00001 SUBROUTINE SSTEMR( 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 3.2.2) -- 00007 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00008 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00009 * -- June 2010 -- 00010 * 00011 * .. Scalar Arguments .. 00012 CHARACTER JOBZ, RANGE 00013 LOGICAL TRYRAC 00014 INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N 00015 REAL VL, VU 00016 * .. 00017 * .. Array Arguments .. 00018 INTEGER ISUPPZ( * ), IWORK( * ) 00019 REAL D( * ), E( * ), W( * ), WORK( * ) 00020 REAL Z( LDZ, * ) 00021 * .. 00022 * 00023 * Purpose 00024 * ======= 00025 * 00026 * SSTEMR computes selected eigenvalues and, optionally, eigenvectors 00027 * of a real symmetric tridiagonal matrix T. Any such unreduced matrix has 00028 * a well defined set of pairwise different real eigenvalues, the corresponding 00029 * real eigenvectors are pairwise orthogonal. 00030 * 00031 * The spectrum may be computed either completely or partially by specifying 00032 * either an interval (VL,VU] or a range of indices IL:IU for the desired 00033 * eigenvalues. 00034 * 00035 * Depending on the number of desired eigenvalues, these are computed either 00036 * by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are 00037 * computed by the use of various suitable L D L^T factorizations near clusters 00038 * of close eigenvalues (referred to as RRRs, Relatively Robust 00039 * Representations). An informal sketch of the algorithm follows. 00040 * 00041 * For each unreduced block (submatrix) of T, 00042 * (a) Compute T - sigma I = L D L^T, so that L and D 00043 * define all the wanted eigenvalues to high relative accuracy. 00044 * This means that small relative changes in the entries of D and L 00045 * cause only small relative changes in the eigenvalues and 00046 * eigenvectors. The standard (unfactored) representation of the 00047 * tridiagonal matrix T does not have this property in general. 00048 * (b) Compute the eigenvalues to suitable accuracy. 00049 * If the eigenvectors are desired, the algorithm attains full 00050 * accuracy of the computed eigenvalues only right before 00051 * the corresponding vectors have to be computed, see steps c) and d). 00052 * (c) For each cluster of close eigenvalues, select a new 00053 * shift close to the cluster, find a new factorization, and refine 00054 * the shifted eigenvalues to suitable accuracy. 00055 * (d) For each eigenvalue with a large enough relative separation compute 00056 * the corresponding eigenvector by forming a rank revealing twisted 00057 * factorization. Go back to (c) for any clusters that remain. 00058 * 00059 * For more details, see: 00060 * - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations 00061 * to compute orthogonal eigenvectors of symmetric tridiagonal matrices," 00062 * Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. 00063 * - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and 00064 * Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, 00065 * 2004. Also LAPACK Working Note 154. 00066 * - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric 00067 * tridiagonal eigenvalue/eigenvector problem", 00068 * Computer Science Division Technical Report No. UCB/CSD-97-971, 00069 * UC Berkeley, May 1997. 00070 * 00071 * Further Details 00072 * 1.SSTEMR works only on machines which follow IEEE-754 00073 * floating-point standard in their handling of infinities and NaNs. 00074 * This permits the use of efficient inner loops avoiding a check for 00075 * zero divisors. 00076 * 00077 * Arguments 00078 * ========= 00079 * 00080 * JOBZ (input) CHARACTER*1 00081 * = 'N': Compute eigenvalues only; 00082 * = 'V': Compute eigenvalues and eigenvectors. 00083 * 00084 * RANGE (input) CHARACTER*1 00085 * = 'A': all eigenvalues will be found. 00086 * = 'V': all eigenvalues in the half-open interval (VL,VU] 00087 * will be found. 00088 * = 'I': the IL-th through IU-th eigenvalues will be found. 00089 * 00090 * N (input) INTEGER 00091 * The order of the matrix. N >= 0. 00092 * 00093 * D (input/output) REAL array, dimension (N) 00094 * On entry, the N diagonal elements of the tridiagonal matrix 00095 * T. On exit, D is overwritten. 00096 * 00097 * E (input/output) REAL array, dimension (N) 00098 * On entry, the (N-1) subdiagonal elements of the tridiagonal 00099 * matrix T in elements 1 to N-1 of E. E(N) need not be set on 00100 * input, but is used internally as workspace. 00101 * On exit, E is overwritten. 00102 * 00103 * VL (input) REAL 00104 * VU (input) REAL 00105 * If RANGE='V', the lower and upper bounds of the interval to 00106 * be searched for eigenvalues. VL < VU. 00107 * Not referenced if RANGE = 'A' or 'I'. 00108 * 00109 * IL (input) INTEGER 00110 * IU (input) INTEGER 00111 * If RANGE='I', the indices (in ascending order) of the 00112 * smallest and largest eigenvalues to be returned. 00113 * 1 <= IL <= IU <= N, if N > 0. 00114 * Not referenced if RANGE = 'A' or 'V'. 00115 * 00116 * M (output) INTEGER 00117 * The total number of eigenvalues found. 0 <= M <= N. 00118 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 00119 * 00120 * W (output) REAL array, dimension (N) 00121 * The first M elements contain the selected eigenvalues in 00122 * ascending order. 00123 * 00124 * Z (output) REAL array, dimension (LDZ, max(1,M) ) 00125 * If JOBZ = 'V', and if INFO = 0, then the first M columns of Z 00126 * contain the orthonormal eigenvectors of the matrix T 00127 * corresponding to the selected eigenvalues, with the i-th 00128 * column of Z holding the eigenvector associated with W(i). 00129 * If JOBZ = 'N', then Z is not referenced. 00130 * Note: the user must ensure that at least max(1,M) columns are 00131 * supplied in the array Z; if RANGE = 'V', the exact value of M 00132 * is not known in advance and can be computed with a workspace 00133 * query by setting NZC = -1, see below. 00134 * 00135 * LDZ (input) INTEGER 00136 * The leading dimension of the array Z. LDZ >= 1, and if 00137 * JOBZ = 'V', then LDZ >= max(1,N). 00138 * 00139 * NZC (input) INTEGER 00140 * The number of eigenvectors to be held in the array Z. 00141 * If RANGE = 'A', then NZC >= max(1,N). 00142 * If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]. 00143 * If RANGE = 'I', then NZC >= IU-IL+1. 00144 * If NZC = -1, then a workspace query is assumed; the 00145 * routine calculates the number of columns of the array Z that 00146 * are needed to hold the eigenvectors. 00147 * This value is returned as the first entry of the Z array, and 00148 * no error message related to NZC is issued by XERBLA. 00149 * 00150 * ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) ) 00151 * The support of the eigenvectors in Z, i.e., the indices 00152 * indicating the nonzero elements in Z. The i-th computed eigenvector 00153 * is nonzero only in elements ISUPPZ( 2*i-1 ) through 00154 * ISUPPZ( 2*i ). This is relevant in the case when the matrix 00155 * is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0. 00156 * 00157 * TRYRAC (input/output) LOGICAL 00158 * If TRYRAC.EQ..TRUE., indicates that the code should check whether 00159 * the tridiagonal matrix defines its eigenvalues to high relative 00160 * accuracy. If so, the code uses relative-accuracy preserving 00161 * algorithms that might be (a bit) slower depending on the matrix. 00162 * If the matrix does not define its eigenvalues to high relative 00163 * accuracy, the code can uses possibly faster algorithms. 00164 * If TRYRAC.EQ..FALSE., the code is not required to guarantee 00165 * relatively accurate eigenvalues and can use the fastest possible 00166 * techniques. 00167 * On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix 00168 * does not define its eigenvalues to high relative accuracy. 00169 * 00170 * WORK (workspace/output) REAL array, dimension (LWORK) 00171 * On exit, if INFO = 0, WORK(1) returns the optimal 00172 * (and minimal) LWORK. 00173 * 00174 * LWORK (input) INTEGER 00175 * The dimension of the array WORK. LWORK >= max(1,18*N) 00176 * if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'. 00177 * If LWORK = -1, then a workspace query is assumed; the routine 00178 * only calculates the optimal size of the WORK array, returns 00179 * this value as the first entry of the WORK array, and no error 00180 * message related to LWORK is issued by XERBLA. 00181 * 00182 * IWORK (workspace/output) INTEGER array, dimension (LIWORK) 00183 * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 00184 * 00185 * LIWORK (input) INTEGER 00186 * The dimension of the array IWORK. LIWORK >= max(1,10*N) 00187 * if the eigenvectors are desired, and LIWORK >= max(1,8*N) 00188 * if only the eigenvalues are to be computed. 00189 * If LIWORK = -1, then a workspace query is assumed; the 00190 * routine only calculates the optimal size of the IWORK array, 00191 * returns this value as the first entry of the IWORK array, and 00192 * no error message related to LIWORK is issued by XERBLA. 00193 * 00194 * INFO (output) INTEGER 00195 * On exit, INFO 00196 * = 0: successful exit 00197 * < 0: if INFO = -i, the i-th argument had an illegal value 00198 * > 0: if INFO = 1X, internal error in SLARRE, 00199 * if INFO = 2X, internal error in SLARRV. 00200 * Here, the digit X = ABS( IINFO ) < 10, where IINFO is 00201 * the nonzero error code returned by SLARRE or 00202 * SLARRV, respectively. 00203 * 00204 * 00205 * Further Details 00206 * =============== 00207 * 00208 * Based on contributions by 00209 * Beresford Parlett, University of California, Berkeley, USA 00210 * Jim Demmel, University of California, Berkeley, USA 00211 * Inderjit Dhillon, University of Texas, Austin, USA 00212 * Osni Marques, LBNL/NERSC, USA 00213 * Christof Voemel, University of California, Berkeley, USA 00214 * 00215 * ===================================================================== 00216 * 00217 * .. Parameters .. 00218 REAL ZERO, ONE, FOUR, MINRGP 00219 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, 00220 $ FOUR = 4.0E0, 00221 $ MINRGP = 3.0E-3 ) 00222 * .. 00223 * .. Local Scalars .. 00224 LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY 00225 INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW, 00226 $ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD, 00227 $ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP, 00228 $ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT, 00229 $ NZCMIN, OFFSET, WBEGIN, WEND 00230 REAL BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN, 00231 $ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN, 00232 $ THRESH, TMP, TNRM, WL, WU 00233 * .. 00234 * .. 00235 * .. External Functions .. 00236 LOGICAL LSAME 00237 REAL SLAMCH, SLANST 00238 EXTERNAL LSAME, SLAMCH, SLANST 00239 * .. 00240 * .. External Subroutines .. 00241 EXTERNAL SCOPY, SLAE2, SLAEV2, SLARRC, SLARRE, SLARRJ, 00242 $ SLARRR, SLARRV, SLASRT, SSCAL, SSWAP, XERBLA 00243 * .. 00244 * .. Intrinsic Functions .. 00245 INTRINSIC MAX, MIN, SQRT 00246 * .. 00247 * .. Executable Statements .. 00248 * 00249 * Test the input parameters. 00250 * 00251 WANTZ = LSAME( JOBZ, 'V' ) 00252 ALLEIG = LSAME( RANGE, 'A' ) 00253 VALEIG = LSAME( RANGE, 'V' ) 00254 INDEIG = LSAME( RANGE, 'I' ) 00255 * 00256 LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) ) 00257 ZQUERY = ( NZC.EQ.-1 ) 00258 00259 * SSTEMR needs WORK of size 6*N, IWORK of size 3*N. 00260 * In addition, SLARRE needs WORK of size 6*N, IWORK of size 5*N. 00261 * Furthermore, SLARRV needs WORK of size 12*N, IWORK of size 7*N. 00262 IF( WANTZ ) THEN 00263 LWMIN = 18*N 00264 LIWMIN = 10*N 00265 ELSE 00266 * need less workspace if only the eigenvalues are wanted 00267 LWMIN = 12*N 00268 LIWMIN = 8*N 00269 ENDIF 00270 00271 WL = ZERO 00272 WU = ZERO 00273 IIL = 0 00274 IIU = 0 00275 00276 IF( VALEIG ) THEN 00277 * We do not reference VL, VU in the cases RANGE = 'I','A' 00278 * The interval (WL, WU] contains all the wanted eigenvalues. 00279 * It is either given by the user or computed in SLARRE. 00280 WL = VL 00281 WU = VU 00282 ELSEIF( INDEIG ) THEN 00283 * We do not reference IL, IU in the cases RANGE = 'V','A' 00284 IIL = IL 00285 IIU = IU 00286 ENDIF 00287 * 00288 INFO = 0 00289 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 00290 INFO = -1 00291 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 00292 INFO = -2 00293 ELSE IF( N.LT.0 ) THEN 00294 INFO = -3 00295 ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN 00296 INFO = -7 00297 ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN 00298 INFO = -8 00299 ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN 00300 INFO = -9 00301 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 00302 INFO = -13 00303 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 00304 INFO = -17 00305 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 00306 INFO = -19 00307 END IF 00308 * 00309 * Get machine constants. 00310 * 00311 SAFMIN = SLAMCH( 'Safe minimum' ) 00312 EPS = SLAMCH( 'Precision' ) 00313 SMLNUM = SAFMIN / EPS 00314 BIGNUM = ONE / SMLNUM 00315 RMIN = SQRT( SMLNUM ) 00316 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 00317 * 00318 IF( INFO.EQ.0 ) THEN 00319 WORK( 1 ) = LWMIN 00320 IWORK( 1 ) = LIWMIN 00321 * 00322 IF( WANTZ .AND. ALLEIG ) THEN 00323 NZCMIN = N 00324 ELSE IF( WANTZ .AND. VALEIG ) THEN 00325 CALL SLARRC( 'T', N, VL, VU, D, E, SAFMIN, 00326 $ NZCMIN, ITMP, ITMP2, INFO ) 00327 ELSE IF( WANTZ .AND. INDEIG ) THEN 00328 NZCMIN = IIU-IIL+1 00329 ELSE 00330 * WANTZ .EQ. FALSE. 00331 NZCMIN = 0 00332 ENDIF 00333 IF( ZQUERY .AND. INFO.EQ.0 ) THEN 00334 Z( 1,1 ) = NZCMIN 00335 ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN 00336 INFO = -14 00337 END IF 00338 END IF 00339 00340 IF( INFO.NE.0 ) THEN 00341 * 00342 CALL XERBLA( 'SSTEMR', -INFO ) 00343 * 00344 RETURN 00345 ELSE IF( LQUERY .OR. ZQUERY ) THEN 00346 RETURN 00347 END IF 00348 * 00349 * Handle N = 0, 1, and 2 cases immediately 00350 * 00351 M = 0 00352 IF( N.EQ.0 ) 00353 $ RETURN 00354 * 00355 IF( N.EQ.1 ) THEN 00356 IF( ALLEIG .OR. INDEIG ) THEN 00357 M = 1 00358 W( 1 ) = D( 1 ) 00359 ELSE 00360 IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN 00361 M = 1 00362 W( 1 ) = D( 1 ) 00363 END IF 00364 END IF 00365 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 00366 Z( 1, 1 ) = ONE 00367 ISUPPZ(1) = 1 00368 ISUPPZ(2) = 1 00369 END IF 00370 RETURN 00371 END IF 00372 * 00373 IF( N.EQ.2 ) THEN 00374 IF( .NOT.WANTZ ) THEN 00375 CALL SLAE2( D(1), E(1), D(2), R1, R2 ) 00376 ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 00377 CALL SLAEV2( D(1), E(1), D(2), R1, R2, CS, SN ) 00378 END IF 00379 IF( ALLEIG.OR. 00380 $ (VALEIG.AND.(R2.GT.WL).AND. 00381 $ (R2.LE.WU)).OR. 00382 $ (INDEIG.AND.(IIL.EQ.1)) ) THEN 00383 M = M+1 00384 W( M ) = R2 00385 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 00386 Z( 1, M ) = -SN 00387 Z( 2, M ) = CS 00388 * Note: At most one of SN and CS can be zero. 00389 IF (SN.NE.ZERO) THEN 00390 IF (CS.NE.ZERO) THEN 00391 ISUPPZ(2*M-1) = 1 00392 ISUPPZ(2*M) = 2 00393 ELSE 00394 ISUPPZ(2*M-1) = 1 00395 ISUPPZ(2*M) = 1 00396 END IF 00397 ELSE 00398 ISUPPZ(2*M-1) = 2 00399 ISUPPZ(2*M) = 2 00400 END IF 00401 ENDIF 00402 ENDIF 00403 IF( ALLEIG.OR. 00404 $ (VALEIG.AND.(R1.GT.WL).AND. 00405 $ (R1.LE.WU)).OR. 00406 $ (INDEIG.AND.(IIU.EQ.2)) ) THEN 00407 M = M+1 00408 W( M ) = R1 00409 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 00410 Z( 1, M ) = CS 00411 Z( 2, M ) = SN 00412 * Note: At most one of SN and CS can be zero. 00413 IF (SN.NE.ZERO) THEN 00414 IF (CS.NE.ZERO) THEN 00415 ISUPPZ(2*M-1) = 1 00416 ISUPPZ(2*M) = 2 00417 ELSE 00418 ISUPPZ(2*M-1) = 1 00419 ISUPPZ(2*M) = 1 00420 END IF 00421 ELSE 00422 ISUPPZ(2*M-1) = 2 00423 ISUPPZ(2*M) = 2 00424 END IF 00425 ENDIF 00426 ENDIF 00427 RETURN 00428 END IF 00429 00430 * Continue with general N 00431 00432 INDGRS = 1 00433 INDERR = 2*N + 1 00434 INDGP = 3*N + 1 00435 INDD = 4*N + 1 00436 INDE2 = 5*N + 1 00437 INDWRK = 6*N + 1 00438 * 00439 IINSPL = 1 00440 IINDBL = N + 1 00441 IINDW = 2*N + 1 00442 IINDWK = 3*N + 1 00443 * 00444 * Scale matrix to allowable range, if necessary. 00445 * The allowable range is related to the PIVMIN parameter; see the 00446 * comments in SLARRD. The preference for scaling small values 00447 * up is heuristic; we expect users' matrices not to be close to the 00448 * RMAX threshold. 00449 * 00450 SCALE = ONE 00451 TNRM = SLANST( 'M', N, D, E ) 00452 IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN 00453 SCALE = RMIN / TNRM 00454 ELSE IF( TNRM.GT.RMAX ) THEN 00455 SCALE = RMAX / TNRM 00456 END IF 00457 IF( SCALE.NE.ONE ) THEN 00458 CALL SSCAL( N, SCALE, D, 1 ) 00459 CALL SSCAL( N-1, SCALE, E, 1 ) 00460 TNRM = TNRM*SCALE 00461 IF( VALEIG ) THEN 00462 * If eigenvalues in interval have to be found, 00463 * scale (WL, WU] accordingly 00464 WL = WL*SCALE 00465 WU = WU*SCALE 00466 ENDIF 00467 END IF 00468 * 00469 * Compute the desired eigenvalues of the tridiagonal after splitting 00470 * into smaller subblocks if the corresponding off-diagonal elements 00471 * are small 00472 * THRESH is the splitting parameter for SLARRE 00473 * A negative THRESH forces the old splitting criterion based on the 00474 * size of the off-diagonal. A positive THRESH switches to splitting 00475 * which preserves relative accuracy. 00476 * 00477 IF( TRYRAC ) THEN 00478 * Test whether the matrix warrants the more expensive relative approach. 00479 CALL SLARRR( N, D, E, IINFO ) 00480 ELSE 00481 * The user does not care about relative accurately eigenvalues 00482 IINFO = -1 00483 ENDIF 00484 * Set the splitting criterion 00485 IF (IINFO.EQ.0) THEN 00486 THRESH = EPS 00487 ELSE 00488 THRESH = -EPS 00489 * relative accuracy is desired but T does not guarantee it 00490 TRYRAC = .FALSE. 00491 ENDIF 00492 * 00493 IF( TRYRAC ) THEN 00494 * Copy original diagonal, needed to guarantee relative accuracy 00495 CALL SCOPY(N,D,1,WORK(INDD),1) 00496 ENDIF 00497 * Store the squares of the offdiagonal values of T 00498 DO 5 J = 1, N-1 00499 WORK( INDE2+J-1 ) = E(J)**2 00500 5 CONTINUE 00501 00502 * Set the tolerance parameters for bisection 00503 IF( .NOT.WANTZ ) THEN 00504 * SLARRE computes the eigenvalues to full precision. 00505 RTOL1 = FOUR * EPS 00506 RTOL2 = FOUR * EPS 00507 ELSE 00508 * SLARRE computes the eigenvalues to less than full precision. 00509 * SLARRV will refine the eigenvalue approximations, and we can 00510 * need less accurate initial bisection in SLARRE. 00511 * Note: these settings do only affect the subset case and SLARRE 00512 RTOL1 = MAX( SQRT(EPS)*5.0E-2, FOUR * EPS ) 00513 RTOL2 = MAX( SQRT(EPS)*5.0E-3, FOUR * EPS ) 00514 ENDIF 00515 CALL SLARRE( RANGE, N, WL, WU, IIL, IIU, D, E, 00516 $ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT, 00517 $ IWORK( IINSPL ), M, W, WORK( INDERR ), 00518 $ WORK( INDGP ), IWORK( IINDBL ), 00519 $ IWORK( IINDW ), WORK( INDGRS ), PIVMIN, 00520 $ WORK( INDWRK ), IWORK( IINDWK ), IINFO ) 00521 IF( IINFO.NE.0 ) THEN 00522 INFO = 10 + ABS( IINFO ) 00523 RETURN 00524 END IF 00525 * Note that if RANGE .NE. 'V', SLARRE computes bounds on the desired 00526 * part of the spectrum. All desired eigenvalues are contained in 00527 * (WL,WU] 00528 00529 00530 IF( WANTZ ) THEN 00531 * 00532 * Compute the desired eigenvectors corresponding to the computed 00533 * eigenvalues 00534 * 00535 CALL SLARRV( N, WL, WU, D, E, 00536 $ PIVMIN, IWORK( IINSPL ), M, 00537 $ 1, M, MINRGP, RTOL1, RTOL2, 00538 $ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ), 00539 $ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ, 00540 $ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO ) 00541 IF( IINFO.NE.0 ) THEN 00542 INFO = 20 + ABS( IINFO ) 00543 RETURN 00544 END IF 00545 ELSE 00546 * SLARRE computes eigenvalues of the (shifted) root representation 00547 * SLARRV returns the eigenvalues of the unshifted matrix. 00548 * However, if the eigenvectors are not desired by the user, we need 00549 * to apply the corresponding shifts from SLARRE to obtain the 00550 * eigenvalues of the original matrix. 00551 DO 20 J = 1, M 00552 ITMP = IWORK( IINDBL+J-1 ) 00553 W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) ) 00554 20 CONTINUE 00555 END IF 00556 * 00557 00558 IF ( TRYRAC ) THEN 00559 * Refine computed eigenvalues so that they are relatively accurate 00560 * with respect to the original matrix T. 00561 IBEGIN = 1 00562 WBEGIN = 1 00563 DO 39 JBLK = 1, IWORK( IINDBL+M-1 ) 00564 IEND = IWORK( IINSPL+JBLK-1 ) 00565 IN = IEND - IBEGIN + 1 00566 WEND = WBEGIN - 1 00567 * check if any eigenvalues have to be refined in this block 00568 36 CONTINUE 00569 IF( WEND.LT.M ) THEN 00570 IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN 00571 WEND = WEND + 1 00572 GO TO 36 00573 END IF 00574 END IF 00575 IF( WEND.LT.WBEGIN ) THEN 00576 IBEGIN = IEND + 1 00577 GO TO 39 00578 END IF 00579 00580 OFFSET = IWORK(IINDW+WBEGIN-1)-1 00581 IFIRST = IWORK(IINDW+WBEGIN-1) 00582 ILAST = IWORK(IINDW+WEND-1) 00583 RTOL2 = FOUR * EPS 00584 CALL SLARRJ( IN, 00585 $ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1), 00586 $ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN), 00587 $ WORK( INDERR+WBEGIN-1 ), 00588 $ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN, 00589 $ TNRM, IINFO ) 00590 IBEGIN = IEND + 1 00591 WBEGIN = WEND + 1 00592 39 CONTINUE 00593 ENDIF 00594 * 00595 * If matrix was scaled, then rescale eigenvalues appropriately. 00596 * 00597 IF( SCALE.NE.ONE ) THEN 00598 CALL SSCAL( M, ONE / SCALE, W, 1 ) 00599 END IF 00600 * 00601 * If eigenvalues are not in increasing order, then sort them, 00602 * possibly along with eigenvectors. 00603 * 00604 IF( NSPLIT.GT.1 ) THEN 00605 IF( .NOT. WANTZ ) THEN 00606 CALL SLASRT( 'I', M, W, IINFO ) 00607 IF( IINFO.NE.0 ) THEN 00608 INFO = 3 00609 RETURN 00610 END IF 00611 ELSE 00612 DO 60 J = 1, M - 1 00613 I = 0 00614 TMP = W( J ) 00615 DO 50 JJ = J + 1, M 00616 IF( W( JJ ).LT.TMP ) THEN 00617 I = JJ 00618 TMP = W( JJ ) 00619 END IF 00620 50 CONTINUE 00621 IF( I.NE.0 ) THEN 00622 W( I ) = W( J ) 00623 W( J ) = TMP 00624 IF( WANTZ ) THEN 00625 CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 00626 ITMP = ISUPPZ( 2*I-1 ) 00627 ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 ) 00628 ISUPPZ( 2*J-1 ) = ITMP 00629 ITMP = ISUPPZ( 2*I ) 00630 ISUPPZ( 2*I ) = ISUPPZ( 2*J ) 00631 ISUPPZ( 2*J ) = ITMP 00632 END IF 00633 END IF 00634 60 CONTINUE 00635 END IF 00636 ENDIF 00637 * 00638 * 00639 WORK( 1 ) = LWMIN 00640 IWORK( 1 ) = LIWMIN 00641 RETURN 00642 * 00643 * End of SSTEMR 00644 * 00645 END
1.7.2