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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DSTEMR 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download DSTEMR + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstemr.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstemr.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstemr.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, 00022 * M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, 00023 * IWORK, LIWORK, INFO ) 00024 * 00025 * .. Scalar Arguments .. 00026 * CHARACTER JOBZ, RANGE 00027 * LOGICAL TRYRAC 00028 * INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N 00029 * DOUBLE PRECISION VL, VU 00030 * .. 00031 * .. Array Arguments .. 00032 * INTEGER ISUPPZ( * ), IWORK( * ) 00033 * DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ) 00034 * DOUBLE PRECISION Z( LDZ, * ) 00035 * .. 00036 * 00037 * 00038 *> \par Purpose: 00039 * ============= 00040 *> 00041 *> \verbatim 00042 *> 00043 *> DSTEMR computes selected eigenvalues and, optionally, eigenvectors 00044 *> of a real symmetric tridiagonal matrix T. Any such unreduced matrix has 00045 *> a well defined set of pairwise different real eigenvalues, the corresponding 00046 *> real eigenvectors are pairwise orthogonal. 00047 *> 00048 *> The spectrum may be computed either completely or partially by specifying 00049 *> either an interval (VL,VU] or a range of indices IL:IU for the desired 00050 *> eigenvalues. 00051 *> 00052 *> Depending on the number of desired eigenvalues, these are computed either 00053 *> by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are 00054 *> computed by the use of various suitable L D L^T factorizations near clusters 00055 *> of close eigenvalues (referred to as RRRs, Relatively Robust 00056 *> Representations). An informal sketch of the algorithm follows. 00057 *> 00058 *> For each unreduced block (submatrix) of T, 00059 *> (a) Compute T - sigma I = L D L^T, so that L and D 00060 *> define all the wanted eigenvalues to high relative accuracy. 00061 *> This means that small relative changes in the entries of D and L 00062 *> cause only small relative changes in the eigenvalues and 00063 *> eigenvectors. The standard (unfactored) representation of the 00064 *> tridiagonal matrix T does not have this property in general. 00065 *> (b) Compute the eigenvalues to suitable accuracy. 00066 *> If the eigenvectors are desired, the algorithm attains full 00067 *> accuracy of the computed eigenvalues only right before 00068 *> the corresponding vectors have to be computed, see steps c) and d). 00069 *> (c) For each cluster of close eigenvalues, select a new 00070 *> shift close to the cluster, find a new factorization, and refine 00071 *> the shifted eigenvalues to suitable accuracy. 00072 *> (d) For each eigenvalue with a large enough relative separation compute 00073 *> the corresponding eigenvector by forming a rank revealing twisted 00074 *> factorization. Go back to (c) for any clusters that remain. 00075 *> 00076 *> For more details, see: 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 *> Further Details 00089 *> 1.DSTEMR works only on machines which follow IEEE-754 00090 *> floating-point standard in their handling of infinities and NaNs. 00091 *> This permits the use of efficient inner loops avoiding a check for 00092 *> zero divisors. 00093 *> \endverbatim 00094 * 00095 * Arguments: 00096 * ========== 00097 * 00098 *> \param[in] JOBZ 00099 *> \verbatim 00100 *> JOBZ is CHARACTER*1 00101 *> = 'N': Compute eigenvalues only; 00102 *> = 'V': Compute eigenvalues and eigenvectors. 00103 *> \endverbatim 00104 *> 00105 *> \param[in] RANGE 00106 *> \verbatim 00107 *> RANGE is CHARACTER*1 00108 *> = 'A': all eigenvalues will be found. 00109 *> = 'V': all eigenvalues in the half-open interval (VL,VU] 00110 *> will be found. 00111 *> = 'I': the IL-th through IU-th eigenvalues will be found. 00112 *> \endverbatim 00113 *> 00114 *> \param[in] N 00115 *> \verbatim 00116 *> N is INTEGER 00117 *> The order of the matrix. N >= 0. 00118 *> \endverbatim 00119 *> 00120 *> \param[in,out] D 00121 *> \verbatim 00122 *> D is DOUBLE PRECISION array, dimension (N) 00123 *> On entry, the N diagonal elements of the tridiagonal matrix 00124 *> T. On exit, D is overwritten. 00125 *> \endverbatim 00126 *> 00127 *> \param[in,out] E 00128 *> \verbatim 00129 *> E is DOUBLE PRECISION array, dimension (N) 00130 *> On entry, the (N-1) subdiagonal elements of the tridiagonal 00131 *> matrix T in elements 1 to N-1 of E. E(N) need not be set on 00132 *> input, but is used internally as workspace. 00133 *> On exit, E is overwritten. 00134 *> \endverbatim 00135 *> 00136 *> \param[in] VL 00137 *> \verbatim 00138 *> VL is DOUBLE PRECISION 00139 *> \endverbatim 00140 *> 00141 *> \param[in] VU 00142 *> \verbatim 00143 *> VU is DOUBLE PRECISION 00144 *> 00145 *> If RANGE='V', the lower and upper bounds of the interval to 00146 *> be searched for eigenvalues. VL < VU. 00147 *> Not referenced if RANGE = 'A' or 'I'. 00148 *> \endverbatim 00149 *> 00150 *> \param[in] IL 00151 *> \verbatim 00152 *> IL is INTEGER 00153 *> \endverbatim 00154 *> 00155 *> \param[in] IU 00156 *> \verbatim 00157 *> IU is INTEGER 00158 *> 00159 *> If RANGE='I', the indices (in ascending order) of the 00160 *> smallest and largest eigenvalues to be returned. 00161 *> 1 <= IL <= IU <= N, if N > 0. 00162 *> Not referenced if RANGE = 'A' or 'V'. 00163 *> \endverbatim 00164 *> 00165 *> \param[out] M 00166 *> \verbatim 00167 *> M is INTEGER 00168 *> The total number of eigenvalues found. 0 <= M <= N. 00169 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 00170 *> \endverbatim 00171 *> 00172 *> \param[out] W 00173 *> \verbatim 00174 *> W is DOUBLE PRECISION array, dimension (N) 00175 *> The first M elements contain the selected eigenvalues in 00176 *> ascending order. 00177 *> \endverbatim 00178 *> 00179 *> \param[out] Z 00180 *> \verbatim 00181 *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) ) 00182 *> If JOBZ = 'V', and if INFO = 0, then the first M columns of Z 00183 *> contain the orthonormal eigenvectors of the matrix T 00184 *> corresponding to the selected eigenvalues, with the i-th 00185 *> column of Z holding the eigenvector associated with W(i). 00186 *> If JOBZ = 'N', then Z is not referenced. 00187 *> Note: the user must ensure that at least max(1,M) columns are 00188 *> supplied in the array Z; if RANGE = 'V', the exact value of M 00189 *> is not known in advance and can be computed with a workspace 00190 *> query by setting NZC = -1, see below. 00191 *> \endverbatim 00192 *> 00193 *> \param[in] LDZ 00194 *> \verbatim 00195 *> LDZ is INTEGER 00196 *> The leading dimension of the array Z. LDZ >= 1, and if 00197 *> JOBZ = 'V', then LDZ >= max(1,N). 00198 *> \endverbatim 00199 *> 00200 *> \param[in] NZC 00201 *> \verbatim 00202 *> NZC is INTEGER 00203 *> The number of eigenvectors to be held in the array Z. 00204 *> If RANGE = 'A', then NZC >= max(1,N). 00205 *> If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]. 00206 *> If RANGE = 'I', then NZC >= IU-IL+1. 00207 *> If NZC = -1, then a workspace query is assumed; the 00208 *> routine calculates the number of columns of the array Z that 00209 *> are needed to hold the eigenvectors. 00210 *> This value is returned as the first entry of the Z array, and 00211 *> no error message related to NZC is issued by XERBLA. 00212 *> \endverbatim 00213 *> 00214 *> \param[out] ISUPPZ 00215 *> \verbatim 00216 *> ISUPPZ is INTEGER ARRAY, dimension ( 2*max(1,M) ) 00217 *> The support of the eigenvectors in Z, i.e., the indices 00218 *> indicating the nonzero elements in Z. The i-th computed eigenvector 00219 *> is nonzero only in elements ISUPPZ( 2*i-1 ) through 00220 *> ISUPPZ( 2*i ). This is relevant in the case when the matrix 00221 *> is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0. 00222 *> \endverbatim 00223 *> 00224 *> \param[in,out] TRYRAC 00225 *> \verbatim 00226 *> TRYRAC is LOGICAL 00227 *> If TRYRAC.EQ..TRUE., indicates that the code should check whether 00228 *> the tridiagonal matrix defines its eigenvalues to high relative 00229 *> accuracy. If so, the code uses relative-accuracy preserving 00230 *> algorithms that might be (a bit) slower depending on the matrix. 00231 *> If the matrix does not define its eigenvalues to high relative 00232 *> accuracy, the code can uses possibly faster algorithms. 00233 *> If TRYRAC.EQ..FALSE., the code is not required to guarantee 00234 *> relatively accurate eigenvalues and can use the fastest possible 00235 *> techniques. 00236 *> On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix 00237 *> does not define its eigenvalues to high relative accuracy. 00238 *> \endverbatim 00239 *> 00240 *> \param[out] WORK 00241 *> \verbatim 00242 *> WORK is DOUBLE PRECISION array, dimension (LWORK) 00243 *> On exit, if INFO = 0, WORK(1) returns the optimal 00244 *> (and minimal) LWORK. 00245 *> \endverbatim 00246 *> 00247 *> \param[in] LWORK 00248 *> \verbatim 00249 *> LWORK is INTEGER 00250 *> The dimension of the array WORK. LWORK >= max(1,18*N) 00251 *> if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'. 00252 *> If LWORK = -1, then a workspace query is assumed; the routine 00253 *> only calculates the optimal size of the WORK array, returns 00254 *> this value as the first entry of the WORK array, and no error 00255 *> message related to LWORK is issued by XERBLA. 00256 *> \endverbatim 00257 *> 00258 *> \param[out] IWORK 00259 *> \verbatim 00260 *> IWORK is INTEGER array, dimension (LIWORK) 00261 *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 00262 *> \endverbatim 00263 *> 00264 *> \param[in] LIWORK 00265 *> \verbatim 00266 *> LIWORK is INTEGER 00267 *> The dimension of the array IWORK. LIWORK >= max(1,10*N) 00268 *> if the eigenvectors are desired, and LIWORK >= max(1,8*N) 00269 *> if only the eigenvalues are to be computed. 00270 *> If LIWORK = -1, then a workspace query is assumed; the 00271 *> routine only calculates the optimal size of the IWORK array, 00272 *> returns this value as the first entry of the IWORK array, and 00273 *> no error message related to LIWORK is issued by XERBLA. 00274 *> \endverbatim 00275 *> 00276 *> \param[out] INFO 00277 *> \verbatim 00278 *> INFO is INTEGER 00279 *> On exit, INFO 00280 *> = 0: successful exit 00281 *> < 0: if INFO = -i, the i-th argument had an illegal value 00282 *> > 0: if INFO = 1X, internal error in DLARRE, 00283 *> if INFO = 2X, internal error in DLARRV. 00284 *> Here, the digit X = ABS( IINFO ) < 10, where IINFO is 00285 *> the nonzero error code returned by DLARRE or 00286 *> DLARRV, respectively. 00287 *> \endverbatim 00288 * 00289 * Authors: 00290 * ======== 00291 * 00292 *> \author Univ. of Tennessee 00293 *> \author Univ. of California Berkeley 00294 *> \author Univ. of Colorado Denver 00295 *> \author NAG Ltd. 00296 * 00297 *> \date November 2011 00298 * 00299 *> \ingroup doubleOTHERcomputational 00300 * 00301 *> \par Contributors: 00302 * ================== 00303 *> 00304 *> Beresford Parlett, University of California, Berkeley, USA \n 00305 *> Jim Demmel, University of California, Berkeley, USA \n 00306 *> Inderjit Dhillon, University of Texas, Austin, USA \n 00307 *> Osni Marques, LBNL/NERSC, USA \n 00308 *> Christof Voemel, University of California, Berkeley, USA 00309 * 00310 * ===================================================================== 00311 SUBROUTINE DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, 00312 $ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, 00313 $ IWORK, LIWORK, INFO ) 00314 * 00315 * -- LAPACK computational routine (version 3.4.0) -- 00316 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00317 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00318 * November 2011 00319 * 00320 * .. Scalar Arguments .. 00321 CHARACTER JOBZ, RANGE 00322 LOGICAL TRYRAC 00323 INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N 00324 DOUBLE PRECISION VL, VU 00325 * .. 00326 * .. Array Arguments .. 00327 INTEGER ISUPPZ( * ), IWORK( * ) 00328 DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ) 00329 DOUBLE PRECISION Z( LDZ, * ) 00330 * .. 00331 * 00332 * ===================================================================== 00333 * 00334 * .. Parameters .. 00335 DOUBLE PRECISION ZERO, ONE, FOUR, MINRGP 00336 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, 00337 $ FOUR = 4.0D0, 00338 $ MINRGP = 1.0D-3 ) 00339 * .. 00340 * .. Local Scalars .. 00341 LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY 00342 INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW, 00343 $ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD, 00344 $ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP, 00345 $ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT, 00346 $ NZCMIN, OFFSET, WBEGIN, WEND 00347 DOUBLE PRECISION BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN, 00348 $ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN, 00349 $ THRESH, TMP, TNRM, WL, WU 00350 * .. 00351 * .. 00352 * .. External Functions .. 00353 LOGICAL LSAME 00354 DOUBLE PRECISION DLAMCH, DLANST 00355 EXTERNAL LSAME, DLAMCH, DLANST 00356 * .. 00357 * .. External Subroutines .. 00358 EXTERNAL DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ, 00359 $ DLARRR, DLARRV, DLASRT, DSCAL, DSWAP, XERBLA 00360 * .. 00361 * .. Intrinsic Functions .. 00362 INTRINSIC MAX, MIN, SQRT 00363 00364 00365 * .. 00366 * .. Executable Statements .. 00367 * 00368 * Test the input parameters. 00369 * 00370 WANTZ = LSAME( JOBZ, 'V' ) 00371 ALLEIG = LSAME( RANGE, 'A' ) 00372 VALEIG = LSAME( RANGE, 'V' ) 00373 INDEIG = LSAME( RANGE, 'I' ) 00374 * 00375 LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) ) 00376 ZQUERY = ( NZC.EQ.-1 ) 00377 00378 * DSTEMR needs WORK of size 6*N, IWORK of size 3*N. 00379 * In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N. 00380 * Furthermore, DLARRV needs WORK of size 12*N, IWORK of size 7*N. 00381 IF( WANTZ ) THEN 00382 LWMIN = 18*N 00383 LIWMIN = 10*N 00384 ELSE 00385 * need less workspace if only the eigenvalues are wanted 00386 LWMIN = 12*N 00387 LIWMIN = 8*N 00388 ENDIF 00389 00390 WL = ZERO 00391 WU = ZERO 00392 IIL = 0 00393 IIU = 0 00394 00395 IF( VALEIG ) THEN 00396 * We do not reference VL, VU in the cases RANGE = 'I','A' 00397 * The interval (WL, WU] contains all the wanted eigenvalues. 00398 * It is either given by the user or computed in DLARRE. 00399 WL = VL 00400 WU = VU 00401 ELSEIF( INDEIG ) THEN 00402 * We do not reference IL, IU in the cases RANGE = 'V','A' 00403 IIL = IL 00404 IIU = IU 00405 ENDIF 00406 * 00407 INFO = 0 00408 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 00409 INFO = -1 00410 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 00411 INFO = -2 00412 ELSE IF( N.LT.0 ) THEN 00413 INFO = -3 00414 ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN 00415 INFO = -7 00416 ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN 00417 INFO = -8 00418 ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN 00419 INFO = -9 00420 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 00421 INFO = -13 00422 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 00423 INFO = -17 00424 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 00425 INFO = -19 00426 END IF 00427 * 00428 * Get machine constants. 00429 * 00430 SAFMIN = DLAMCH( 'Safe minimum' ) 00431 EPS = DLAMCH( 'Precision' ) 00432 SMLNUM = SAFMIN / EPS 00433 BIGNUM = ONE / SMLNUM 00434 RMIN = SQRT( SMLNUM ) 00435 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 00436 * 00437 IF( INFO.EQ.0 ) THEN 00438 WORK( 1 ) = LWMIN 00439 IWORK( 1 ) = LIWMIN 00440 * 00441 IF( WANTZ .AND. ALLEIG ) THEN 00442 NZCMIN = N 00443 ELSE IF( WANTZ .AND. VALEIG ) THEN 00444 CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN, 00445 $ NZCMIN, ITMP, ITMP2, INFO ) 00446 ELSE IF( WANTZ .AND. INDEIG ) THEN 00447 NZCMIN = IIU-IIL+1 00448 ELSE 00449 * WANTZ .EQ. FALSE. 00450 NZCMIN = 0 00451 ENDIF 00452 IF( ZQUERY .AND. INFO.EQ.0 ) THEN 00453 Z( 1,1 ) = NZCMIN 00454 ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN 00455 INFO = -14 00456 END IF 00457 END IF 00458 00459 IF( INFO.NE.0 ) THEN 00460 * 00461 CALL XERBLA( 'DSTEMR', -INFO ) 00462 * 00463 RETURN 00464 ELSE IF( LQUERY .OR. ZQUERY ) THEN 00465 RETURN 00466 END IF 00467 * 00468 * Handle N = 0, 1, and 2 cases immediately 00469 * 00470 M = 0 00471 IF( N.EQ.0 ) 00472 $ RETURN 00473 * 00474 IF( N.EQ.1 ) THEN 00475 IF( ALLEIG .OR. INDEIG ) THEN 00476 M = 1 00477 W( 1 ) = D( 1 ) 00478 ELSE 00479 IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN 00480 M = 1 00481 W( 1 ) = D( 1 ) 00482 END IF 00483 END IF 00484 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 00485 Z( 1, 1 ) = ONE 00486 ISUPPZ(1) = 1 00487 ISUPPZ(2) = 1 00488 END IF 00489 RETURN 00490 END IF 00491 * 00492 IF( N.EQ.2 ) THEN 00493 IF( .NOT.WANTZ ) THEN 00494 CALL DLAE2( D(1), E(1), D(2), R1, R2 ) 00495 ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 00496 CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN ) 00497 END IF 00498 IF( ALLEIG.OR. 00499 $ (VALEIG.AND.(R2.GT.WL).AND. 00500 $ (R2.LE.WU)).OR. 00501 $ (INDEIG.AND.(IIL.EQ.1)) ) THEN 00502 M = M+1 00503 W( M ) = R2 00504 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 00505 Z( 1, M ) = -SN 00506 Z( 2, M ) = CS 00507 * Note: At most one of SN and CS can be zero. 00508 IF (SN.NE.ZERO) THEN 00509 IF (CS.NE.ZERO) THEN 00510 ISUPPZ(2*M-1) = 1 00511 ISUPPZ(2*M) = 2 00512 ELSE 00513 ISUPPZ(2*M-1) = 1 00514 ISUPPZ(2*M) = 1 00515 END IF 00516 ELSE 00517 ISUPPZ(2*M-1) = 2 00518 ISUPPZ(2*M) = 2 00519 END IF 00520 ENDIF 00521 ENDIF 00522 IF( ALLEIG.OR. 00523 $ (VALEIG.AND.(R1.GT.WL).AND. 00524 $ (R1.LE.WU)).OR. 00525 $ (INDEIG.AND.(IIU.EQ.2)) ) THEN 00526 M = M+1 00527 W( M ) = R1 00528 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 00529 Z( 1, M ) = CS 00530 Z( 2, M ) = SN 00531 * Note: At most one of SN and CS can be zero. 00532 IF (SN.NE.ZERO) THEN 00533 IF (CS.NE.ZERO) THEN 00534 ISUPPZ(2*M-1) = 1 00535 ISUPPZ(2*M) = 2 00536 ELSE 00537 ISUPPZ(2*M-1) = 1 00538 ISUPPZ(2*M) = 1 00539 END IF 00540 ELSE 00541 ISUPPZ(2*M-1) = 2 00542 ISUPPZ(2*M) = 2 00543 END IF 00544 ENDIF 00545 ENDIF 00546 RETURN 00547 END IF 00548 00549 * Continue with general N 00550 00551 INDGRS = 1 00552 INDERR = 2*N + 1 00553 INDGP = 3*N + 1 00554 INDD = 4*N + 1 00555 INDE2 = 5*N + 1 00556 INDWRK = 6*N + 1 00557 * 00558 IINSPL = 1 00559 IINDBL = N + 1 00560 IINDW = 2*N + 1 00561 IINDWK = 3*N + 1 00562 * 00563 * Scale matrix to allowable range, if necessary. 00564 * The allowable range is related to the PIVMIN parameter; see the 00565 * comments in DLARRD. The preference for scaling small values 00566 * up is heuristic; we expect users' matrices not to be close to the 00567 * RMAX threshold. 00568 * 00569 SCALE = ONE 00570 TNRM = DLANST( 'M', N, D, E ) 00571 IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN 00572 SCALE = RMIN / TNRM 00573 ELSE IF( TNRM.GT.RMAX ) THEN 00574 SCALE = RMAX / TNRM 00575 END IF 00576 IF( SCALE.NE.ONE ) THEN 00577 CALL DSCAL( N, SCALE, D, 1 ) 00578 CALL DSCAL( N-1, SCALE, E, 1 ) 00579 TNRM = TNRM*SCALE 00580 IF( VALEIG ) THEN 00581 * If eigenvalues in interval have to be found, 00582 * scale (WL, WU] accordingly 00583 WL = WL*SCALE 00584 WU = WU*SCALE 00585 ENDIF 00586 END IF 00587 * 00588 * Compute the desired eigenvalues of the tridiagonal after splitting 00589 * into smaller subblocks if the corresponding off-diagonal elements 00590 * are small 00591 * THRESH is the splitting parameter for DLARRE 00592 * A negative THRESH forces the old splitting criterion based on the 00593 * size of the off-diagonal. A positive THRESH switches to splitting 00594 * which preserves relative accuracy. 00595 * 00596 IF( TRYRAC ) THEN 00597 * Test whether the matrix warrants the more expensive relative approach. 00598 CALL DLARRR( N, D, E, IINFO ) 00599 ELSE 00600 * The user does not care about relative accurately eigenvalues 00601 IINFO = -1 00602 ENDIF 00603 * Set the splitting criterion 00604 IF (IINFO.EQ.0) THEN 00605 THRESH = EPS 00606 ELSE 00607 THRESH = -EPS 00608 * relative accuracy is desired but T does not guarantee it 00609 TRYRAC = .FALSE. 00610 ENDIF 00611 * 00612 IF( TRYRAC ) THEN 00613 * Copy original diagonal, needed to guarantee relative accuracy 00614 CALL DCOPY(N,D,1,WORK(INDD),1) 00615 ENDIF 00616 * Store the squares of the offdiagonal values of T 00617 DO 5 J = 1, N-1 00618 WORK( INDE2+J-1 ) = E(J)**2 00619 5 CONTINUE 00620 00621 * Set the tolerance parameters for bisection 00622 IF( .NOT.WANTZ ) THEN 00623 * DLARRE computes the eigenvalues to full precision. 00624 RTOL1 = FOUR * EPS 00625 RTOL2 = FOUR * EPS 00626 ELSE 00627 * DLARRE computes the eigenvalues to less than full precision. 00628 * DLARRV will refine the eigenvalue approximations, and we can 00629 * need less accurate initial bisection in DLARRE. 00630 * Note: these settings do only affect the subset case and DLARRE 00631 RTOL1 = SQRT(EPS) 00632 RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS ) 00633 ENDIF 00634 CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E, 00635 $ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT, 00636 $ IWORK( IINSPL ), M, W, WORK( INDERR ), 00637 $ WORK( INDGP ), IWORK( IINDBL ), 00638 $ IWORK( IINDW ), WORK( INDGRS ), PIVMIN, 00639 $ WORK( INDWRK ), IWORK( IINDWK ), IINFO ) 00640 IF( IINFO.NE.0 ) THEN 00641 INFO = 10 + ABS( IINFO ) 00642 RETURN 00643 END IF 00644 * Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired 00645 * part of the spectrum. All desired eigenvalues are contained in 00646 * (WL,WU] 00647 00648 00649 IF( WANTZ ) THEN 00650 * 00651 * Compute the desired eigenvectors corresponding to the computed 00652 * eigenvalues 00653 * 00654 CALL DLARRV( N, WL, WU, D, E, 00655 $ PIVMIN, IWORK( IINSPL ), M, 00656 $ 1, M, MINRGP, RTOL1, RTOL2, 00657 $ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ), 00658 $ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ, 00659 $ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO ) 00660 IF( IINFO.NE.0 ) THEN 00661 INFO = 20 + ABS( IINFO ) 00662 RETURN 00663 END IF 00664 ELSE 00665 * DLARRE computes eigenvalues of the (shifted) root representation 00666 * DLARRV returns the eigenvalues of the unshifted matrix. 00667 * However, if the eigenvectors are not desired by the user, we need 00668 * to apply the corresponding shifts from DLARRE to obtain the 00669 * eigenvalues of the original matrix. 00670 DO 20 J = 1, M 00671 ITMP = IWORK( IINDBL+J-1 ) 00672 W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) ) 00673 20 CONTINUE 00674 END IF 00675 * 00676 00677 IF ( TRYRAC ) THEN 00678 * Refine computed eigenvalues so that they are relatively accurate 00679 * with respect to the original matrix T. 00680 IBEGIN = 1 00681 WBEGIN = 1 00682 DO 39 JBLK = 1, IWORK( IINDBL+M-1 ) 00683 IEND = IWORK( IINSPL+JBLK-1 ) 00684 IN = IEND - IBEGIN + 1 00685 WEND = WBEGIN - 1 00686 * check if any eigenvalues have to be refined in this block 00687 36 CONTINUE 00688 IF( WEND.LT.M ) THEN 00689 IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN 00690 WEND = WEND + 1 00691 GO TO 36 00692 END IF 00693 END IF 00694 IF( WEND.LT.WBEGIN ) THEN 00695 IBEGIN = IEND + 1 00696 GO TO 39 00697 END IF 00698 00699 OFFSET = IWORK(IINDW+WBEGIN-1)-1 00700 IFIRST = IWORK(IINDW+WBEGIN-1) 00701 ILAST = IWORK(IINDW+WEND-1) 00702 RTOL2 = FOUR * EPS 00703 CALL DLARRJ( IN, 00704 $ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1), 00705 $ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN), 00706 $ WORK( INDERR+WBEGIN-1 ), 00707 $ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN, 00708 $ TNRM, IINFO ) 00709 IBEGIN = IEND + 1 00710 WBEGIN = WEND + 1 00711 39 CONTINUE 00712 ENDIF 00713 * 00714 * If matrix was scaled, then rescale eigenvalues appropriately. 00715 * 00716 IF( SCALE.NE.ONE ) THEN 00717 CALL DSCAL( M, ONE / SCALE, W, 1 ) 00718 END IF 00719 * 00720 * If eigenvalues are not in increasing order, then sort them, 00721 * possibly along with eigenvectors. 00722 * 00723 IF( NSPLIT.GT.1 ) THEN 00724 IF( .NOT. WANTZ ) THEN 00725 CALL DLASRT( 'I', M, W, IINFO ) 00726 IF( IINFO.NE.0 ) THEN 00727 INFO = 3 00728 RETURN 00729 END IF 00730 ELSE 00731 DO 60 J = 1, M - 1 00732 I = 0 00733 TMP = W( J ) 00734 DO 50 JJ = J + 1, M 00735 IF( W( JJ ).LT.TMP ) THEN 00736 I = JJ 00737 TMP = W( JJ ) 00738 END IF 00739 50 CONTINUE 00740 IF( I.NE.0 ) THEN 00741 W( I ) = W( J ) 00742 W( J ) = TMP 00743 IF( WANTZ ) THEN 00744 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 00745 ITMP = ISUPPZ( 2*I-1 ) 00746 ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 ) 00747 ISUPPZ( 2*J-1 ) = ITMP 00748 ITMP = ISUPPZ( 2*I ) 00749 ISUPPZ( 2*I ) = ISUPPZ( 2*J ) 00750 ISUPPZ( 2*J ) = ITMP 00751 END IF 00752 END IF 00753 60 CONTINUE 00754 END IF 00755 ENDIF 00756 * 00757 * 00758 WORK( 1 ) = LWMIN 00759 IWORK( 1 ) = LIWMIN 00760 RETURN 00761 * 00762 * End of DSTEMR 00763 * 00764 END