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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DSPGST 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download DSPGVX + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspgvx.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspgvx.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspgvx.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, 00022 * IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, 00023 * IFAIL, INFO ) 00024 * 00025 * .. Scalar Arguments .. 00026 * CHARACTER JOBZ, RANGE, UPLO 00027 * INTEGER IL, INFO, ITYPE, IU, LDZ, M, N 00028 * DOUBLE PRECISION ABSTOL, VL, VU 00029 * .. 00030 * .. Array Arguments .. 00031 * INTEGER IFAIL( * ), IWORK( * ) 00032 * DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ), 00033 * $ Z( LDZ, * ) 00034 * .. 00035 * 00036 * 00037 *> \par Purpose: 00038 * ============= 00039 *> 00040 *> \verbatim 00041 *> 00042 *> DSPGVX computes selected eigenvalues, and optionally, eigenvectors 00043 *> of a real generalized symmetric-definite eigenproblem, of the form 00044 *> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A 00045 *> and B are assumed to be symmetric, stored in packed storage, and B 00046 *> is also positive definite. Eigenvalues and eigenvectors can be 00047 *> selected by specifying either a range of values or a range of indices 00048 *> 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 triangle of A and B are stored; 00083 *> = 'L': Lower triangle of A and B are stored. 00084 *> \endverbatim 00085 *> 00086 *> \param[in] N 00087 *> \verbatim 00088 *> N is INTEGER 00089 *> The order of the matrix pencil (A,B). N >= 0. 00090 *> \endverbatim 00091 *> 00092 *> \param[in,out] AP 00093 *> \verbatim 00094 *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) 00095 *> On entry, the upper or lower triangle of the symmetric matrix 00096 *> A, packed columnwise in a linear array. The j-th column of A 00097 *> is stored in the array AP as follows: 00098 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00099 *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. 00100 *> 00101 *> On exit, the contents of AP are destroyed. 00102 *> \endverbatim 00103 *> 00104 *> \param[in,out] BP 00105 *> \verbatim 00106 *> BP is DOUBLE PRECISION array, dimension (N*(N+1)/2) 00107 *> On entry, the upper or lower triangle of the symmetric matrix 00108 *> B, packed columnwise in a linear array. The j-th column of B 00109 *> is stored in the array BP as follows: 00110 *> if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; 00111 *> if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. 00112 *> 00113 *> On exit, the triangular factor U or L from the Cholesky 00114 *> factorization B = U**T*U or B = L*L**T, in the same storage 00115 *> format as B. 00116 *> \endverbatim 00117 *> 00118 *> \param[in] VL 00119 *> \verbatim 00120 *> VL is DOUBLE PRECISION 00121 *> \endverbatim 00122 *> 00123 *> \param[in] VU 00124 *> \verbatim 00125 *> VU is DOUBLE PRECISION 00126 *> 00127 *> If RANGE='V', the lower and upper bounds of the interval to 00128 *> be searched for eigenvalues. VL < VU. 00129 *> Not referenced if RANGE = 'A' or 'I'. 00130 *> \endverbatim 00131 *> 00132 *> \param[in] IL 00133 *> \verbatim 00134 *> IL is INTEGER 00135 *> \endverbatim 00136 *> 00137 *> \param[in] IU 00138 *> \verbatim 00139 *> IU is INTEGER 00140 *> 00141 *> If RANGE='I', the indices (in ascending order) of the 00142 *> smallest and largest eigenvalues to be returned. 00143 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 00144 *> Not referenced if RANGE = 'A' or 'V'. 00145 *> \endverbatim 00146 *> 00147 *> \param[in] ABSTOL 00148 *> \verbatim 00149 *> ABSTOL is DOUBLE PRECISION 00150 *> The absolute error tolerance for the eigenvalues. 00151 *> An approximate eigenvalue is accepted as converged 00152 *> when it is determined to lie in an interval [a,b] 00153 *> of width less than or equal to 00154 *> 00155 *> ABSTOL + EPS * max( |a|,|b| ) , 00156 *> 00157 *> where EPS is the machine precision. If ABSTOL is less than 00158 *> or equal to zero, then EPS*|T| will be used in its place, 00159 *> where |T| is the 1-norm of the tridiagonal matrix obtained 00160 *> by reducing A to tridiagonal form. 00161 *> 00162 *> Eigenvalues will be computed most accurately when ABSTOL is 00163 *> set to twice the underflow threshold 2*DLAMCH('S'), not zero. 00164 *> If this routine returns with INFO>0, indicating that some 00165 *> eigenvectors did not converge, try setting ABSTOL to 00166 *> 2*DLAMCH('S'). 00167 *> \endverbatim 00168 *> 00169 *> \param[out] M 00170 *> \verbatim 00171 *> M is INTEGER 00172 *> The total number of eigenvalues found. 0 <= M <= N. 00173 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 00174 *> \endverbatim 00175 *> 00176 *> \param[out] W 00177 *> \verbatim 00178 *> W is DOUBLE PRECISION array, dimension (N) 00179 *> On normal exit, the first M elements contain the selected 00180 *> eigenvalues in ascending order. 00181 *> \endverbatim 00182 *> 00183 *> \param[out] Z 00184 *> \verbatim 00185 *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M)) 00186 *> If JOBZ = 'N', then Z is not referenced. 00187 *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z 00188 *> contain the orthonormal eigenvectors of the matrix A 00189 *> corresponding to the selected eigenvalues, with the i-th 00190 *> column of Z holding the eigenvector associated with W(i). 00191 *> The eigenvectors are normalized as follows: 00192 *> if ITYPE = 1 or 2, Z**T*B*Z = I; 00193 *> if ITYPE = 3, Z**T*inv(B)*Z = I. 00194 *> 00195 *> If an eigenvector fails to converge, then that column of Z 00196 *> contains the latest approximation to the eigenvector, and the 00197 *> index of the eigenvector is returned in IFAIL. 00198 *> Note: the user must ensure that at least max(1,M) columns are 00199 *> supplied in the array Z; if RANGE = 'V', the exact value of M 00200 *> is not known in advance and an upper bound must be used. 00201 *> \endverbatim 00202 *> 00203 *> \param[in] LDZ 00204 *> \verbatim 00205 *> LDZ is INTEGER 00206 *> The leading dimension of the array Z. LDZ >= 1, and if 00207 *> JOBZ = 'V', LDZ >= max(1,N). 00208 *> \endverbatim 00209 *> 00210 *> \param[out] WORK 00211 *> \verbatim 00212 *> WORK is DOUBLE PRECISION array, dimension (8*N) 00213 *> \endverbatim 00214 *> 00215 *> \param[out] IWORK 00216 *> \verbatim 00217 *> IWORK is INTEGER array, dimension (5*N) 00218 *> \endverbatim 00219 *> 00220 *> \param[out] IFAIL 00221 *> \verbatim 00222 *> IFAIL is INTEGER array, dimension (N) 00223 *> If JOBZ = 'V', then if INFO = 0, the first M elements of 00224 *> IFAIL are zero. If INFO > 0, then IFAIL contains the 00225 *> indices of the eigenvectors that failed to converge. 00226 *> If JOBZ = 'N', then IFAIL is not referenced. 00227 *> \endverbatim 00228 *> 00229 *> \param[out] INFO 00230 *> \verbatim 00231 *> INFO is INTEGER 00232 *> = 0: successful exit 00233 *> < 0: if INFO = -i, the i-th argument had an illegal value 00234 *> > 0: DPPTRF or DSPEVX returned an error code: 00235 *> <= N: if INFO = i, DSPEVX failed to converge; 00236 *> i eigenvectors failed to converge. Their indices 00237 *> are stored in array IFAIL. 00238 *> > N: if INFO = N + i, for 1 <= i <= N, then the leading 00239 *> minor of order i of B is not positive definite. 00240 *> The factorization of B could not be completed and 00241 *> no eigenvalues or eigenvectors were computed. 00242 *> \endverbatim 00243 * 00244 * Authors: 00245 * ======== 00246 * 00247 *> \author Univ. of Tennessee 00248 *> \author Univ. of California Berkeley 00249 *> \author Univ. of Colorado Denver 00250 *> \author NAG Ltd. 00251 * 00252 *> \date November 2011 00253 * 00254 *> \ingroup doubleOTHEReigen 00255 * 00256 *> \par Contributors: 00257 * ================== 00258 *> 00259 *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA 00260 * 00261 * ===================================================================== 00262 SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, 00263 $ IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, 00264 $ IFAIL, INFO ) 00265 * 00266 * -- LAPACK driver routine (version 3.4.0) -- 00267 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00268 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00269 * November 2011 00270 * 00271 * .. Scalar Arguments .. 00272 CHARACTER JOBZ, RANGE, UPLO 00273 INTEGER IL, INFO, ITYPE, IU, LDZ, M, N 00274 DOUBLE PRECISION ABSTOL, VL, VU 00275 * .. 00276 * .. Array Arguments .. 00277 INTEGER IFAIL( * ), IWORK( * ) 00278 DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ), 00279 $ Z( LDZ, * ) 00280 * .. 00281 * 00282 * ===================================================================== 00283 * 00284 * .. Local Scalars .. 00285 LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ 00286 CHARACTER TRANS 00287 INTEGER J 00288 * .. 00289 * .. External Functions .. 00290 LOGICAL LSAME 00291 EXTERNAL LSAME 00292 * .. 00293 * .. External Subroutines .. 00294 EXTERNAL DPPTRF, DSPEVX, DSPGST, DTPMV, DTPSV, XERBLA 00295 * .. 00296 * .. Intrinsic Functions .. 00297 INTRINSIC MIN 00298 * .. 00299 * .. Executable Statements .. 00300 * 00301 * Test the input parameters. 00302 * 00303 UPPER = LSAME( UPLO, 'U' ) 00304 WANTZ = LSAME( JOBZ, 'V' ) 00305 ALLEIG = LSAME( RANGE, 'A' ) 00306 VALEIG = LSAME( RANGE, 'V' ) 00307 INDEIG = LSAME( RANGE, 'I' ) 00308 * 00309 INFO = 0 00310 IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN 00311 INFO = -1 00312 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 00313 INFO = -2 00314 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 00315 INFO = -3 00316 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN 00317 INFO = -4 00318 ELSE IF( N.LT.0 ) THEN 00319 INFO = -5 00320 ELSE 00321 IF( VALEIG ) THEN 00322 IF( N.GT.0 .AND. VU.LE.VL ) THEN 00323 INFO = -9 00324 END IF 00325 ELSE IF( INDEIG ) THEN 00326 IF( IL.LT.1 ) THEN 00327 INFO = -10 00328 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 00329 INFO = -11 00330 END IF 00331 END IF 00332 END IF 00333 IF( INFO.EQ.0 ) THEN 00334 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 00335 INFO = -16 00336 END IF 00337 END IF 00338 * 00339 IF( INFO.NE.0 ) THEN 00340 CALL XERBLA( 'DSPGVX', -INFO ) 00341 RETURN 00342 END IF 00343 * 00344 * Quick return if possible 00345 * 00346 M = 0 00347 IF( N.EQ.0 ) 00348 $ RETURN 00349 * 00350 * Form a Cholesky factorization of B. 00351 * 00352 CALL DPPTRF( UPLO, N, BP, INFO ) 00353 IF( INFO.NE.0 ) THEN 00354 INFO = N + INFO 00355 RETURN 00356 END IF 00357 * 00358 * Transform problem to standard eigenvalue problem and solve. 00359 * 00360 CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO ) 00361 CALL DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, 00362 $ W, Z, LDZ, WORK, IWORK, IFAIL, INFO ) 00363 * 00364 IF( WANTZ ) THEN 00365 * 00366 * Backtransform eigenvectors to the original problem. 00367 * 00368 IF( INFO.GT.0 ) 00369 $ M = INFO - 1 00370 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN 00371 * 00372 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x; 00373 * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y 00374 * 00375 IF( UPPER ) THEN 00376 TRANS = 'N' 00377 ELSE 00378 TRANS = 'T' 00379 END IF 00380 * 00381 DO 10 J = 1, M 00382 CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), 00383 $ 1 ) 00384 10 CONTINUE 00385 * 00386 ELSE IF( ITYPE.EQ.3 ) THEN 00387 * 00388 * For B*A*x=(lambda)*x; 00389 * backtransform eigenvectors: x = L*y or U**T*y 00390 * 00391 IF( UPPER ) THEN 00392 TRANS = 'T' 00393 ELSE 00394 TRANS = 'N' 00395 END IF 00396 * 00397 DO 20 J = 1, M 00398 CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), 00399 $ 1 ) 00400 20 CONTINUE 00401 END IF 00402 END IF 00403 * 00404 RETURN 00405 * 00406 * End of DSPGVX 00407 * 00408 END