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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE SLARRV( N, VL, VU, D, L, PIVMIN, 00002 $ ISPLIT, M, DOL, DOU, MINRGP, 00003 $ RTOL1, RTOL2, W, WERR, WGAP, 00004 $ IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ, 00005 $ WORK, IWORK, INFO ) 00006 * 00007 * -- LAPACK auxiliary routine (version 3.3.1) -- 00008 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00009 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00010 * -- April 2011 -- 00011 * 00012 * .. Scalar Arguments .. 00013 INTEGER DOL, DOU, INFO, LDZ, M, N 00014 REAL MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU 00015 * .. 00016 * .. Array Arguments .. 00017 INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ), 00018 $ ISUPPZ( * ), IWORK( * ) 00019 REAL D( * ), GERS( * ), L( * ), W( * ), WERR( * ), 00020 $ WGAP( * ), WORK( * ) 00021 REAL Z( LDZ, * ) 00022 * .. 00023 * 00024 * Purpose 00025 * ======= 00026 * 00027 * SLARRV computes the eigenvectors of the tridiagonal matrix 00028 * T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T. 00029 * The input eigenvalues should have been computed by SLARRE. 00030 * 00031 * Arguments 00032 * ========= 00033 * 00034 * N (input) INTEGER 00035 * The order of the matrix. N >= 0. 00036 * 00037 * VL (input) REAL 00038 * VU (input) REAL 00039 * Lower and upper bounds of the interval that contains the desired 00040 * eigenvalues. VL < VU. Needed to compute gaps on the left or right 00041 * end of the extremal eigenvalues in the desired RANGE. 00042 * 00043 * D (input/output) REAL array, dimension (N) 00044 * On entry, the N diagonal elements of the diagonal matrix D. 00045 * On exit, D may be overwritten. 00046 * 00047 * L (input/output) REAL array, dimension (N) 00048 * On entry, the (N-1) subdiagonal elements of the unit 00049 * bidiagonal matrix L are in elements 1 to N-1 of L 00050 * (if the matrix is not splitted.) At the end of each block 00051 * is stored the corresponding shift as given by SLARRE. 00052 * On exit, L is overwritten. 00053 * 00054 * PIVMIN (input) REAL 00055 * The minimum pivot allowed in the Sturm sequence. 00056 * 00057 * ISPLIT (input) INTEGER array, dimension (N) 00058 * The splitting points, at which T breaks up into blocks. 00059 * The first block consists of rows/columns 1 to 00060 * ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 00061 * through ISPLIT( 2 ), etc. 00062 * 00063 * M (input) INTEGER 00064 * The total number of input eigenvalues. 0 <= M <= N. 00065 * 00066 * DOL (input) INTEGER 00067 * DOU (input) INTEGER 00068 * If the user wants to compute only selected eigenvectors from all 00069 * the eigenvalues supplied, he can specify an index range DOL:DOU. 00070 * Or else the setting DOL=1, DOU=M should be applied. 00071 * Note that DOL and DOU refer to the order in which the eigenvalues 00072 * are stored in W. 00073 * If the user wants to compute only selected eigenpairs, then 00074 * the columns DOL-1 to DOU+1 of the eigenvector space Z contain the 00075 * computed eigenvectors. All other columns of Z are set to zero. 00076 * 00077 * MINRGP (input) REAL 00078 * 00079 * RTOL1 (input) REAL 00080 * RTOL2 (input) REAL 00081 * Parameters for bisection. 00082 * An interval [LEFT,RIGHT] has converged if 00083 * RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) 00084 * 00085 * W (input/output) REAL array, dimension (N) 00086 * The first M elements of W contain the APPROXIMATE eigenvalues for 00087 * which eigenvectors are to be computed. The eigenvalues 00088 * should be grouped by split-off block and ordered from 00089 * smallest to largest within the block ( The output array 00090 * W from SLARRE is expected here ). Furthermore, they are with 00091 * respect to the shift of the corresponding root representation 00092 * for their block. On exit, W holds the eigenvalues of the 00093 * UNshifted matrix. 00094 * 00095 * WERR (input/output) REAL array, dimension (N) 00096 * The first M elements contain the semiwidth of the uncertainty 00097 * interval of the corresponding eigenvalue in W 00098 * 00099 * WGAP (input/output) REAL array, dimension (N) 00100 * The separation from the right neighbor eigenvalue in W. 00101 * 00102 * IBLOCK (input) INTEGER array, dimension (N) 00103 * The indices of the blocks (submatrices) associated with the 00104 * corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue 00105 * W(i) belongs to the first block from the top, =2 if W(i) 00106 * belongs to the second block, etc. 00107 * 00108 * INDEXW (input) INTEGER array, dimension (N) 00109 * The indices of the eigenvalues within each block (submatrix); 00110 * for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the 00111 * i-th eigenvalue W(i) is the 10-th eigenvalue in the second block. 00112 * 00113 * GERS (input) REAL array, dimension (2*N) 00114 * The N Gerschgorin intervals (the i-th Gerschgorin interval 00115 * is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should 00116 * be computed from the original UNshifted matrix. 00117 * 00118 * Z (output) REAL array, dimension (LDZ, max(1,M) ) 00119 * If INFO = 0, the first M columns of Z contain the 00120 * orthonormal eigenvectors of the matrix T 00121 * corresponding to the input eigenvalues, with the i-th 00122 * column of Z holding the eigenvector associated with W(i). 00123 * Note: the user must ensure that at least max(1,M) columns are 00124 * supplied in the array Z. 00125 * 00126 * LDZ (input) INTEGER 00127 * The leading dimension of the array Z. LDZ >= 1, and if 00128 * JOBZ = 'V', LDZ >= max(1,N). 00129 * 00130 * ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) ) 00131 * The support of the eigenvectors in Z, i.e., the indices 00132 * indicating the nonzero elements in Z. The I-th eigenvector 00133 * is nonzero only in elements ISUPPZ( 2*I-1 ) through 00134 * ISUPPZ( 2*I ). 00135 * 00136 * WORK (workspace) REAL array, dimension (12*N) 00137 * 00138 * IWORK (workspace) INTEGER array, dimension (7*N) 00139 * 00140 * INFO (output) INTEGER 00141 * = 0: successful exit 00142 * 00143 * > 0: A problem occured in SLARRV. 00144 * < 0: One of the called subroutines signaled an internal problem. 00145 * Needs inspection of the corresponding parameter IINFO 00146 * for further information. 00147 * 00148 * =-1: Problem in SLARRB when refining a child's eigenvalues. 00149 * =-2: Problem in SLARRF when computing the RRR of a child. 00150 * When a child is inside a tight cluster, it can be difficult 00151 * to find an RRR. A partial remedy from the user's point of 00152 * view is to make the parameter MINRGP smaller and recompile. 00153 * However, as the orthogonality of the computed vectors is 00154 * proportional to 1/MINRGP, the user should be aware that 00155 * he might be trading in precision when he decreases MINRGP. 00156 * =-3: Problem in SLARRB when refining a single eigenvalue 00157 * after the Rayleigh correction was rejected. 00158 * = 5: The Rayleigh Quotient Iteration failed to converge to 00159 * full accuracy in MAXITR steps. 00160 * 00161 * Further Details 00162 * =============== 00163 * 00164 * Based on contributions by 00165 * Beresford Parlett, University of California, Berkeley, USA 00166 * Jim Demmel, University of California, Berkeley, USA 00167 * Inderjit Dhillon, University of Texas, Austin, USA 00168 * Osni Marques, LBNL/NERSC, USA 00169 * Christof Voemel, University of California, Berkeley, USA 00170 * 00171 * ===================================================================== 00172 * 00173 * .. Parameters .. 00174 INTEGER MAXITR 00175 PARAMETER ( MAXITR = 10 ) 00176 REAL ZERO, ONE, TWO, THREE, FOUR, HALF 00177 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, 00178 $ TWO = 2.0E0, THREE = 3.0E0, 00179 $ FOUR = 4.0E0, HALF = 0.5E0) 00180 * .. 00181 * .. Local Scalars .. 00182 LOGICAL ESKIP, NEEDBS, STP2II, TRYRQC, USEDBS, USEDRQ 00183 INTEGER DONE, I, IBEGIN, IDONE, IEND, II, IINDC1, 00184 $ IINDC2, IINDR, IINDWK, IINFO, IM, IN, INDEIG, 00185 $ INDLD, INDLLD, INDWRK, ISUPMN, ISUPMX, ITER, 00186 $ ITMP1, J, JBLK, K, MINIWSIZE, MINWSIZE, NCLUS, 00187 $ NDEPTH, NEGCNT, NEWCLS, NEWFST, NEWFTT, NEWLST, 00188 $ NEWSIZ, OFFSET, OLDCLS, OLDFST, OLDIEN, OLDLST, 00189 $ OLDNCL, P, PARITY, Q, WBEGIN, WEND, WINDEX, 00190 $ WINDMN, WINDPL, ZFROM, ZTO, ZUSEDL, ZUSEDU, 00191 $ ZUSEDW 00192 REAL BSTRES, BSTW, EPS, FUDGE, GAP, GAPTOL, GL, GU, 00193 $ LAMBDA, LEFT, LGAP, MINGMA, NRMINV, RESID, 00194 $ RGAP, RIGHT, RQCORR, RQTOL, SAVGAP, SGNDEF, 00195 $ SIGMA, SPDIAM, SSIGMA, TAU, TMP, TOL, ZTZ 00196 * .. 00197 * .. External Functions .. 00198 REAL SLAMCH 00199 EXTERNAL SLAMCH 00200 * .. 00201 * .. External Subroutines .. 00202 EXTERNAL SCOPY, SLAR1V, SLARRB, SLARRF, SLASET, 00203 $ SSCAL 00204 * .. 00205 * .. Intrinsic Functions .. 00206 INTRINSIC ABS, REAL, MAX, MIN 00207 * .. 00208 * .. Executable Statements .. 00209 * .. 00210 00211 * The first N entries of WORK are reserved for the eigenvalues 00212 INDLD = N+1 00213 INDLLD= 2*N+1 00214 INDWRK= 3*N+1 00215 MINWSIZE = 12 * N 00216 00217 DO 5 I= 1,MINWSIZE 00218 WORK( I ) = ZERO 00219 5 CONTINUE 00220 00221 * IWORK(IINDR+1:IINDR+N) hold the twist indices R for the 00222 * factorization used to compute the FP vector 00223 IINDR = 0 00224 * IWORK(IINDC1+1:IINC2+N) are used to store the clusters of the current 00225 * layer and the one above. 00226 IINDC1 = N 00227 IINDC2 = 2*N 00228 IINDWK = 3*N + 1 00229 00230 MINIWSIZE = 7 * N 00231 DO 10 I= 1,MINIWSIZE 00232 IWORK( I ) = 0 00233 10 CONTINUE 00234 00235 ZUSEDL = 1 00236 IF(DOL.GT.1) THEN 00237 * Set lower bound for use of Z 00238 ZUSEDL = DOL-1 00239 ENDIF 00240 ZUSEDU = M 00241 IF(DOU.LT.M) THEN 00242 * Set lower bound for use of Z 00243 ZUSEDU = DOU+1 00244 ENDIF 00245 * The width of the part of Z that is used 00246 ZUSEDW = ZUSEDU - ZUSEDL + 1 00247 00248 00249 CALL SLASET( 'Full', N, ZUSEDW, ZERO, ZERO, 00250 $ Z(1,ZUSEDL), LDZ ) 00251 00252 EPS = SLAMCH( 'Precision' ) 00253 RQTOL = TWO * EPS 00254 * 00255 * Set expert flags for standard code. 00256 TRYRQC = .TRUE. 00257 00258 IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN 00259 ELSE 00260 * Only selected eigenpairs are computed. Since the other evalues 00261 * are not refined by RQ iteration, bisection has to compute to full 00262 * accuracy. 00263 RTOL1 = FOUR * EPS 00264 RTOL2 = FOUR * EPS 00265 ENDIF 00266 00267 * The entries WBEGIN:WEND in W, WERR, WGAP correspond to the 00268 * desired eigenvalues. The support of the nonzero eigenvector 00269 * entries is contained in the interval IBEGIN:IEND. 00270 * Remark that if k eigenpairs are desired, then the eigenvectors 00271 * are stored in k contiguous columns of Z. 00272 00273 * DONE is the number of eigenvectors already computed 00274 DONE = 0 00275 IBEGIN = 1 00276 WBEGIN = 1 00277 DO 170 JBLK = 1, IBLOCK( M ) 00278 IEND = ISPLIT( JBLK ) 00279 SIGMA = L( IEND ) 00280 * Find the eigenvectors of the submatrix indexed IBEGIN 00281 * through IEND. 00282 WEND = WBEGIN - 1 00283 15 CONTINUE 00284 IF( WEND.LT.M ) THEN 00285 IF( IBLOCK( WEND+1 ).EQ.JBLK ) THEN 00286 WEND = WEND + 1 00287 GO TO 15 00288 END IF 00289 END IF 00290 IF( WEND.LT.WBEGIN ) THEN 00291 IBEGIN = IEND + 1 00292 GO TO 170 00293 ELSEIF( (WEND.LT.DOL).OR.(WBEGIN.GT.DOU) ) THEN 00294 IBEGIN = IEND + 1 00295 WBEGIN = WEND + 1 00296 GO TO 170 00297 END IF 00298 00299 * Find local spectral diameter of the block 00300 GL = GERS( 2*IBEGIN-1 ) 00301 GU = GERS( 2*IBEGIN ) 00302 DO 20 I = IBEGIN+1 , IEND 00303 GL = MIN( GERS( 2*I-1 ), GL ) 00304 GU = MAX( GERS( 2*I ), GU ) 00305 20 CONTINUE 00306 SPDIAM = GU - GL 00307 00308 * OLDIEN is the last index of the previous block 00309 OLDIEN = IBEGIN - 1 00310 * Calculate the size of the current block 00311 IN = IEND - IBEGIN + 1 00312 * The number of eigenvalues in the current block 00313 IM = WEND - WBEGIN + 1 00314 00315 * This is for a 1x1 block 00316 IF( IBEGIN.EQ.IEND ) THEN 00317 DONE = DONE+1 00318 Z( IBEGIN, WBEGIN ) = ONE 00319 ISUPPZ( 2*WBEGIN-1 ) = IBEGIN 00320 ISUPPZ( 2*WBEGIN ) = IBEGIN 00321 W( WBEGIN ) = W( WBEGIN ) + SIGMA 00322 WORK( WBEGIN ) = W( WBEGIN ) 00323 IBEGIN = IEND + 1 00324 WBEGIN = WBEGIN + 1 00325 GO TO 170 00326 END IF 00327 00328 * The desired (shifted) eigenvalues are stored in W(WBEGIN:WEND) 00329 * Note that these can be approximations, in this case, the corresp. 00330 * entries of WERR give the size of the uncertainty interval. 00331 * The eigenvalue approximations will be refined when necessary as 00332 * high relative accuracy is required for the computation of the 00333 * corresponding eigenvectors. 00334 CALL SCOPY( IM, W( WBEGIN ), 1, 00335 $ WORK( WBEGIN ), 1 ) 00336 00337 * We store in W the eigenvalue approximations w.r.t. the original 00338 * matrix T. 00339 DO 30 I=1,IM 00340 W(WBEGIN+I-1) = W(WBEGIN+I-1)+SIGMA 00341 30 CONTINUE 00342 00343 00344 * NDEPTH is the current depth of the representation tree 00345 NDEPTH = 0 00346 * PARITY is either 1 or 0 00347 PARITY = 1 00348 * NCLUS is the number of clusters for the next level of the 00349 * representation tree, we start with NCLUS = 1 for the root 00350 NCLUS = 1 00351 IWORK( IINDC1+1 ) = 1 00352 IWORK( IINDC1+2 ) = IM 00353 00354 * IDONE is the number of eigenvectors already computed in the current 00355 * block 00356 IDONE = 0 00357 * loop while( IDONE.LT.IM ) 00358 * generate the representation tree for the current block and 00359 * compute the eigenvectors 00360 40 CONTINUE 00361 IF( IDONE.LT.IM ) THEN 00362 * This is a crude protection against infinitely deep trees 00363 IF( NDEPTH.GT.M ) THEN 00364 INFO = -2 00365 RETURN 00366 ENDIF 00367 * breadth first processing of the current level of the representation 00368 * tree: OLDNCL = number of clusters on current level 00369 OLDNCL = NCLUS 00370 * reset NCLUS to count the number of child clusters 00371 NCLUS = 0 00372 * 00373 PARITY = 1 - PARITY 00374 IF( PARITY.EQ.0 ) THEN 00375 OLDCLS = IINDC1 00376 NEWCLS = IINDC2 00377 ELSE 00378 OLDCLS = IINDC2 00379 NEWCLS = IINDC1 00380 END IF 00381 * Process the clusters on the current level 00382 DO 150 I = 1, OLDNCL 00383 J = OLDCLS + 2*I 00384 * OLDFST, OLDLST = first, last index of current cluster. 00385 * cluster indices start with 1 and are relative 00386 * to WBEGIN when accessing W, WGAP, WERR, Z 00387 OLDFST = IWORK( J-1 ) 00388 OLDLST = IWORK( J ) 00389 IF( NDEPTH.GT.0 ) THEN 00390 * Retrieve relatively robust representation (RRR) of cluster 00391 * that has been computed at the previous level 00392 * The RRR is stored in Z and overwritten once the eigenvectors 00393 * have been computed or when the cluster is refined 00394 00395 IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN 00396 * Get representation from location of the leftmost evalue 00397 * of the cluster 00398 J = WBEGIN + OLDFST - 1 00399 ELSE 00400 IF(WBEGIN+OLDFST-1.LT.DOL) THEN 00401 * Get representation from the left end of Z array 00402 J = DOL - 1 00403 ELSEIF(WBEGIN+OLDFST-1.GT.DOU) THEN 00404 * Get representation from the right end of Z array 00405 J = DOU 00406 ELSE 00407 J = WBEGIN + OLDFST - 1 00408 ENDIF 00409 ENDIF 00410 CALL SCOPY( IN, Z( IBEGIN, J ), 1, D( IBEGIN ), 1 ) 00411 CALL SCOPY( IN-1, Z( IBEGIN, J+1 ), 1, L( IBEGIN ), 00412 $ 1 ) 00413 SIGMA = Z( IEND, J+1 ) 00414 00415 * Set the corresponding entries in Z to zero 00416 CALL SLASET( 'Full', IN, 2, ZERO, ZERO, 00417 $ Z( IBEGIN, J), LDZ ) 00418 END IF 00419 00420 * Compute DL and DLL of current RRR 00421 DO 50 J = IBEGIN, IEND-1 00422 TMP = D( J )*L( J ) 00423 WORK( INDLD-1+J ) = TMP 00424 WORK( INDLLD-1+J ) = TMP*L( J ) 00425 50 CONTINUE 00426 00427 IF( NDEPTH.GT.0 ) THEN 00428 * P and Q are index of the first and last eigenvalue to compute 00429 * within the current block 00430 P = INDEXW( WBEGIN-1+OLDFST ) 00431 Q = INDEXW( WBEGIN-1+OLDLST ) 00432 * Offset for the arrays WORK, WGAP and WERR, i.e., the P-OFFSET 00433 * through the Q-OFFSET elements of these arrays are to be used. 00434 * OFFSET = P-OLDFST 00435 OFFSET = INDEXW( WBEGIN ) - 1 00436 * perform limited bisection (if necessary) to get approximate 00437 * eigenvalues to the precision needed. 00438 CALL SLARRB( IN, D( IBEGIN ), 00439 $ WORK(INDLLD+IBEGIN-1), 00440 $ P, Q, RTOL1, RTOL2, OFFSET, 00441 $ WORK(WBEGIN),WGAP(WBEGIN),WERR(WBEGIN), 00442 $ WORK( INDWRK ), IWORK( IINDWK ), 00443 $ PIVMIN, SPDIAM, IN, IINFO ) 00444 IF( IINFO.NE.0 ) THEN 00445 INFO = -1 00446 RETURN 00447 ENDIF 00448 * We also recompute the extremal gaps. W holds all eigenvalues 00449 * of the unshifted matrix and must be used for computation 00450 * of WGAP, the entries of WORK might stem from RRRs with 00451 * different shifts. The gaps from WBEGIN-1+OLDFST to 00452 * WBEGIN-1+OLDLST are correctly computed in SLARRB. 00453 * However, we only allow the gaps to become greater since 00454 * this is what should happen when we decrease WERR 00455 IF( OLDFST.GT.1) THEN 00456 WGAP( WBEGIN+OLDFST-2 ) = 00457 $ MAX(WGAP(WBEGIN+OLDFST-2), 00458 $ W(WBEGIN+OLDFST-1)-WERR(WBEGIN+OLDFST-1) 00459 $ - W(WBEGIN+OLDFST-2)-WERR(WBEGIN+OLDFST-2) ) 00460 ENDIF 00461 IF( WBEGIN + OLDLST -1 .LT. WEND ) THEN 00462 WGAP( WBEGIN+OLDLST-1 ) = 00463 $ MAX(WGAP(WBEGIN+OLDLST-1), 00464 $ W(WBEGIN+OLDLST)-WERR(WBEGIN+OLDLST) 00465 $ - W(WBEGIN+OLDLST-1)-WERR(WBEGIN+OLDLST-1) ) 00466 ENDIF 00467 * Each time the eigenvalues in WORK get refined, we store 00468 * the newly found approximation with all shifts applied in W 00469 DO 53 J=OLDFST,OLDLST 00470 W(WBEGIN+J-1) = WORK(WBEGIN+J-1)+SIGMA 00471 53 CONTINUE 00472 END IF 00473 00474 * Process the current node. 00475 NEWFST = OLDFST 00476 DO 140 J = OLDFST, OLDLST 00477 IF( J.EQ.OLDLST ) THEN 00478 * we are at the right end of the cluster, this is also the 00479 * boundary of the child cluster 00480 NEWLST = J 00481 ELSE IF ( WGAP( WBEGIN + J -1).GE. 00482 $ MINRGP* ABS( WORK(WBEGIN + J -1) ) ) THEN 00483 * the right relative gap is big enough, the child cluster 00484 * (NEWFST,..,NEWLST) is well separated from the following 00485 NEWLST = J 00486 ELSE 00487 * inside a child cluster, the relative gap is not 00488 * big enough. 00489 GOTO 140 00490 END IF 00491 00492 * Compute size of child cluster found 00493 NEWSIZ = NEWLST - NEWFST + 1 00494 00495 * NEWFTT is the place in Z where the new RRR or the computed 00496 * eigenvector is to be stored 00497 IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN 00498 * Store representation at location of the leftmost evalue 00499 * of the cluster 00500 NEWFTT = WBEGIN + NEWFST - 1 00501 ELSE 00502 IF(WBEGIN+NEWFST-1.LT.DOL) THEN 00503 * Store representation at the left end of Z array 00504 NEWFTT = DOL - 1 00505 ELSEIF(WBEGIN+NEWFST-1.GT.DOU) THEN 00506 * Store representation at the right end of Z array 00507 NEWFTT = DOU 00508 ELSE 00509 NEWFTT = WBEGIN + NEWFST - 1 00510 ENDIF 00511 ENDIF 00512 00513 IF( NEWSIZ.GT.1) THEN 00514 * 00515 * Current child is not a singleton but a cluster. 00516 * Compute and store new representation of child. 00517 * 00518 * 00519 * Compute left and right cluster gap. 00520 * 00521 * LGAP and RGAP are not computed from WORK because 00522 * the eigenvalue approximations may stem from RRRs 00523 * different shifts. However, W hold all eigenvalues 00524 * of the unshifted matrix. Still, the entries in WGAP 00525 * have to be computed from WORK since the entries 00526 * in W might be of the same order so that gaps are not 00527 * exhibited correctly for very close eigenvalues. 00528 IF( NEWFST.EQ.1 ) THEN 00529 LGAP = MAX( ZERO, 00530 $ W(WBEGIN)-WERR(WBEGIN) - VL ) 00531 ELSE 00532 LGAP = WGAP( WBEGIN+NEWFST-2 ) 00533 ENDIF 00534 RGAP = WGAP( WBEGIN+NEWLST-1 ) 00535 * 00536 * Compute left- and rightmost eigenvalue of child 00537 * to high precision in order to shift as close 00538 * as possible and obtain as large relative gaps 00539 * as possible 00540 * 00541 DO 55 K =1,2 00542 IF(K.EQ.1) THEN 00543 P = INDEXW( WBEGIN-1+NEWFST ) 00544 ELSE 00545 P = INDEXW( WBEGIN-1+NEWLST ) 00546 ENDIF 00547 OFFSET = INDEXW( WBEGIN ) - 1 00548 CALL SLARRB( IN, D(IBEGIN), 00549 $ WORK( INDLLD+IBEGIN-1 ),P,P, 00550 $ RQTOL, RQTOL, OFFSET, 00551 $ WORK(WBEGIN),WGAP(WBEGIN), 00552 $ WERR(WBEGIN),WORK( INDWRK ), 00553 $ IWORK( IINDWK ), PIVMIN, SPDIAM, 00554 $ IN, IINFO ) 00555 55 CONTINUE 00556 * 00557 IF((WBEGIN+NEWLST-1.LT.DOL).OR. 00558 $ (WBEGIN+NEWFST-1.GT.DOU)) THEN 00559 * if the cluster contains no desired eigenvalues 00560 * skip the computation of that branch of the rep. tree 00561 * 00562 * We could skip before the refinement of the extremal 00563 * eigenvalues of the child, but then the representation 00564 * tree could be different from the one when nothing is 00565 * skipped. For this reason we skip at this place. 00566 IDONE = IDONE + NEWLST - NEWFST + 1 00567 GOTO 139 00568 ENDIF 00569 * 00570 * Compute RRR of child cluster. 00571 * Note that the new RRR is stored in Z 00572 * 00573 * SLARRF needs LWORK = 2*N 00574 CALL SLARRF( IN, D( IBEGIN ), L( IBEGIN ), 00575 $ WORK(INDLD+IBEGIN-1), 00576 $ NEWFST, NEWLST, WORK(WBEGIN), 00577 $ WGAP(WBEGIN), WERR(WBEGIN), 00578 $ SPDIAM, LGAP, RGAP, PIVMIN, TAU, 00579 $ Z(IBEGIN, NEWFTT),Z(IBEGIN, NEWFTT+1), 00580 $ WORK( INDWRK ), IINFO ) 00581 IF( IINFO.EQ.0 ) THEN 00582 * a new RRR for the cluster was found by SLARRF 00583 * update shift and store it 00584 SSIGMA = SIGMA + TAU 00585 Z( IEND, NEWFTT+1 ) = SSIGMA 00586 * WORK() are the midpoints and WERR() the semi-width 00587 * Note that the entries in W are unchanged. 00588 DO 116 K = NEWFST, NEWLST 00589 FUDGE = 00590 $ THREE*EPS*ABS(WORK(WBEGIN+K-1)) 00591 WORK( WBEGIN + K - 1 ) = 00592 $ WORK( WBEGIN + K - 1) - TAU 00593 FUDGE = FUDGE + 00594 $ FOUR*EPS*ABS(WORK(WBEGIN+K-1)) 00595 * Fudge errors 00596 WERR( WBEGIN + K - 1 ) = 00597 $ WERR( WBEGIN + K - 1 ) + FUDGE 00598 * Gaps are not fudged. Provided that WERR is small 00599 * when eigenvalues are close, a zero gap indicates 00600 * that a new representation is needed for resolving 00601 * the cluster. A fudge could lead to a wrong decision 00602 * of judging eigenvalues 'separated' which in 00603 * reality are not. This could have a negative impact 00604 * on the orthogonality of the computed eigenvectors. 00605 116 CONTINUE 00606 00607 NCLUS = NCLUS + 1 00608 K = NEWCLS + 2*NCLUS 00609 IWORK( K-1 ) = NEWFST 00610 IWORK( K ) = NEWLST 00611 ELSE 00612 INFO = -2 00613 RETURN 00614 ENDIF 00615 ELSE 00616 * 00617 * Compute eigenvector of singleton 00618 * 00619 ITER = 0 00620 * 00621 TOL = FOUR * LOG(REAL(IN)) * EPS 00622 * 00623 K = NEWFST 00624 WINDEX = WBEGIN + K - 1 00625 WINDMN = MAX(WINDEX - 1,1) 00626 WINDPL = MIN(WINDEX + 1,M) 00627 LAMBDA = WORK( WINDEX ) 00628 DONE = DONE + 1 00629 * Check if eigenvector computation is to be skipped 00630 IF((WINDEX.LT.DOL).OR. 00631 $ (WINDEX.GT.DOU)) THEN 00632 ESKIP = .TRUE. 00633 GOTO 125 00634 ELSE 00635 ESKIP = .FALSE. 00636 ENDIF 00637 LEFT = WORK( WINDEX ) - WERR( WINDEX ) 00638 RIGHT = WORK( WINDEX ) + WERR( WINDEX ) 00639 INDEIG = INDEXW( WINDEX ) 00640 * Note that since we compute the eigenpairs for a child, 00641 * all eigenvalue approximations are w.r.t the same shift. 00642 * In this case, the entries in WORK should be used for 00643 * computing the gaps since they exhibit even very small 00644 * differences in the eigenvalues, as opposed to the 00645 * entries in W which might "look" the same. 00646 00647 IF( K .EQ. 1) THEN 00648 * In the case RANGE='I' and with not much initial 00649 * accuracy in LAMBDA and VL, the formula 00650 * LGAP = MAX( ZERO, (SIGMA - VL) + LAMBDA ) 00651 * can lead to an overestimation of the left gap and 00652 * thus to inadequately early RQI 'convergence'. 00653 * Prevent this by forcing a small left gap. 00654 LGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT)) 00655 ELSE 00656 LGAP = WGAP(WINDMN) 00657 ENDIF 00658 IF( K .EQ. IM) THEN 00659 * In the case RANGE='I' and with not much initial 00660 * accuracy in LAMBDA and VU, the formula 00661 * can lead to an overestimation of the right gap and 00662 * thus to inadequately early RQI 'convergence'. 00663 * Prevent this by forcing a small right gap. 00664 RGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT)) 00665 ELSE 00666 RGAP = WGAP(WINDEX) 00667 ENDIF 00668 GAP = MIN( LGAP, RGAP ) 00669 IF(( K .EQ. 1).OR.(K .EQ. IM)) THEN 00670 * The eigenvector support can become wrong 00671 * because significant entries could be cut off due to a 00672 * large GAPTOL parameter in LAR1V. Prevent this. 00673 GAPTOL = ZERO 00674 ELSE 00675 GAPTOL = GAP * EPS 00676 ENDIF 00677 ISUPMN = IN 00678 ISUPMX = 1 00679 * Update WGAP so that it holds the minimum gap 00680 * to the left or the right. This is crucial in the 00681 * case where bisection is used to ensure that the 00682 * eigenvalue is refined up to the required precision. 00683 * The correct value is restored afterwards. 00684 SAVGAP = WGAP(WINDEX) 00685 WGAP(WINDEX) = GAP 00686 * We want to use the Rayleigh Quotient Correction 00687 * as often as possible since it converges quadratically 00688 * when we are close enough to the desired eigenvalue. 00689 * However, the Rayleigh Quotient can have the wrong sign 00690 * and lead us away from the desired eigenvalue. In this 00691 * case, the best we can do is to use bisection. 00692 USEDBS = .FALSE. 00693 USEDRQ = .FALSE. 00694 * Bisection is initially turned off unless it is forced 00695 NEEDBS = .NOT.TRYRQC 00696 120 CONTINUE 00697 * Check if bisection should be used to refine eigenvalue 00698 IF(NEEDBS) THEN 00699 * Take the bisection as new iterate 00700 USEDBS = .TRUE. 00701 ITMP1 = IWORK( IINDR+WINDEX ) 00702 OFFSET = INDEXW( WBEGIN ) - 1 00703 CALL SLARRB( IN, D(IBEGIN), 00704 $ WORK(INDLLD+IBEGIN-1),INDEIG,INDEIG, 00705 $ ZERO, TWO*EPS, OFFSET, 00706 $ WORK(WBEGIN),WGAP(WBEGIN), 00707 $ WERR(WBEGIN),WORK( INDWRK ), 00708 $ IWORK( IINDWK ), PIVMIN, SPDIAM, 00709 $ ITMP1, IINFO ) 00710 IF( IINFO.NE.0 ) THEN 00711 INFO = -3 00712 RETURN 00713 ENDIF 00714 LAMBDA = WORK( WINDEX ) 00715 * Reset twist index from inaccurate LAMBDA to 00716 * force computation of true MINGMA 00717 IWORK( IINDR+WINDEX ) = 0 00718 ENDIF 00719 * Given LAMBDA, compute the eigenvector. 00720 CALL SLAR1V( IN, 1, IN, LAMBDA, D( IBEGIN ), 00721 $ L( IBEGIN ), WORK(INDLD+IBEGIN-1), 00722 $ WORK(INDLLD+IBEGIN-1), 00723 $ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ), 00724 $ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA, 00725 $ IWORK( IINDR+WINDEX ), ISUPPZ( 2*WINDEX-1 ), 00726 $ NRMINV, RESID, RQCORR, WORK( INDWRK ) ) 00727 IF(ITER .EQ. 0) THEN 00728 BSTRES = RESID 00729 BSTW = LAMBDA 00730 ELSEIF(RESID.LT.BSTRES) THEN 00731 BSTRES = RESID 00732 BSTW = LAMBDA 00733 ENDIF 00734 ISUPMN = MIN(ISUPMN,ISUPPZ( 2*WINDEX-1 )) 00735 ISUPMX = MAX(ISUPMX,ISUPPZ( 2*WINDEX )) 00736 ITER = ITER + 1 00737 00738 * sin alpha <= |resid|/gap 00739 * Note that both the residual and the gap are 00740 * proportional to the matrix, so ||T|| doesn't play 00741 * a role in the quotient 00742 00743 * 00744 * Convergence test for Rayleigh-Quotient iteration 00745 * (omitted when Bisection has been used) 00746 * 00747 IF( RESID.GT.TOL*GAP .AND. ABS( RQCORR ).GT. 00748 $ RQTOL*ABS( LAMBDA ) .AND. .NOT. USEDBS) 00749 $ THEN 00750 * We need to check that the RQCORR update doesn't 00751 * move the eigenvalue away from the desired one and 00752 * towards a neighbor. -> protection with bisection 00753 IF(INDEIG.LE.NEGCNT) THEN 00754 * The wanted eigenvalue lies to the left 00755 SGNDEF = -ONE 00756 ELSE 00757 * The wanted eigenvalue lies to the right 00758 SGNDEF = ONE 00759 ENDIF 00760 * We only use the RQCORR if it improves the 00761 * the iterate reasonably. 00762 IF( ( RQCORR*SGNDEF.GE.ZERO ) 00763 $ .AND.( LAMBDA + RQCORR.LE. RIGHT) 00764 $ .AND.( LAMBDA + RQCORR.GE. LEFT) 00765 $ ) THEN 00766 USEDRQ = .TRUE. 00767 * Store new midpoint of bisection interval in WORK 00768 IF(SGNDEF.EQ.ONE) THEN 00769 * The current LAMBDA is on the left of the true 00770 * eigenvalue 00771 LEFT = LAMBDA 00772 * We prefer to assume that the error estimate 00773 * is correct. We could make the interval not 00774 * as a bracket but to be modified if the RQCORR 00775 * chooses to. In this case, the RIGHT side should 00776 * be modified as follows: 00777 * RIGHT = MAX(RIGHT, LAMBDA + RQCORR) 00778 ELSE 00779 * The current LAMBDA is on the right of the true 00780 * eigenvalue 00781 RIGHT = LAMBDA 00782 * See comment about assuming the error estimate is 00783 * correct above. 00784 * LEFT = MIN(LEFT, LAMBDA + RQCORR) 00785 ENDIF 00786 WORK( WINDEX ) = 00787 $ HALF * (RIGHT + LEFT) 00788 * Take RQCORR since it has the correct sign and 00789 * improves the iterate reasonably 00790 LAMBDA = LAMBDA + RQCORR 00791 * Update width of error interval 00792 WERR( WINDEX ) = 00793 $ HALF * (RIGHT-LEFT) 00794 ELSE 00795 NEEDBS = .TRUE. 00796 ENDIF 00797 IF(RIGHT-LEFT.LT.RQTOL*ABS(LAMBDA)) THEN 00798 * The eigenvalue is computed to bisection accuracy 00799 * compute eigenvector and stop 00800 USEDBS = .TRUE. 00801 GOTO 120 00802 ELSEIF( ITER.LT.MAXITR ) THEN 00803 GOTO 120 00804 ELSEIF( ITER.EQ.MAXITR ) THEN 00805 NEEDBS = .TRUE. 00806 GOTO 120 00807 ELSE 00808 INFO = 5 00809 RETURN 00810 END IF 00811 ELSE 00812 STP2II = .FALSE. 00813 IF(USEDRQ .AND. USEDBS .AND. 00814 $ BSTRES.LE.RESID) THEN 00815 LAMBDA = BSTW 00816 STP2II = .TRUE. 00817 ENDIF 00818 IF (STP2II) THEN 00819 * improve error angle by second step 00820 CALL SLAR1V( IN, 1, IN, LAMBDA, 00821 $ D( IBEGIN ), L( IBEGIN ), 00822 $ WORK(INDLD+IBEGIN-1), 00823 $ WORK(INDLLD+IBEGIN-1), 00824 $ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ), 00825 $ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA, 00826 $ IWORK( IINDR+WINDEX ), 00827 $ ISUPPZ( 2*WINDEX-1 ), 00828 $ NRMINV, RESID, RQCORR, WORK( INDWRK ) ) 00829 ENDIF 00830 WORK( WINDEX ) = LAMBDA 00831 END IF 00832 * 00833 * Compute FP-vector support w.r.t. whole matrix 00834 * 00835 ISUPPZ( 2*WINDEX-1 ) = ISUPPZ( 2*WINDEX-1 )+OLDIEN 00836 ISUPPZ( 2*WINDEX ) = ISUPPZ( 2*WINDEX )+OLDIEN 00837 ZFROM = ISUPPZ( 2*WINDEX-1 ) 00838 ZTO = ISUPPZ( 2*WINDEX ) 00839 ISUPMN = ISUPMN + OLDIEN 00840 ISUPMX = ISUPMX + OLDIEN 00841 * Ensure vector is ok if support in the RQI has changed 00842 IF(ISUPMN.LT.ZFROM) THEN 00843 DO 122 II = ISUPMN,ZFROM-1 00844 Z( II, WINDEX ) = ZERO 00845 122 CONTINUE 00846 ENDIF 00847 IF(ISUPMX.GT.ZTO) THEN 00848 DO 123 II = ZTO+1,ISUPMX 00849 Z( II, WINDEX ) = ZERO 00850 123 CONTINUE 00851 ENDIF 00852 CALL SSCAL( ZTO-ZFROM+1, NRMINV, 00853 $ Z( ZFROM, WINDEX ), 1 ) 00854 125 CONTINUE 00855 * Update W 00856 W( WINDEX ) = LAMBDA+SIGMA 00857 * Recompute the gaps on the left and right 00858 * But only allow them to become larger and not 00859 * smaller (which can only happen through "bad" 00860 * cancellation and doesn't reflect the theory 00861 * where the initial gaps are underestimated due 00862 * to WERR being too crude.) 00863 IF(.NOT.ESKIP) THEN 00864 IF( K.GT.1) THEN 00865 WGAP( WINDMN ) = MAX( WGAP(WINDMN), 00866 $ W(WINDEX)-WERR(WINDEX) 00867 $ - W(WINDMN)-WERR(WINDMN) ) 00868 ENDIF 00869 IF( WINDEX.LT.WEND ) THEN 00870 WGAP( WINDEX ) = MAX( SAVGAP, 00871 $ W( WINDPL )-WERR( WINDPL ) 00872 $ - W( WINDEX )-WERR( WINDEX) ) 00873 ENDIF 00874 ENDIF 00875 IDONE = IDONE + 1 00876 ENDIF 00877 * here ends the code for the current child 00878 * 00879 139 CONTINUE 00880 * Proceed to any remaining child nodes 00881 NEWFST = J + 1 00882 140 CONTINUE 00883 150 CONTINUE 00884 NDEPTH = NDEPTH + 1 00885 GO TO 40 00886 END IF 00887 IBEGIN = IEND + 1 00888 WBEGIN = WEND + 1 00889 170 CONTINUE 00890 * 00891 00892 RETURN 00893 * 00894 * End of SLARRV 00895 * 00896 END
1.7.3 
