IFAIL. 00214 *> Note: the user must ensure that at least max(1,M) columns are 00215 *> supplied in the array Z; if RANGE = 'V', the exact value of M 00216 *> is not known in advance and an upper bound must be used. 00217 *> \endverbatim 00218 *> 00219 *> \param[in] LDZ 00220 *> \verbatim 00221 *> LDZ is INTEGER 00222 *> The leading dimension of the array Z. LDZ >= 1, and if 00223 *> JOBZ = 'V', LDZ >= max(1,N). 00224 *> \endverbatim 00225 *> 00226 *> \param[out] WORK 00227 *> \verbatim 00228 *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) 00229 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00230 *> \endverbatim 00231 *> 00232 *> \param[in] LWORK 00233 *> \verbatim 00234 *> LWORK is INTEGER 00235 *> The length of the array WORK. LWORK >= max(1,2*N). 00236 *> For optimal efficiency, LWORK >= (NB+1)*N, 00237 *> where NB is the blocksize for ZHETRD returned by ILAENV. 00238 *> 00239 *> If LWORK = -1, then a workspace query is assumed; the routine 00240 *> only calculates the optimal size of the WORK array, returns 00241 *> this value as the first entry of the WORK array, and no error 00242 *> message related to LWORK is issued by XERBLA. 00243 *> \endverbatim 00244 *> 00245 *> \param[out] RWORK 00246 *> \verbatim 00247 *> RWORK is DOUBLE PRECISION array, dimension (7*N) 00248 *> \endverbatim 00249 *> 00250 *> \param[out] IWORK 00251 *> \verbatim 00252 *> IWORK is INTEGER array, dimension (5*N) 00253 *> \endverbatim 00254 *> 00255 *> \param[out] IFAIL 00256 *> \verbatim 00257 *> IFAIL is INTEGER array, dimension (N) 00258 *> If JOBZ = 'V', then if INFO = 0, the first M elements of 00259 *> IFAIL are zero. If INFO > 0, then IFAIL contains the 00260 *> indices of the eigenvectors that failed to converge. 00261 *> If JOBZ = 'N', then IFAIL is not referenced. 00262 *> \endverbatim 00263 *> 00264 *> \param[out] INFO 00265 *> \verbatim 00266 *> INFO is INTEGER 00267 *> = 0: successful exit 00268 *> < 0: if INFO = -i, the i-th argument had an illegal value 00269 *> > 0: ZPOTRF or ZHEEVX returned an error code: 00270 *> <= N: if INFO = i, ZHEEVX failed to converge; 00271 *> i eigenvectors failed to converge. Their indices 00272 *> are stored in array IFAIL. 00273 *> > N: if INFO = N + i, for 1 <= i <= N, then the leading 00274 *> minor of order i of B is not positive definite. 00275 *> The factorization of B could not be completed and 00276 *> no eigenvalues or eigenvectors were computed. 00277 *> \endverbatim 00278 * 00279 * Authors: 00280 * ======== 00281 * 00282 *> \author Univ. of Tennessee 00283 *> \author Univ. of California Berkeley 00284 *> \author Univ. of Colorado Denver 00285 *> \author NAG Ltd. 00286 * 00287 *> \date November 2011 00288 * 00289 *> \ingroup complex16HEeigen 00290 * 00291 *> \par Contributors: 00292 * ================== 00293 *> 00294 *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA 00295 * 00296 * ===================================================================== 00297 SUBROUTINE ZHEGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, 00298 $ VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, 00299 $ LWORK, RWORK, IWORK, IFAIL, INFO ) 00300 * 00301 * -- LAPACK driver routine (version 3.4.0) -- 00302 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00303 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00304 * November 2011 00305 * 00306 * .. Scalar Arguments .. 00307 CHARACTER JOBZ, RANGE, UPLO 00308 INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N 00309 DOUBLE PRECISION ABSTOL, VL, VU 00310 * .. 00311 * .. Array Arguments .. 00312 INTEGER IFAIL( * ), IWORK( * ) 00313 DOUBLE PRECISION RWORK( * ), W( * ) 00314 COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ), 00315 $ Z( LDZ, * ) 00316 * .. 00317 * 00318 * ===================================================================== 00319 * 00320 * .. Parameters .. 00321 COMPLEX*16 CONE 00322 PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) ) 00323 * .. 00324 * .. Local Scalars .. 00325 LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ 00326 CHARACTER TRANS 00327 INTEGER LWKOPT, NB 00328 * .. 00329 * .. External Functions .. 00330 LOGICAL LSAME 00331 INTEGER ILAENV 00332 EXTERNAL LSAME, ILAENV 00333 * .. 00334 * .. External Subroutines .. 00335 EXTERNAL XERBLA, ZHEEVX, ZHEGST, ZPOTRF, ZTRMM, ZTRSM 00336 * .. 00337 * .. Intrinsic Functions .. 00338 INTRINSIC MAX, MIN 00339 * .. 00340 * .. Executable Statements .. 00341 * 00342 * Test the input parameters. 00343 * 00344 WANTZ = LSAME( JOBZ, 'V' ) 00345 UPPER = LSAME( UPLO, 'U' ) 00346 ALLEIG = LSAME( RANGE, 'A' ) 00347 VALEIG = LSAME( RANGE, 'V' ) 00348 INDEIG = LSAME( RANGE, 'I' ) 00349 LQUERY = ( LWORK.EQ.-1 ) 00350 * 00351 INFO = 0 00352 IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN 00353 INFO = -1 00354 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 00355 INFO = -2 00356 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 00357 INFO = -3 00358 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN 00359 INFO = -4 00360 ELSE IF( N.LT.0 ) THEN 00361 INFO = -5 00362 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00363 INFO = -7 00364 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00365 INFO = -9 00366 ELSE 00367 IF( VALEIG ) THEN 00368 IF( N.GT.0 .AND. VU.LE.VL ) 00369 $ INFO = -11 00370 ELSE IF( INDEIG ) THEN 00371 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 00372 INFO = -12 00373 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 00374 INFO = -13 00375 END IF 00376 END IF 00377 END IF 00378 IF (INFO.EQ.0) THEN 00379 IF (LDZ.LT.1 .OR. (WANTZ .AND. LDZ.LT.N)) THEN 00380 INFO = -18 00381 END IF 00382 END IF 00383 * 00384 IF( INFO.EQ.0 ) THEN 00385 NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 ) 00386 LWKOPT = MAX( 1, ( NB + 1 )*N ) 00387 WORK( 1 ) = LWKOPT 00388 * 00389 IF( LWORK.LT.MAX( 1, 2*N ) .AND. .NOT.LQUERY ) THEN 00390 INFO = -20 00391 END IF 00392 END IF 00393 * 00394 IF( INFO.NE.0 ) THEN 00395 CALL XERBLA( 'ZHEGVX', -INFO ) 00396 RETURN 00397 ELSE IF( LQUERY ) THEN 00398 RETURN 00399 END IF 00400 * 00401 * Quick return if possible 00402 * 00403 M = 0 00404 IF( N.EQ.0 ) THEN 00405 RETURN 00406 END IF 00407 * 00408 * Form a Cholesky factorization of B. 00409 * 00410 CALL ZPOTRF( UPLO, N, B, LDB, INFO ) 00411 IF( INFO.NE.0 ) THEN 00412 INFO = N + INFO 00413 RETURN 00414 END IF 00415 * 00416 * Transform problem to standard eigenvalue problem and solve. 00417 * 00418 CALL ZHEGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO ) 00419 CALL ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, 00420 $ M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL, 00421 $ INFO ) 00422 * 00423 IF( WANTZ ) THEN 00424 * 00425 * Backtransform eigenvectors to the original problem. 00426 * 00427 IF( INFO.GT.0 ) 00428 $ M = INFO - 1 00429 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN 00430 * 00431 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x; 00432 * backtransform eigenvectors: x = inv(L)**H *y or inv(U)*y 00433 * 00434 IF( UPPER ) THEN 00435 TRANS = 'N' 00436 ELSE 00437 TRANS = 'C' 00438 END IF 00439 * 00440 CALL ZTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, M, CONE, B, 00441 $ LDB, Z, LDZ ) 00442 * 00443 ELSE IF( ITYPE.EQ.3 ) THEN 00444 * 00445 * For B*A*x=(lambda)*x; 00446 * backtransform eigenvectors: x = L*y or U**H *y 00447 * 00448 IF( UPPER ) THEN 00449 TRANS = 'C' 00450 ELSE 00451 TRANS = 'N' 00452 END IF 00453 * 00454 CALL ZTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, M, CONE, B, 00455 $ LDB, Z, LDZ ) 00456 END IF 00457 END IF 00458 * 00459 * Set WORK(1) to optimal complex workspace size. 00460 * 00461 WORK( 1 ) = LWKOPT 00462 * 00463 RETURN 00464 * 00465 * End of ZHEGVX 00466 * 00467 END