![]() |
LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
|
00001 *> \brief <b> SSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b> 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download SSYEVR + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ssyevr.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ssyevr.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ssyevr.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE SSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, 00022 * ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, 00023 * IWORK, LIWORK, INFO ) 00024 * 00025 * .. Scalar Arguments .. 00026 * CHARACTER JOBZ, RANGE, UPLO 00027 * INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N 00028 * REAL ABSTOL, VL, VU 00029 * .. 00030 * .. Array Arguments .. 00031 * INTEGER ISUPPZ( * ), IWORK( * ) 00032 * REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * ) 00033 * .. 00034 * 00035 * 00036 *> \par Purpose: 00037 * ============= 00038 *> 00039 *> \verbatim 00040 *> 00041 *> SSYEVR computes selected eigenvalues and, optionally, eigenvectors 00042 *> of a real symmetric matrix A. Eigenvalues and eigenvectors can be 00043 *> selected by specifying either a range of values or a range of 00044 *> indices for the desired eigenvalues. 00045 *> 00046 *> SSYEVR first reduces the matrix A to tridiagonal form T with a call 00047 *> to SSYTRD. Then, whenever possible, SSYEVR calls SSTEMR to compute 00048 *> the eigenspectrum using Relatively Robust Representations. SSTEMR 00049 *> computes eigenvalues by the dqds algorithm, while orthogonal 00050 *> eigenvectors are computed from various "good" L D L^T representations 00051 *> (also known as Relatively Robust Representations). Gram-Schmidt 00052 *> orthogonalization is avoided as far as possible. More specifically, 00053 *> the various steps of the algorithm are as follows. 00054 *> 00055 *> For each unreduced block (submatrix) of T, 00056 *> (a) Compute T - sigma I = L D L^T, so that L and D 00057 *> define all the wanted eigenvalues to high relative accuracy. 00058 *> This means that small relative changes in the entries of D and L 00059 *> cause only small relative changes in the eigenvalues and 00060 *> eigenvectors. The standard (unfactored) representation of the 00061 *> tridiagonal matrix T does not have this property in general. 00062 *> (b) Compute the eigenvalues to suitable accuracy. 00063 *> If the eigenvectors are desired, the algorithm attains full 00064 *> accuracy of the computed eigenvalues only right before 00065 *> the corresponding vectors have to be computed, see steps c) and d). 00066 *> (c) For each cluster of close eigenvalues, select a new 00067 *> shift close to the cluster, find a new factorization, and refine 00068 *> the shifted eigenvalues to suitable accuracy. 00069 *> (d) For each eigenvalue with a large enough relative separation compute 00070 *> the corresponding eigenvector by forming a rank revealing twisted 00071 *> factorization. Go back to (c) for any clusters that remain. 00072 *> 00073 *> The desired accuracy of the output can be specified by the input 00074 *> parameter ABSTOL. 00075 *> 00076 *> For more details, see SSTEMR's documentation and: 00077 *> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations 00078 *> to compute orthogonal eigenvectors of symmetric tridiagonal matrices," 00079 *> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. 00080 *> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and 00081 *> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, 00082 *> 2004. Also LAPACK Working Note 154. 00083 *> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric 00084 *> tridiagonal eigenvalue/eigenvector problem", 00085 *> Computer Science Division Technical Report No. UCB/CSD-97-971, 00086 *> UC Berkeley, May 1997. 00087 *> 00088 *> 00089 *> Note 1 : SSYEVR calls SSTEMR when the full spectrum is requested 00090 *> on machines which conform to the ieee-754 floating point standard. 00091 *> SSYEVR calls SSTEBZ and SSTEIN on non-ieee machines and 00092 *> when partial spectrum requests are made. 00093 *> 00094 *> Normal execution of SSTEMR may create NaNs and infinities and 00095 *> hence may abort due to a floating point exception in environments 00096 *> which do not handle NaNs and infinities in the ieee standard default 00097 *> manner. 00098 *> \endverbatim 00099 * 00100 * Arguments: 00101 * ========== 00102 * 00103 *> \param[in] JOBZ 00104 *> \verbatim 00105 *> JOBZ is CHARACTER*1 00106 *> = 'N': Compute eigenvalues only; 00107 *> = 'V': Compute eigenvalues and eigenvectors. 00108 *> \endverbatim 00109 *> 00110 *> \param[in] RANGE 00111 *> \verbatim 00112 *> RANGE is CHARACTER*1 00113 *> = 'A': all eigenvalues will be found. 00114 *> = 'V': all eigenvalues in the half-open interval (VL,VU] 00115 *> will be found. 00116 *> = 'I': the IL-th through IU-th eigenvalues will be found. 00117 *> For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and 00118 *> SSTEIN are called 00119 *> \endverbatim 00120 *> 00121 *> \param[in] UPLO 00122 *> \verbatim 00123 *> UPLO is CHARACTER*1 00124 *> = 'U': Upper triangle of A is stored; 00125 *> = 'L': Lower triangle of A is stored. 00126 *> \endverbatim 00127 *> 00128 *> \param[in] N 00129 *> \verbatim 00130 *> N is INTEGER 00131 *> The order of the matrix A. N >= 0. 00132 *> \endverbatim 00133 *> 00134 *> \param[in,out] A 00135 *> \verbatim 00136 *> A is REAL array, dimension (LDA, N) 00137 *> On entry, the symmetric matrix A. If UPLO = 'U', the 00138 *> leading N-by-N upper triangular part of A contains the 00139 *> upper triangular part of the matrix A. If UPLO = 'L', 00140 *> the leading N-by-N lower triangular part of A contains 00141 *> the lower triangular part of the matrix A. 00142 *> On exit, the lower triangle (if UPLO='L') or the upper 00143 *> triangle (if UPLO='U') of A, including the diagonal, is 00144 *> destroyed. 00145 *> \endverbatim 00146 *> 00147 *> \param[in] LDA 00148 *> \verbatim 00149 *> LDA is INTEGER 00150 *> The leading dimension of the array A. LDA >= max(1,N). 00151 *> \endverbatim 00152 *> 00153 *> \param[in] VL 00154 *> \verbatim 00155 *> VL is REAL 00156 *> \endverbatim 00157 *> 00158 *> \param[in] VU 00159 *> \verbatim 00160 *> VU is REAL 00161 *> If RANGE='V', the lower and upper bounds of the interval to 00162 *> be searched for eigenvalues. VL < VU. 00163 *> Not referenced if RANGE = 'A' or 'I'. 00164 *> \endverbatim 00165 *> 00166 *> \param[in] IL 00167 *> \verbatim 00168 *> IL is INTEGER 00169 *> \endverbatim 00170 *> 00171 *> \param[in] IU 00172 *> \verbatim 00173 *> IU is INTEGER 00174 *> If RANGE='I', the indices (in ascending order) of the 00175 *> smallest and largest eigenvalues to be returned. 00176 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 00177 *> Not referenced if RANGE = 'A' or 'V'. 00178 *> \endverbatim 00179 *> 00180 *> \param[in] ABSTOL 00181 *> \verbatim 00182 *> ABSTOL is REAL 00183 *> The absolute error tolerance for the eigenvalues. 00184 *> An approximate eigenvalue is accepted as converged 00185 *> when it is determined to lie in an interval [a,b] 00186 *> of width less than or equal to 00187 *> 00188 *> ABSTOL + EPS * max( |a|,|b| ) , 00189 *> 00190 *> where EPS is the machine precision. If ABSTOL is less than 00191 *> or equal to zero, then EPS*|T| will be used in its place, 00192 *> where |T| is the 1-norm of the tridiagonal matrix obtained 00193 *> by reducing A to tridiagonal form. 00194 *> 00195 *> See "Computing Small Singular Values of Bidiagonal Matrices 00196 *> with Guaranteed High Relative Accuracy," by Demmel and 00197 *> Kahan, LAPACK Working Note #3. 00198 *> 00199 *> If high relative accuracy is important, set ABSTOL to 00200 *> SLAMCH( 'Safe minimum' ). Doing so will guarantee that 00201 *> eigenvalues are computed to high relative accuracy when 00202 *> possible in future releases. The current code does not 00203 *> make any guarantees about high relative accuracy, but 00204 *> future releases will. See J. Barlow and J. Demmel, 00205 *> "Computing Accurate Eigensystems of Scaled Diagonally 00206 *> Dominant Matrices", LAPACK Working Note #7, for a discussion 00207 *> of which matrices define their eigenvalues to high relative 00208 *> accuracy. 00209 *> \endverbatim 00210 *> 00211 *> \param[out] M 00212 *> \verbatim 00213 *> M is INTEGER 00214 *> The total number of eigenvalues found. 0 <= M <= N. 00215 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 00216 *> \endverbatim 00217 *> 00218 *> \param[out] W 00219 *> \verbatim 00220 *> W is REAL array, dimension (N) 00221 *> The first M elements contain the selected eigenvalues in 00222 *> ascending order. 00223 *> \endverbatim 00224 *> 00225 *> \param[out] Z 00226 *> \verbatim 00227 *> Z is REAL array, dimension (LDZ, max(1,M)) 00228 *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z 00229 *> contain the orthonormal eigenvectors of the matrix A 00230 *> corresponding to the selected eigenvalues, with the i-th 00231 *> column of Z holding the eigenvector associated with W(i). 00232 *> If JOBZ = 'N', then Z is not referenced. 00233 *> Note: the user must ensure that at least max(1,M) columns are 00234 *> supplied in the array Z; if RANGE = 'V', the exact value of M 00235 *> is not known in advance and an upper bound must be used. 00236 *> Supplying N columns is always safe. 00237 *> \endverbatim 00238 *> 00239 *> \param[in] LDZ 00240 *> \verbatim 00241 *> LDZ is INTEGER 00242 *> The leading dimension of the array Z. LDZ >= 1, and if 00243 *> JOBZ = 'V', LDZ >= max(1,N). 00244 *> \endverbatim 00245 *> 00246 *> \param[out] ISUPPZ 00247 *> \verbatim 00248 *> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) 00249 *> The support of the eigenvectors in Z, i.e., the indices 00250 *> indicating the nonzero elements in Z. The i-th eigenvector 00251 *> is nonzero only in elements ISUPPZ( 2*i-1 ) through 00252 *> ISUPPZ( 2*i ). 00253 *> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 00254 *> \endverbatim 00255 *> 00256 *> \param[out] WORK 00257 *> \verbatim 00258 *> WORK is REAL array, dimension (MAX(1,LWORK)) 00259 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00260 *> \endverbatim 00261 *> 00262 *> \param[in] LWORK 00263 *> \verbatim 00264 *> LWORK is INTEGER 00265 *> The dimension of the array WORK. LWORK >= max(1,26*N). 00266 *> For optimal efficiency, LWORK >= (NB+6)*N, 00267 *> where NB is the max of the blocksize for SSYTRD and SORMTR 00268 *> returned by ILAENV. 00269 *> 00270 *> If LWORK = -1, then a workspace query is assumed; the routine 00271 *> only calculates the optimal sizes of the WORK and IWORK 00272 *> arrays, returns these values as the first entries of the WORK 00273 *> and IWORK arrays, and no error message related to LWORK or 00274 *> LIWORK is issued by XERBLA. 00275 *> \endverbatim 00276 *> 00277 *> \param[out] IWORK 00278 *> \verbatim 00279 *> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) 00280 *> On exit, if INFO = 0, IWORK(1) returns the optimal LWORK. 00281 *> \endverbatim 00282 *> 00283 *> \param[in] LIWORK 00284 *> \verbatim 00285 *> LIWORK is INTEGER 00286 *> The dimension of the array IWORK. LIWORK >= max(1,10*N). 00287 *> 00288 *> If LIWORK = -1, then a workspace query is assumed; the 00289 *> routine only calculates the optimal sizes of the WORK and 00290 *> IWORK arrays, returns these values as the first entries of 00291 *> the WORK and IWORK arrays, and no error message related to 00292 *> LWORK or LIWORK is issued by XERBLA. 00293 *> \endverbatim 00294 *> 00295 *> \param[out] INFO 00296 *> \verbatim 00297 *> INFO is INTEGER 00298 *> = 0: successful exit 00299 *> < 0: if INFO = -i, the i-th argument had an illegal value 00300 *> > 0: Internal error 00301 *> \endverbatim 00302 * 00303 * Authors: 00304 * ======== 00305 * 00306 *> \author Univ. of Tennessee 00307 *> \author Univ. of California Berkeley 00308 *> \author Univ. of Colorado Denver 00309 *> \author NAG Ltd. 00310 * 00311 *> \date November 2011 00312 * 00313 *> \ingroup realSYeigen 00314 * 00315 *> \par Contributors: 00316 * ================== 00317 *> 00318 *> Inderjit Dhillon, IBM Almaden, USA \n 00319 *> Osni Marques, LBNL/NERSC, USA \n 00320 *> Ken Stanley, Computer Science Division, University of 00321 *> California at Berkeley, USA \n 00322 *> Jason Riedy, Computer Science Division, University of 00323 *> California at Berkeley, USA \n 00324 *> 00325 * ===================================================================== 00326 SUBROUTINE SSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, 00327 $ ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, 00328 $ IWORK, LIWORK, INFO ) 00329 * 00330 * -- LAPACK driver routine (version 3.4.0) -- 00331 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00332 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00333 * November 2011 00334 * 00335 * .. Scalar Arguments .. 00336 CHARACTER JOBZ, RANGE, UPLO 00337 INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N 00338 REAL ABSTOL, VL, VU 00339 * .. 00340 * .. Array Arguments .. 00341 INTEGER ISUPPZ( * ), IWORK( * ) 00342 REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * ) 00343 * .. 00344 * 00345 * ===================================================================== 00346 * 00347 * .. Parameters .. 00348 REAL ZERO, ONE, TWO 00349 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 ) 00350 * .. 00351 * .. Local Scalars .. 00352 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG, 00353 $ WANTZ, TRYRAC 00354 CHARACTER ORDER 00355 INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE, 00356 $ INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU, 00357 $ INDWK, INDWKN, ISCALE, J, JJ, LIWMIN, 00358 $ LLWORK, LLWRKN, LWKOPT, LWMIN, NB, NSPLIT 00359 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, 00360 $ SIGMA, SMLNUM, TMP1, VLL, VUU 00361 * .. 00362 * .. External Functions .. 00363 LOGICAL LSAME 00364 INTEGER ILAENV 00365 REAL SLAMCH, SLANSY 00366 EXTERNAL LSAME, ILAENV, SLAMCH, SLANSY 00367 * .. 00368 * .. External Subroutines .. 00369 EXTERNAL SCOPY, SORMTR, SSCAL, SSTEBZ, SSTEMR, SSTEIN, 00370 $ SSTERF, SSWAP, SSYTRD, XERBLA 00371 * .. 00372 * .. Intrinsic Functions .. 00373 INTRINSIC MAX, MIN, SQRT 00374 * .. 00375 * .. Executable Statements .. 00376 * 00377 * Test the input parameters. 00378 * 00379 IEEEOK = ILAENV( 10, 'SSYEVR', 'N', 1, 2, 3, 4 ) 00380 * 00381 LOWER = LSAME( UPLO, 'L' ) 00382 WANTZ = LSAME( JOBZ, 'V' ) 00383 ALLEIG = LSAME( RANGE, 'A' ) 00384 VALEIG = LSAME( RANGE, 'V' ) 00385 INDEIG = LSAME( RANGE, 'I' ) 00386 * 00387 LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) ) 00388 * 00389 LWMIN = MAX( 1, 26*N ) 00390 LIWMIN = MAX( 1, 10*N ) 00391 * 00392 INFO = 0 00393 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 00394 INFO = -1 00395 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 00396 INFO = -2 00397 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN 00398 INFO = -3 00399 ELSE IF( N.LT.0 ) THEN 00400 INFO = -4 00401 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00402 INFO = -6 00403 ELSE 00404 IF( VALEIG ) THEN 00405 IF( N.GT.0 .AND. VU.LE.VL ) 00406 $ INFO = -8 00407 ELSE IF( INDEIG ) THEN 00408 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 00409 INFO = -9 00410 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 00411 INFO = -10 00412 END IF 00413 END IF 00414 END IF 00415 IF( INFO.EQ.0 ) THEN 00416 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 00417 INFO = -15 00418 END IF 00419 END IF 00420 * 00421 IF( INFO.EQ.0 ) THEN 00422 NB = ILAENV( 1, 'SSYTRD', UPLO, N, -1, -1, -1 ) 00423 NB = MAX( NB, ILAENV( 1, 'SORMTR', UPLO, N, -1, -1, -1 ) ) 00424 LWKOPT = MAX( ( NB+1 )*N, LWMIN ) 00425 WORK( 1 ) = LWKOPT 00426 IWORK( 1 ) = LIWMIN 00427 * 00428 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 00429 INFO = -18 00430 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 00431 INFO = -20 00432 END IF 00433 END IF 00434 * 00435 IF( INFO.NE.0 ) THEN 00436 CALL XERBLA( 'SSYEVR', -INFO ) 00437 RETURN 00438 ELSE IF( LQUERY ) THEN 00439 RETURN 00440 END IF 00441 * 00442 * Quick return if possible 00443 * 00444 M = 0 00445 IF( N.EQ.0 ) THEN 00446 WORK( 1 ) = 1 00447 RETURN 00448 END IF 00449 * 00450 IF( N.EQ.1 ) THEN 00451 WORK( 1 ) = 26 00452 IF( ALLEIG .OR. INDEIG ) THEN 00453 M = 1 00454 W( 1 ) = A( 1, 1 ) 00455 ELSE 00456 IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN 00457 M = 1 00458 W( 1 ) = A( 1, 1 ) 00459 END IF 00460 END IF 00461 IF( WANTZ ) THEN 00462 Z( 1, 1 ) = ONE 00463 ISUPPZ( 1 ) = 1 00464 ISUPPZ( 2 ) = 1 00465 END IF 00466 RETURN 00467 END IF 00468 * 00469 * Get machine constants. 00470 * 00471 SAFMIN = SLAMCH( 'Safe minimum' ) 00472 EPS = SLAMCH( 'Precision' ) 00473 SMLNUM = SAFMIN / EPS 00474 BIGNUM = ONE / SMLNUM 00475 RMIN = SQRT( SMLNUM ) 00476 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 00477 * 00478 * Scale matrix to allowable range, if necessary. 00479 * 00480 ISCALE = 0 00481 ABSTLL = ABSTOL 00482 IF (VALEIG) THEN 00483 VLL = VL 00484 VUU = VU 00485 END IF 00486 ANRM = SLANSY( 'M', UPLO, N, A, LDA, WORK ) 00487 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN 00488 ISCALE = 1 00489 SIGMA = RMIN / ANRM 00490 ELSE IF( ANRM.GT.RMAX ) THEN 00491 ISCALE = 1 00492 SIGMA = RMAX / ANRM 00493 END IF 00494 IF( ISCALE.EQ.1 ) THEN 00495 IF( LOWER ) THEN 00496 DO 10 J = 1, N 00497 CALL SSCAL( N-J+1, SIGMA, A( J, J ), 1 ) 00498 10 CONTINUE 00499 ELSE 00500 DO 20 J = 1, N 00501 CALL SSCAL( J, SIGMA, A( 1, J ), 1 ) 00502 20 CONTINUE 00503 END IF 00504 IF( ABSTOL.GT.0 ) 00505 $ ABSTLL = ABSTOL*SIGMA 00506 IF( VALEIG ) THEN 00507 VLL = VL*SIGMA 00508 VUU = VU*SIGMA 00509 END IF 00510 END IF 00511 00512 * Initialize indices into workspaces. Note: The IWORK indices are 00513 * used only if SSTERF or SSTEMR fail. 00514 00515 * WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the 00516 * elementary reflectors used in SSYTRD. 00517 INDTAU = 1 00518 * WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries. 00519 INDD = INDTAU + N 00520 * WORK(INDE:INDE+N-1) stores the off-diagonal entries of the 00521 * tridiagonal matrix from SSYTRD. 00522 INDE = INDD + N 00523 * WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over 00524 * -written by SSTEMR (the SSTERF path copies the diagonal to W). 00525 INDDD = INDE + N 00526 * WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over 00527 * -written while computing the eigenvalues in SSTERF and SSTEMR. 00528 INDEE = INDDD + N 00529 * INDWK is the starting offset of the left-over workspace, and 00530 * LLWORK is the remaining workspace size. 00531 INDWK = INDEE + N 00532 LLWORK = LWORK - INDWK + 1 00533 00534 * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and 00535 * stores the block indices of each of the M<=N eigenvalues. 00536 INDIBL = 1 00537 * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and 00538 * stores the starting and finishing indices of each block. 00539 INDISP = INDIBL + N 00540 * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors 00541 * that corresponding to eigenvectors that fail to converge in 00542 * SSTEIN. This information is discarded; if any fail, the driver 00543 * returns INFO > 0. 00544 INDIFL = INDISP + N 00545 * INDIWO is the offset of the remaining integer workspace. 00546 INDIWO = INDISP + N 00547 00548 * 00549 * Call SSYTRD to reduce symmetric matrix to tridiagonal form. 00550 * 00551 CALL SSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ), 00552 $ WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO ) 00553 * 00554 * If all eigenvalues are desired 00555 * then call SSTERF or SSTEMR and SORMTR. 00556 * 00557 TEST = .FALSE. 00558 IF( INDEIG ) THEN 00559 IF( IL.EQ.1 .AND. IU.EQ.N ) THEN 00560 TEST = .TRUE. 00561 END IF 00562 END IF 00563 IF( ( ALLEIG.OR.TEST ) .AND. ( IEEEOK.EQ.1 ) ) THEN 00564 IF( .NOT.WANTZ ) THEN 00565 CALL SCOPY( N, WORK( INDD ), 1, W, 1 ) 00566 CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) 00567 CALL SSTERF( N, W, WORK( INDEE ), INFO ) 00568 ELSE 00569 CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) 00570 CALL SCOPY( N, WORK( INDD ), 1, WORK( INDDD ), 1 ) 00571 * 00572 IF (ABSTOL .LE. TWO*N*EPS) THEN 00573 TRYRAC = .TRUE. 00574 ELSE 00575 TRYRAC = .FALSE. 00576 END IF 00577 CALL SSTEMR( JOBZ, 'A', N, WORK( INDDD ), WORK( INDEE ), 00578 $ VL, VU, IL, IU, M, W, Z, LDZ, N, ISUPPZ, 00579 $ TRYRAC, WORK( INDWK ), LWORK, IWORK, LIWORK, 00580 $ INFO ) 00581 * 00582 * 00583 * 00584 * Apply orthogonal matrix used in reduction to tridiagonal 00585 * form to eigenvectors returned by SSTEIN. 00586 * 00587 IF( WANTZ .AND. INFO.EQ.0 ) THEN 00588 INDWKN = INDE 00589 LLWRKN = LWORK - INDWKN + 1 00590 CALL SORMTR( 'L', UPLO, 'N', N, M, A, LDA, 00591 $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ), 00592 $ LLWRKN, IINFO ) 00593 END IF 00594 END IF 00595 * 00596 * 00597 IF( INFO.EQ.0 ) THEN 00598 * Everything worked. Skip SSTEBZ/SSTEIN. IWORK(:) are 00599 * undefined. 00600 M = N 00601 GO TO 30 00602 END IF 00603 INFO = 0 00604 END IF 00605 * 00606 * Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. 00607 * Also call SSTEBZ and SSTEIN if SSTEMR fails. 00608 * 00609 IF( WANTZ ) THEN 00610 ORDER = 'B' 00611 ELSE 00612 ORDER = 'E' 00613 END IF 00614 00615 CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, 00616 $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W, 00617 $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWK ), 00618 $ IWORK( INDIWO ), INFO ) 00619 * 00620 IF( WANTZ ) THEN 00621 CALL SSTEIN( N, WORK( INDD ), WORK( INDE ), M, W, 00622 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, 00623 $ WORK( INDWK ), IWORK( INDIWO ), IWORK( INDIFL ), 00624 $ INFO ) 00625 * 00626 * Apply orthogonal matrix used in reduction to tridiagonal 00627 * form to eigenvectors returned by SSTEIN. 00628 * 00629 INDWKN = INDE 00630 LLWRKN = LWORK - INDWKN + 1 00631 CALL SORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z, 00632 $ LDZ, WORK( INDWKN ), LLWRKN, IINFO ) 00633 END IF 00634 * 00635 * If matrix was scaled, then rescale eigenvalues appropriately. 00636 * 00637 * Jump here if SSTEMR/SSTEIN succeeded. 00638 30 CONTINUE 00639 IF( ISCALE.EQ.1 ) THEN 00640 IF( INFO.EQ.0 ) THEN 00641 IMAX = M 00642 ELSE 00643 IMAX = INFO - 1 00644 END IF 00645 CALL SSCAL( IMAX, ONE / SIGMA, W, 1 ) 00646 END IF 00647 * 00648 * If eigenvalues are not in order, then sort them, along with 00649 * eigenvectors. Note: We do not sort the IFAIL portion of IWORK. 00650 * It may not be initialized (if SSTEMR/SSTEIN succeeded), and we do 00651 * not return this detailed information to the user. 00652 * 00653 IF( WANTZ ) THEN 00654 DO 50 J = 1, M - 1 00655 I = 0 00656 TMP1 = W( J ) 00657 DO 40 JJ = J + 1, M 00658 IF( W( JJ ).LT.TMP1 ) THEN 00659 I = JJ 00660 TMP1 = W( JJ ) 00661 END IF 00662 40 CONTINUE 00663 * 00664 IF( I.NE.0 ) THEN 00665 W( I ) = W( J ) 00666 W( J ) = TMP1 00667 CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 00668 END IF 00669 50 CONTINUE 00670 END IF 00671 * 00672 * Set WORK(1) to optimal workspace size. 00673 * 00674 WORK( 1 ) = LWKOPT 00675 IWORK( 1 ) = LIWMIN 00676 * 00677 RETURN 00678 * 00679 * End of SSYEVR 00680 * 00681 END