00001 SUBROUTINE ZSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, 00002 $ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, 00003 $ IWORK, LIWORK, INFO ) 00004 IMPLICIT NONE 00005 * 00006 * -- LAPACK computational routine (version 3.2.1) -- 00007 * 00008 * -- April 2009 -- 00009 * 00010 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00011 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00012 * 00013 * .. Scalar Arguments .. 00014 CHARACTER JOBZ, RANGE 00015 LOGICAL TRYRAC 00016 INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N 00017 DOUBLE PRECISION VL, VU 00018 * .. 00019 * .. Array Arguments .. 00020 INTEGER ISUPPZ( * ), IWORK( * ) 00021 DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ) 00022 COMPLEX*16 Z( LDZ, * ) 00023 * .. 00024 * 00025 * Purpose 00026 * ======= 00027 * 00028 * ZSTEMR computes selected eigenvalues and, optionally, eigenvectors 00029 * of a real symmetric tridiagonal matrix T. Any such unreduced matrix has 00030 * a well defined set of pairwise different real eigenvalues, the corresponding 00031 * real eigenvectors are pairwise orthogonal. 00032 * 00033 * The spectrum may be computed either completely or partially by specifying 00034 * either an interval (VL,VU] or a range of indices IL:IU for the desired 00035 * eigenvalues. 00036 * 00037 * Depending on the number of desired eigenvalues, these are computed either 00038 * by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are 00039 * computed by the use of various suitable L D L^T factorizations near clusters 00040 * of close eigenvalues (referred to as RRRs, Relatively Robust 00041 * Representations). An informal sketch of the algorithm follows. 00042 * 00043 * For each unreduced block (submatrix) of T, 00044 * (a) Compute T - sigma I = L D L^T, so that L and D 00045 * define all the wanted eigenvalues to high relative accuracy. 00046 * This means that small relative changes in the entries of D and L 00047 * cause only small relative changes in the eigenvalues and 00048 * eigenvectors. The standard (unfactored) representation of the 00049 * tridiagonal matrix T does not have this property in general. 00050 * (b) Compute the eigenvalues to suitable accuracy. 00051 * If the eigenvectors are desired, the algorithm attains full 00052 * accuracy of the computed eigenvalues only right before 00053 * the corresponding vectors have to be computed, see steps c) and d). 00054 * (c) For each cluster of close eigenvalues, select a new 00055 * shift close to the cluster, find a new factorization, and refine 00056 * the shifted eigenvalues to suitable accuracy. 00057 * (d) For each eigenvalue with a large enough relative separation compute 00058 * the corresponding eigenvector by forming a rank revealing twisted 00059 * factorization. Go back to (c) for any clusters that remain. 00060 * 00061 * For more details, see: 00062 * - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations 00063 * to compute orthogonal eigenvectors of symmetric tridiagonal matrices," 00064 * Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. 00065 * - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and 00066 * Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, 00067 * 2004. Also LAPACK Working Note 154. 00068 * - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric 00069 * tridiagonal eigenvalue/eigenvector problem", 00070 * Computer Science Division Technical Report No. UCB/CSD-97-971, 00071 * UC Berkeley, May 1997. 00072 * 00073 * Further Details 00074 * 1.ZSTEMR works only on machines which follow IEEE-754 00075 * floating-point standard in their handling of infinities and NaNs. 00076 * This permits the use of efficient inner loops avoiding a check for 00077 * zero divisors. 00078 * 00079 * 2. LAPACK routines can be used to reduce a complex Hermitean matrix to 00080 * real symmetric tridiagonal form. 00081 * 00082 * (Any complex Hermitean tridiagonal matrix has real values on its diagonal 00083 * and potentially complex numbers on its off-diagonals. By applying a 00084 * similarity transform with an appropriate diagonal matrix 00085 * diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean 00086 * matrix can be transformed into a real symmetric matrix and complex 00087 * arithmetic can be entirely avoided.) 00088 * 00089 * While the eigenvectors of the real symmetric tridiagonal matrix are real, 00090 * the eigenvectors of original complex Hermitean matrix have complex entries 00091 * in general. 00092 * Since LAPACK drivers overwrite the matrix data with the eigenvectors, 00093 * ZSTEMR accepts complex workspace to facilitate interoperability 00094 * with ZUNMTR or ZUPMTR. 00095 * 00096 * Arguments 00097 * ========= 00098 * 00099 * JOBZ (input) CHARACTER*1 00100 * = 'N': Compute eigenvalues only; 00101 * = 'V': Compute eigenvalues and eigenvectors. 00102 * 00103 * RANGE (input) CHARACTER*1 00104 * = 'A': all eigenvalues will be found. 00105 * = 'V': all eigenvalues in the half-open interval (VL,VU] 00106 * will be found. 00107 * = 'I': the IL-th through IU-th eigenvalues will be found. 00108 * 00109 * N (input) INTEGER 00110 * The order of the matrix. N >= 0. 00111 * 00112 * D (input/output) DOUBLE PRECISION array, dimension (N) 00113 * On entry, the N diagonal elements of the tridiagonal matrix 00114 * T. On exit, D is overwritten. 00115 * 00116 * E (input/output) DOUBLE PRECISION array, dimension (N) 00117 * On entry, the (N-1) subdiagonal elements of the tridiagonal 00118 * matrix T in elements 1 to N-1 of E. E(N) need not be set on 00119 * input, but is used internally as workspace. 00120 * On exit, E is overwritten. 00121 * 00122 * VL (input) DOUBLE PRECISION 00123 * VU (input) DOUBLE PRECISION 00124 * If RANGE='V', the lower and upper bounds of the interval to 00125 * be searched for eigenvalues. VL < VU. 00126 * Not referenced if RANGE = 'A' or 'I'. 00127 * 00128 * IL (input) INTEGER 00129 * IU (input) INTEGER 00130 * If RANGE='I', the indices (in ascending order) of the 00131 * smallest and largest eigenvalues to be returned. 00132 * 1 <= IL <= IU <= N, if N > 0. 00133 * Not referenced if RANGE = 'A' or 'V'. 00134 * 00135 * M (output) INTEGER 00136 * The total number of eigenvalues found. 0 <= M <= N. 00137 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 00138 * 00139 * W (output) DOUBLE PRECISION array, dimension (N) 00140 * The first M elements contain the selected eigenvalues in 00141 * ascending order. 00142 * 00143 * Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M) ) 00144 * If JOBZ = 'V', and if INFO = 0, then the first M columns of Z 00145 * contain the orthonormal eigenvectors of the matrix T 00146 * corresponding to the selected eigenvalues, with the i-th 00147 * column of Z holding the eigenvector associated with W(i). 00148 * If JOBZ = 'N', then Z is not referenced. 00149 * Note: the user must ensure that at least max(1,M) columns are 00150 * supplied in the array Z; if RANGE = 'V', the exact value of M 00151 * is not known in advance and can be computed with a workspace 00152 * query by setting NZC = -1, see below. 00153 * 00154 * LDZ (input) INTEGER 00155 * The leading dimension of the array Z. LDZ >= 1, and if 00156 * JOBZ = 'V', then LDZ >= max(1,N). 00157 * 00158 * NZC (input) INTEGER 00159 * The number of eigenvectors to be held in the array Z. 00160 * If RANGE = 'A', then NZC >= max(1,N). 00161 * If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]. 00162 * If RANGE = 'I', then NZC >= IU-IL+1. 00163 * If NZC = -1, then a workspace query is assumed; the 00164 * routine calculates the number of columns of the array Z that 00165 * are needed to hold the eigenvectors. 00166 * This value is returned as the first entry of the Z array, and 00167 * no error message related to NZC is issued by XERBLA. 00168 * 00169 * ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) ) 00170 * The support of the eigenvectors in Z, i.e., the indices 00171 * indicating the nonzero elements in Z. The i-th computed eigenvector 00172 * is nonzero only in elements ISUPPZ( 2*i-1 ) through 00173 * ISUPPZ( 2*i ). This is relevant in the case when the matrix 00174 * is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0. 00175 * 00176 * TRYRAC (input/output) LOGICAL 00177 * If TRYRAC.EQ..TRUE., indicates that the code should check whether 00178 * the tridiagonal matrix defines its eigenvalues to high relative 00179 * accuracy. If so, the code uses relative-accuracy preserving 00180 * algorithms that might be (a bit) slower depending on the matrix. 00181 * If the matrix does not define its eigenvalues to high relative 00182 * accuracy, the code can uses possibly faster algorithms. 00183 * If TRYRAC.EQ..FALSE., the code is not required to guarantee 00184 * relatively accurate eigenvalues and can use the fastest possible 00185 * techniques. 00186 * On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix 00187 * does not define its eigenvalues to high relative accuracy. 00188 * 00189 * WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) 00190 * On exit, if INFO = 0, WORK(1) returns the optimal 00191 * (and minimal) LWORK. 00192 * 00193 * LWORK (input) INTEGER 00194 * The dimension of the array WORK. LWORK >= max(1,18*N) 00195 * if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'. 00196 * If LWORK = -1, then a workspace query is assumed; the routine 00197 * only calculates the optimal size of the WORK array, returns 00198 * this value as the first entry of the WORK array, and no error 00199 * message related to LWORK is issued by XERBLA. 00200 * 00201 * IWORK (workspace/output) INTEGER array, dimension (LIWORK) 00202 * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 00203 * 00204 * LIWORK (input) INTEGER 00205 * The dimension of the array IWORK. LIWORK >= max(1,10*N) 00206 * if the eigenvectors are desired, and LIWORK >= max(1,8*N) 00207 * if only the eigenvalues are to be computed. 00208 * If LIWORK = -1, then a workspace query is assumed; the 00209 * routine only calculates the optimal size of the IWORK array, 00210 * returns this value as the first entry of the IWORK array, and 00211 * no error message related to LIWORK is issued by XERBLA. 00212 * 00213 * INFO (output) INTEGER 00214 * On exit, INFO 00215 * = 0: successful exit 00216 * < 0: if INFO = -i, the i-th argument had an illegal value 00217 * > 0: if INFO = 1X, internal error in DLARRE, 00218 * if INFO = 2X, internal error in ZLARRV. 00219 * Here, the digit X = ABS( IINFO ) < 10, where IINFO is 00220 * the nonzero error code returned by DLARRE or 00221 * ZLARRV, respectively. 00222 * 00223 * 00224 * Further Details 00225 * =============== 00226 * 00227 * Based on contributions by 00228 * Beresford Parlett, University of California, Berkeley, USA 00229 * Jim Demmel, University of California, Berkeley, USA 00230 * Inderjit Dhillon, University of Texas, Austin, USA 00231 * Osni Marques, LBNL/NERSC, USA 00232 * Christof Voemel, University of California, Berkeley, USA 00233 * 00234 * ===================================================================== 00235 * 00236 * .. Parameters .. 00237 DOUBLE PRECISION ZERO, ONE, FOUR, MINRGP 00238 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, 00239 $ FOUR = 4.0D0, 00240 $ MINRGP = 1.0D-3 ) 00241 * .. 00242 * .. Local Scalars .. 00243 LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY 00244 INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW, 00245 $ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD, 00246 $ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP, 00247 $ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT, 00248 $ NZCMIN, OFFSET, WBEGIN, WEND 00249 DOUBLE PRECISION BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN, 00250 $ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN, 00251 $ THRESH, TMP, TNRM, WL, WU 00252 * .. 00253 * .. 00254 * .. External Functions .. 00255 LOGICAL LSAME 00256 DOUBLE PRECISION DLAMCH, DLANST 00257 EXTERNAL LSAME, DLAMCH, DLANST 00258 * .. 00259 * .. External Subroutines .. 00260 EXTERNAL DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ, 00261 $ DLARRR, DLASRT, DSCAL, XERBLA, ZLARRV, ZSWAP 00262 * .. 00263 * .. Intrinsic Functions .. 00264 INTRINSIC MAX, MIN, SQRT 00265 00266 00267 * .. 00268 * .. Executable Statements .. 00269 * 00270 * Test the input parameters. 00271 * 00272 WANTZ = LSAME( JOBZ, 'V' ) 00273 ALLEIG = LSAME( RANGE, 'A' ) 00274 VALEIG = LSAME( RANGE, 'V' ) 00275 INDEIG = LSAME( RANGE, 'I' ) 00276 * 00277 LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) ) 00278 ZQUERY = ( NZC.EQ.-1 ) 00279 00280 * DSTEMR needs WORK of size 6*N, IWORK of size 3*N. 00281 * In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N. 00282 * Furthermore, ZLARRV needs WORK of size 12*N, IWORK of size 7*N. 00283 IF( WANTZ ) THEN 00284 LWMIN = 18*N 00285 LIWMIN = 10*N 00286 ELSE 00287 * need less workspace if only the eigenvalues are wanted 00288 LWMIN = 12*N 00289 LIWMIN = 8*N 00290 ENDIF 00291 00292 WL = ZERO 00293 WU = ZERO 00294 IIL = 0 00295 IIU = 0 00296 00297 IF( VALEIG ) THEN 00298 * We do not reference VL, VU in the cases RANGE = 'I','A' 00299 * The interval (WL, WU] contains all the wanted eigenvalues. 00300 * It is either given by the user or computed in DLARRE. 00301 WL = VL 00302 WU = VU 00303 ELSEIF( INDEIG ) THEN 00304 * We do not reference IL, IU in the cases RANGE = 'V','A' 00305 IIL = IL 00306 IIU = IU 00307 ENDIF 00308 * 00309 INFO = 0 00310 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 00311 INFO = -1 00312 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 00313 INFO = -2 00314 ELSE IF( N.LT.0 ) THEN 00315 INFO = -3 00316 ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN 00317 INFO = -7 00318 ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN 00319 INFO = -8 00320 ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN 00321 INFO = -9 00322 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 00323 INFO = -13 00324 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 00325 INFO = -17 00326 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 00327 INFO = -19 00328 END IF 00329 * 00330 * Get machine constants. 00331 * 00332 SAFMIN = DLAMCH( 'Safe minimum' ) 00333 EPS = DLAMCH( 'Precision' ) 00334 SMLNUM = SAFMIN / EPS 00335 BIGNUM = ONE / SMLNUM 00336 RMIN = SQRT( SMLNUM ) 00337 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 00338 * 00339 IF( INFO.EQ.0 ) THEN 00340 WORK( 1 ) = LWMIN 00341 IWORK( 1 ) = LIWMIN 00342 * 00343 IF( WANTZ .AND. ALLEIG ) THEN 00344 NZCMIN = N 00345 ELSE IF( WANTZ .AND. VALEIG ) THEN 00346 CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN, 00347 $ NZCMIN, ITMP, ITMP2, INFO ) 00348 ELSE IF( WANTZ .AND. INDEIG ) THEN 00349 NZCMIN = IIU-IIL+1 00350 ELSE 00351 * WANTZ .EQ. FALSE. 00352 NZCMIN = 0 00353 ENDIF 00354 IF( ZQUERY .AND. INFO.EQ.0 ) THEN 00355 Z( 1,1 ) = NZCMIN 00356 ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN 00357 INFO = -14 00358 END IF 00359 END IF 00360 00361 IF( INFO.NE.0 ) THEN 00362 * 00363 CALL XERBLA( 'ZSTEMR', -INFO ) 00364 * 00365 RETURN 00366 ELSE IF( LQUERY .OR. ZQUERY ) THEN 00367 RETURN 00368 END IF 00369 * 00370 * Handle N = 0, 1, and 2 cases immediately 00371 * 00372 M = 0 00373 IF( N.EQ.0 ) 00374 $ RETURN 00375 * 00376 IF( N.EQ.1 ) THEN 00377 IF( ALLEIG .OR. INDEIG ) THEN 00378 M = 1 00379 W( 1 ) = D( 1 ) 00380 ELSE 00381 IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN 00382 M = 1 00383 W( 1 ) = D( 1 ) 00384 END IF 00385 END IF 00386 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 00387 Z( 1, 1 ) = ONE 00388 ISUPPZ(1) = 1 00389 ISUPPZ(2) = 1 00390 END IF 00391 RETURN 00392 END IF 00393 * 00394 IF( N.EQ.2 ) THEN 00395 IF( .NOT.WANTZ ) THEN 00396 CALL DLAE2( D(1), E(1), D(2), R1, R2 ) 00397 ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 00398 CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN ) 00399 END IF 00400 IF( ALLEIG.OR. 00401 $ (VALEIG.AND.(R2.GT.WL).AND. 00402 $ (R2.LE.WU)).OR. 00403 $ (INDEIG.AND.(IIL.EQ.1)) ) THEN 00404 M = M+1 00405 W( M ) = R2 00406 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 00407 Z( 1, M ) = -SN 00408 Z( 2, M ) = CS 00409 * Note: At most one of SN and CS can be zero. 00410 IF (SN.NE.ZERO) THEN 00411 IF (CS.NE.ZERO) THEN 00412 ISUPPZ(2*M-1) = 1 00413 ISUPPZ(2*M-1) = 2 00414 ELSE 00415 ISUPPZ(2*M-1) = 1 00416 ISUPPZ(2*M-1) = 1 00417 END IF 00418 ELSE 00419 ISUPPZ(2*M-1) = 2 00420 ISUPPZ(2*M) = 2 00421 END IF 00422 ENDIF 00423 ENDIF 00424 IF( ALLEIG.OR. 00425 $ (VALEIG.AND.(R1.GT.WL).AND. 00426 $ (R1.LE.WU)).OR. 00427 $ (INDEIG.AND.(IIU.EQ.2)) ) THEN 00428 M = M+1 00429 W( M ) = R1 00430 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 00431 Z( 1, M ) = CS 00432 Z( 2, M ) = SN 00433 * Note: At most one of SN and CS can be zero. 00434 IF (SN.NE.ZERO) THEN 00435 IF (CS.NE.ZERO) THEN 00436 ISUPPZ(2*M-1) = 1 00437 ISUPPZ(2*M-1) = 2 00438 ELSE 00439 ISUPPZ(2*M-1) = 1 00440 ISUPPZ(2*M-1) = 1 00441 END IF 00442 ELSE 00443 ISUPPZ(2*M-1) = 2 00444 ISUPPZ(2*M) = 2 00445 END IF 00446 ENDIF 00447 ENDIF 00448 RETURN 00449 END IF 00450 00451 * Continue with general N 00452 00453 INDGRS = 1 00454 INDERR = 2*N + 1 00455 INDGP = 3*N + 1 00456 INDD = 4*N + 1 00457 INDE2 = 5*N + 1 00458 INDWRK = 6*N + 1 00459 * 00460 IINSPL = 1 00461 IINDBL = N + 1 00462 IINDW = 2*N + 1 00463 IINDWK = 3*N + 1 00464 * 00465 * Scale matrix to allowable range, if necessary. 00466 * The allowable range is related to the PIVMIN parameter; see the 00467 * comments in DLARRD. The preference for scaling small values 00468 * up is heuristic; we expect users' matrices not to be close to the 00469 * RMAX threshold. 00470 * 00471 SCALE = ONE 00472 TNRM = DLANST( 'M', N, D, E ) 00473 IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN 00474 SCALE = RMIN / TNRM 00475 ELSE IF( TNRM.GT.RMAX ) THEN 00476 SCALE = RMAX / TNRM 00477 END IF 00478 IF( SCALE.NE.ONE ) THEN 00479 CALL DSCAL( N, SCALE, D, 1 ) 00480 CALL DSCAL( N-1, SCALE, E, 1 ) 00481 TNRM = TNRM*SCALE 00482 IF( VALEIG ) THEN 00483 * If eigenvalues in interval have to be found, 00484 * scale (WL, WU] accordingly 00485 WL = WL*SCALE 00486 WU = WU*SCALE 00487 ENDIF 00488 END IF 00489 * 00490 * Compute the desired eigenvalues of the tridiagonal after splitting 00491 * into smaller subblocks if the corresponding off-diagonal elements 00492 * are small 00493 * THRESH is the splitting parameter for DLARRE 00494 * A negative THRESH forces the old splitting criterion based on the 00495 * size of the off-diagonal. A positive THRESH switches to splitting 00496 * which preserves relative accuracy. 00497 * 00498 IF( TRYRAC ) THEN 00499 * Test whether the matrix warrants the more expensive relative approach. 00500 CALL DLARRR( N, D, E, IINFO ) 00501 ELSE 00502 * The user does not care about relative accurately eigenvalues 00503 IINFO = -1 00504 ENDIF 00505 * Set the splitting criterion 00506 IF (IINFO.EQ.0) THEN 00507 THRESH = EPS 00508 ELSE 00509 THRESH = -EPS 00510 * relative accuracy is desired but T does not guarantee it 00511 TRYRAC = .FALSE. 00512 ENDIF 00513 * 00514 IF( TRYRAC ) THEN 00515 * Copy original diagonal, needed to guarantee relative accuracy 00516 CALL DCOPY(N,D,1,WORK(INDD),1) 00517 ENDIF 00518 * Store the squares of the offdiagonal values of T 00519 DO 5 J = 1, N-1 00520 WORK( INDE2+J-1 ) = E(J)**2 00521 5 CONTINUE 00522 00523 * Set the tolerance parameters for bisection 00524 IF( .NOT.WANTZ ) THEN 00525 * DLARRE computes the eigenvalues to full precision. 00526 RTOL1 = FOUR * EPS 00527 RTOL2 = FOUR * EPS 00528 ELSE 00529 * DLARRE computes the eigenvalues to less than full precision. 00530 * ZLARRV will refine the eigenvalue approximations, and we only 00531 * need less accurate initial bisection in DLARRE. 00532 * Note: these settings do only affect the subset case and DLARRE 00533 RTOL1 = SQRT(EPS) 00534 RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS ) 00535 ENDIF 00536 CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E, 00537 $ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT, 00538 $ IWORK( IINSPL ), M, W, WORK( INDERR ), 00539 $ WORK( INDGP ), IWORK( IINDBL ), 00540 $ IWORK( IINDW ), WORK( INDGRS ), PIVMIN, 00541 $ WORK( INDWRK ), IWORK( IINDWK ), IINFO ) 00542 IF( IINFO.NE.0 ) THEN 00543 INFO = 10 + ABS( IINFO ) 00544 RETURN 00545 END IF 00546 * Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired 00547 * part of the spectrum. All desired eigenvalues are contained in 00548 * (WL,WU] 00549 00550 00551 IF( WANTZ ) THEN 00552 * 00553 * Compute the desired eigenvectors corresponding to the computed 00554 * eigenvalues 00555 * 00556 CALL ZLARRV( N, WL, WU, D, E, 00557 $ PIVMIN, IWORK( IINSPL ), M, 00558 $ 1, M, MINRGP, RTOL1, RTOL2, 00559 $ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ), 00560 $ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ, 00561 $ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO ) 00562 IF( IINFO.NE.0 ) THEN 00563 INFO = 20 + ABS( IINFO ) 00564 RETURN 00565 END IF 00566 ELSE 00567 * DLARRE computes eigenvalues of the (shifted) root representation 00568 * ZLARRV returns the eigenvalues of the unshifted matrix. 00569 * However, if the eigenvectors are not desired by the user, we need 00570 * to apply the corresponding shifts from DLARRE to obtain the 00571 * eigenvalues of the original matrix. 00572 DO 20 J = 1, M 00573 ITMP = IWORK( IINDBL+J-1 ) 00574 W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) ) 00575 20 CONTINUE 00576 END IF 00577 * 00578 00579 IF ( TRYRAC ) THEN 00580 * Refine computed eigenvalues so that they are relatively accurate 00581 * with respect to the original matrix T. 00582 IBEGIN = 1 00583 WBEGIN = 1 00584 DO 39 JBLK = 1, IWORK( IINDBL+M-1 ) 00585 IEND = IWORK( IINSPL+JBLK-1 ) 00586 IN = IEND - IBEGIN + 1 00587 WEND = WBEGIN - 1 00588 * check if any eigenvalues have to be refined in this block 00589 36 CONTINUE 00590 IF( WEND.LT.M ) THEN 00591 IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN 00592 WEND = WEND + 1 00593 GO TO 36 00594 END IF 00595 END IF 00596 IF( WEND.LT.WBEGIN ) THEN 00597 IBEGIN = IEND + 1 00598 GO TO 39 00599 END IF 00600 00601 OFFSET = IWORK(IINDW+WBEGIN-1)-1 00602 IFIRST = IWORK(IINDW+WBEGIN-1) 00603 ILAST = IWORK(IINDW+WEND-1) 00604 RTOL2 = FOUR * EPS 00605 CALL DLARRJ( IN, 00606 $ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1), 00607 $ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN), 00608 $ WORK( INDERR+WBEGIN-1 ), 00609 $ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN, 00610 $ TNRM, IINFO ) 00611 IBEGIN = IEND + 1 00612 WBEGIN = WEND + 1 00613 39 CONTINUE 00614 ENDIF 00615 * 00616 * If matrix was scaled, then rescale eigenvalues appropriately. 00617 * 00618 IF( SCALE.NE.ONE ) THEN 00619 CALL DSCAL( M, ONE / SCALE, W, 1 ) 00620 END IF 00621 * 00622 * If eigenvalues are not in increasing order, then sort them, 00623 * possibly along with eigenvectors. 00624 * 00625 IF( NSPLIT.GT.1 ) THEN 00626 IF( .NOT. WANTZ ) THEN 00627 CALL DLASRT( 'I', M, W, IINFO ) 00628 IF( IINFO.NE.0 ) THEN 00629 INFO = 3 00630 RETURN 00631 END IF 00632 ELSE 00633 DO 60 J = 1, M - 1 00634 I = 0 00635 TMP = W( J ) 00636 DO 50 JJ = J + 1, M 00637 IF( W( JJ ).LT.TMP ) THEN 00638 I = JJ 00639 TMP = W( JJ ) 00640 END IF 00641 50 CONTINUE 00642 IF( I.NE.0 ) THEN 00643 W( I ) = W( J ) 00644 W( J ) = TMP 00645 IF( WANTZ ) THEN 00646 CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 00647 ITMP = ISUPPZ( 2*I-1 ) 00648 ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 ) 00649 ISUPPZ( 2*J-1 ) = ITMP 00650 ITMP = ISUPPZ( 2*I ) 00651 ISUPPZ( 2*I ) = ISUPPZ( 2*J ) 00652 ISUPPZ( 2*J ) = ITMP 00653 END IF 00654 END IF 00655 60 CONTINUE 00656 END IF 00657 ENDIF 00658 * 00659 * 00660 WORK( 1 ) = LWMIN 00661 IWORK( 1 ) = LIWMIN 00662 RETURN 00663 * 00664 * End of ZSTEMR 00665 * 00666 END
1.7.2