1*> \brief \b DSTEMR 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download DSTEMR + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstemr.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstemr.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstemr.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, 22* M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, 23* IWORK, LIWORK, INFO ) 24* 25* .. Scalar Arguments .. 26* CHARACTER JOBZ, RANGE 27* LOGICAL TRYRAC 28* INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N 29* DOUBLE PRECISION VL, VU 30* .. 31* .. Array Arguments .. 32* INTEGER ISUPPZ( * ), IWORK( * ) 33* DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ) 34* DOUBLE PRECISION Z( LDZ, * ) 35* .. 36* 37* 38*> \par Purpose: 39* ============= 40*> 41*> \verbatim 42*> 43*> DSTEMR computes selected eigenvalues and, optionally, eigenvectors 44*> of a real symmetric tridiagonal matrix T. Any such unreduced matrix has 45*> a well defined set of pairwise different real eigenvalues, the corresponding 46*> real eigenvectors are pairwise orthogonal. 47*> 48*> The spectrum may be computed either completely or partially by specifying 49*> either an interval (VL,VU] or a range of indices IL:IU for the desired 50*> eigenvalues. 51*> 52*> Depending on the number of desired eigenvalues, these are computed either 53*> by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are 54*> computed by the use of various suitable L D L^T factorizations near clusters 55*> of close eigenvalues (referred to as RRRs, Relatively Robust 56*> Representations). An informal sketch of the algorithm follows. 57*> 58*> For each unreduced block (submatrix) of T, 59*> (a) Compute T - sigma I = L D L^T, so that L and D 60*> define all the wanted eigenvalues to high relative accuracy. 61*> This means that small relative changes in the entries of D and L 62*> cause only small relative changes in the eigenvalues and 63*> eigenvectors. The standard (unfactored) representation of the 64*> tridiagonal matrix T does not have this property in general. 65*> (b) Compute the eigenvalues to suitable accuracy. 66*> If the eigenvectors are desired, the algorithm attains full 67*> accuracy of the computed eigenvalues only right before 68*> the corresponding vectors have to be computed, see steps c) and d). 69*> (c) For each cluster of close eigenvalues, select a new 70*> shift close to the cluster, find a new factorization, and refine 71*> the shifted eigenvalues to suitable accuracy. 72*> (d) For each eigenvalue with a large enough relative separation compute 73*> the corresponding eigenvector by forming a rank revealing twisted 74*> factorization. Go back to (c) for any clusters that remain. 75*> 76*> For more details, see: 77*> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations 78*> to compute orthogonal eigenvectors of symmetric tridiagonal matrices," 79*> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. 80*> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and 81*> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, 82*> 2004. Also LAPACK Working Note 154. 83*> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric 84*> tridiagonal eigenvalue/eigenvector problem", 85*> Computer Science Division Technical Report No. UCB/CSD-97-971, 86*> UC Berkeley, May 1997. 87*> 88*> Further Details 89*> 1.DSTEMR works only on machines which follow IEEE-754 90*> floating-point standard in their handling of infinities and NaNs. 91*> This permits the use of efficient inner loops avoiding a check for 92*> zero divisors. 93*> \endverbatim 94* 95* Arguments: 96* ========== 97* 98*> \param[in] JOBZ 99*> \verbatim 100*> JOBZ is CHARACTER*1 101*> = 'N': Compute eigenvalues only; 102*> = 'V': Compute eigenvalues and eigenvectors. 103*> \endverbatim 104*> 105*> \param[in] RANGE 106*> \verbatim 107*> RANGE is CHARACTER*1 108*> = 'A': all eigenvalues will be found. 109*> = 'V': all eigenvalues in the half-open interval (VL,VU] 110*> will be found. 111*> = 'I': the IL-th through IU-th eigenvalues will be found. 112*> \endverbatim 113*> 114*> \param[in] N 115*> \verbatim 116*> N is INTEGER 117*> The order of the matrix. N >= 0. 118*> \endverbatim 119*> 120*> \param[in,out] D 121*> \verbatim 122*> D is DOUBLE PRECISION array, dimension (N) 123*> On entry, the N diagonal elements of the tridiagonal matrix 124*> T. On exit, D is overwritten. 125*> \endverbatim 126*> 127*> \param[in,out] E 128*> \verbatim 129*> E is DOUBLE PRECISION array, dimension (N) 130*> On entry, the (N-1) subdiagonal elements of the tridiagonal 131*> matrix T in elements 1 to N-1 of E. E(N) need not be set on 132*> input, but is used internally as workspace. 133*> On exit, E is overwritten. 134*> \endverbatim 135*> 136*> \param[in] VL 137*> \verbatim 138*> VL is DOUBLE PRECISION 139*> 140*> If RANGE='V', the lower bound of the interval to 141*> be searched for eigenvalues. VL < VU. 142*> Not referenced if RANGE = 'A' or 'I'. 143*> \endverbatim 144*> 145*> \param[in] VU 146*> \verbatim 147*> VU is DOUBLE PRECISION 148*> 149*> If RANGE='V', the upper bound of the interval to 150*> be searched for eigenvalues. VL < VU. 151*> Not referenced if RANGE = 'A' or 'I'. 152*> \endverbatim 153*> 154*> \param[in] IL 155*> \verbatim 156*> IL is INTEGER 157*> 158*> If RANGE='I', the index of the 159*> smallest eigenvalue to be returned. 160*> 1 <= IL <= IU <= N, if N > 0. 161*> Not referenced if RANGE = 'A' or 'V'. 162*> \endverbatim 163*> 164*> \param[in] IU 165*> \verbatim 166*> IU is INTEGER 167*> 168*> If RANGE='I', the index of the 169*> largest eigenvalue to be returned. 170*> 1 <= IL <= IU <= N, if N > 0. 171*> Not referenced if RANGE = 'A' or 'V'. 172*> \endverbatim 173*> 174*> \param[out] M 175*> \verbatim 176*> M is INTEGER 177*> The total number of eigenvalues found. 0 <= M <= N. 178*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 179*> \endverbatim 180*> 181*> \param[out] W 182*> \verbatim 183*> W is DOUBLE PRECISION array, dimension (N) 184*> The first M elements contain the selected eigenvalues in 185*> ascending order. 186*> \endverbatim 187*> 188*> \param[out] Z 189*> \verbatim 190*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) ) 191*> If JOBZ = 'V', and if INFO = 0, then the first M columns of Z 192*> contain the orthonormal eigenvectors of the matrix T 193*> corresponding to the selected eigenvalues, with the i-th 194*> column of Z holding the eigenvector associated with W(i). 195*> If JOBZ = 'N', then Z is not referenced. 196*> Note: the user must ensure that at least max(1,M) columns are 197*> supplied in the array Z; if RANGE = 'V', the exact value of M 198*> is not known in advance and can be computed with a workspace 199*> query by setting NZC = -1, see below. 200*> \endverbatim 201*> 202*> \param[in] LDZ 203*> \verbatim 204*> LDZ is INTEGER 205*> The leading dimension of the array Z. LDZ >= 1, and if 206*> JOBZ = 'V', then LDZ >= max(1,N). 207*> \endverbatim 208*> 209*> \param[in] NZC 210*> \verbatim 211*> NZC is INTEGER 212*> The number of eigenvectors to be held in the array Z. 213*> If RANGE = 'A', then NZC >= max(1,N). 214*> If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]. 215*> If RANGE = 'I', then NZC >= IU-IL+1. 216*> If NZC = -1, then a workspace query is assumed; the 217*> routine calculates the number of columns of the array Z that 218*> are needed to hold the eigenvectors. 219*> This value is returned as the first entry of the Z array, and 220*> no error message related to NZC is issued by XERBLA. 221*> \endverbatim 222*> 223*> \param[out] ISUPPZ 224*> \verbatim 225*> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) 226*> The support of the eigenvectors in Z, i.e., the indices 227*> indicating the nonzero elements in Z. The i-th computed eigenvector 228*> is nonzero only in elements ISUPPZ( 2*i-1 ) through 229*> ISUPPZ( 2*i ). This is relevant in the case when the matrix 230*> is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0. 231*> \endverbatim 232*> 233*> \param[in,out] TRYRAC 234*> \verbatim 235*> TRYRAC is LOGICAL 236*> If TRYRAC = .TRUE., indicates that the code should check whether 237*> the tridiagonal matrix defines its eigenvalues to high relative 238*> accuracy. If so, the code uses relative-accuracy preserving 239*> algorithms that might be (a bit) slower depending on the matrix. 240*> If the matrix does not define its eigenvalues to high relative 241*> accuracy, the code can uses possibly faster algorithms. 242*> If TRYRAC = .FALSE., the code is not required to guarantee 243*> relatively accurate eigenvalues and can use the fastest possible 244*> techniques. 245*> On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix 246*> does not define its eigenvalues to high relative accuracy. 247*> \endverbatim 248*> 249*> \param[out] WORK 250*> \verbatim 251*> WORK is DOUBLE PRECISION array, dimension (LWORK) 252*> On exit, if INFO = 0, WORK(1) returns the optimal 253*> (and minimal) LWORK. 254*> \endverbatim 255*> 256*> \param[in] LWORK 257*> \verbatim 258*> LWORK is INTEGER 259*> The dimension of the array WORK. LWORK >= max(1,18*N) 260*> if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'. 261*> If LWORK = -1, then a workspace query is assumed; the routine 262*> only calculates the optimal size of the WORK array, returns 263*> this value as the first entry of the WORK array, and no error 264*> message related to LWORK is issued by XERBLA. 265*> \endverbatim 266*> 267*> \param[out] IWORK 268*> \verbatim 269*> IWORK is INTEGER array, dimension (LIWORK) 270*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 271*> \endverbatim 272*> 273*> \param[in] LIWORK 274*> \verbatim 275*> LIWORK is INTEGER 276*> The dimension of the array IWORK. LIWORK >= max(1,10*N) 277*> if the eigenvectors are desired, and LIWORK >= max(1,8*N) 278*> if only the eigenvalues are to be computed. 279*> If LIWORK = -1, then a workspace query is assumed; the 280*> routine only calculates the optimal size of the IWORK array, 281*> returns this value as the first entry of the IWORK array, and 282*> no error message related to LIWORK is issued by XERBLA. 283*> \endverbatim 284*> 285*> \param[out] INFO 286*> \verbatim 287*> INFO is INTEGER 288*> On exit, INFO 289*> = 0: successful exit 290*> < 0: if INFO = -i, the i-th argument had an illegal value 291*> > 0: if INFO = 1X, internal error in DLARRE, 292*> if INFO = 2X, internal error in DLARRV. 293*> Here, the digit X = ABS( IINFO ) < 10, where IINFO is 294*> the nonzero error code returned by DLARRE or 295*> DLARRV, respectively. 296*> \endverbatim 297* 298* Authors: 299* ======== 300* 301*> \author Univ. of Tennessee 302*> \author Univ. of California Berkeley 303*> \author Univ. of Colorado Denver 304*> \author NAG Ltd. 305* 306*> \ingroup doubleOTHERcomputational 307* 308*> \par Contributors: 309* ================== 310*> 311*> Beresford Parlett, University of California, Berkeley, USA \n 312*> Jim Demmel, University of California, Berkeley, USA \n 313*> Inderjit Dhillon, University of Texas, Austin, USA \n 314*> Osni Marques, LBNL/NERSC, USA \n 315*> Christof Voemel, University of California, Berkeley, USA 316* 317* ===================================================================== 318 SUBROUTINE DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, 319 $ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, 320 $ IWORK, LIWORK, INFO ) 321* 322* -- LAPACK computational routine -- 323* -- LAPACK is a software package provided by Univ. of Tennessee, -- 324* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 325* 326* .. Scalar Arguments .. 327 CHARACTER JOBZ, RANGE 328 LOGICAL TRYRAC 329 INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N 330 DOUBLE PRECISION VL, VU 331* .. 332* .. Array Arguments .. 333 INTEGER ISUPPZ( * ), IWORK( * ) 334 DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ) 335 DOUBLE PRECISION Z( LDZ, * ) 336* .. 337* 338* ===================================================================== 339* 340* .. Parameters .. 341 DOUBLE PRECISION ZERO, ONE, FOUR, MINRGP 342 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, 343 $ FOUR = 4.0D0, 344 $ MINRGP = 1.0D-3 ) 345* .. 346* .. Local Scalars .. 347 LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY 348 INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW, 349 $ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD, 350 $ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP, 351 $ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT, 352 $ NZCMIN, OFFSET, WBEGIN, WEND 353 DOUBLE PRECISION BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN, 354 $ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN, 355 $ THRESH, TMP, TNRM, WL, WU 356* .. 357* .. 358* .. External Functions .. 359 LOGICAL LSAME 360 DOUBLE PRECISION DLAMCH, DLANST 361 EXTERNAL LSAME, DLAMCH, DLANST 362* .. 363* .. External Subroutines .. 364 EXTERNAL DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ, 365 $ DLARRR, DLARRV, DLASRT, DSCAL, DSWAP, XERBLA 366* .. 367* .. Intrinsic Functions .. 368 INTRINSIC MAX, MIN, SQRT 369 370 371* .. 372* .. Executable Statements .. 373* 374* Test the input parameters. 375* 376 WANTZ = LSAME( JOBZ, 'V' ) 377 ALLEIG = LSAME( RANGE, 'A' ) 378 VALEIG = LSAME( RANGE, 'V' ) 379 INDEIG = LSAME( RANGE, 'I' ) 380* 381 LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) ) 382 ZQUERY = ( NZC.EQ.-1 ) 383 384* DSTEMR needs WORK of size 6*N, IWORK of size 3*N. 385* In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N. 386* Furthermore, DLARRV needs WORK of size 12*N, IWORK of size 7*N. 387 IF( WANTZ ) THEN 388 LWMIN = 18*N 389 LIWMIN = 10*N 390 ELSE 391* need less workspace if only the eigenvalues are wanted 392 LWMIN = 12*N 393 LIWMIN = 8*N 394 ENDIF 395 396 WL = ZERO 397 WU = ZERO 398 IIL = 0 399 IIU = 0 400 NSPLIT = 0 401 402 IF( VALEIG ) THEN 403* We do not reference VL, VU in the cases RANGE = 'I','A' 404* The interval (WL, WU] contains all the wanted eigenvalues. 405* It is either given by the user or computed in DLARRE. 406 WL = VL 407 WU = VU 408 ELSEIF( INDEIG ) THEN 409* We do not reference IL, IU in the cases RANGE = 'V','A' 410 IIL = IL 411 IIU = IU 412 ENDIF 413* 414 INFO = 0 415 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 416 INFO = -1 417 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 418 INFO = -2 419 ELSE IF( N.LT.0 ) THEN 420 INFO = -3 421 ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN 422 INFO = -7 423 ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN 424 INFO = -8 425 ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN 426 INFO = -9 427 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 428 INFO = -13 429 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 430 INFO = -17 431 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 432 INFO = -19 433 END IF 434* 435* Get machine constants. 436* 437 SAFMIN = DLAMCH( 'Safe minimum' ) 438 EPS = DLAMCH( 'Precision' ) 439 SMLNUM = SAFMIN / EPS 440 BIGNUM = ONE / SMLNUM 441 RMIN = SQRT( SMLNUM ) 442 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 443* 444 IF( INFO.EQ.0 ) THEN 445 WORK( 1 ) = LWMIN 446 IWORK( 1 ) = LIWMIN 447* 448 IF( WANTZ .AND. ALLEIG ) THEN 449 NZCMIN = N 450 ELSE IF( WANTZ .AND. VALEIG ) THEN 451 CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN, 452 $ NZCMIN, ITMP, ITMP2, INFO ) 453 ELSE IF( WANTZ .AND. INDEIG ) THEN 454 NZCMIN = IIU-IIL+1 455 ELSE 456* WANTZ .EQ. FALSE. 457 NZCMIN = 0 458 ENDIF 459 IF( ZQUERY .AND. INFO.EQ.0 ) THEN 460 Z( 1,1 ) = NZCMIN 461 ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN 462 INFO = -14 463 END IF 464 END IF 465 466 IF( INFO.NE.0 ) THEN 467* 468 CALL XERBLA( 'DSTEMR', -INFO ) 469* 470 RETURN 471 ELSE IF( LQUERY .OR. ZQUERY ) THEN 472 RETURN 473 END IF 474* 475* Handle N = 0, 1, and 2 cases immediately 476* 477 M = 0 478 IF( N.EQ.0 ) 479 $ RETURN 480* 481 IF( N.EQ.1 ) THEN 482 IF( ALLEIG .OR. INDEIG ) THEN 483 M = 1 484 W( 1 ) = D( 1 ) 485 ELSE 486 IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN 487 M = 1 488 W( 1 ) = D( 1 ) 489 END IF 490 END IF 491 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 492 Z( 1, 1 ) = ONE 493 ISUPPZ(1) = 1 494 ISUPPZ(2) = 1 495 END IF 496 RETURN 497 END IF 498* 499 IF( N.EQ.2 ) THEN 500 IF( .NOT.WANTZ ) THEN 501 CALL DLAE2( D(1), E(1), D(2), R1, R2 ) 502 ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 503 CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN ) 504 END IF 505 IF( ALLEIG.OR. 506 $ (VALEIG.AND.(R2.GT.WL).AND. 507 $ (R2.LE.WU)).OR. 508 $ (INDEIG.AND.(IIL.EQ.1)) ) THEN 509 M = M+1 510 W( M ) = R2 511 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 512 Z( 1, M ) = -SN 513 Z( 2, M ) = CS 514* Note: At most one of SN and CS can be zero. 515 IF (SN.NE.ZERO) THEN 516 IF (CS.NE.ZERO) THEN 517 ISUPPZ(2*M-1) = 1 518 ISUPPZ(2*M) = 2 519 ELSE 520 ISUPPZ(2*M-1) = 1 521 ISUPPZ(2*M) = 1 522 END IF 523 ELSE 524 ISUPPZ(2*M-1) = 2 525 ISUPPZ(2*M) = 2 526 END IF 527 ENDIF 528 ENDIF 529 IF( ALLEIG.OR. 530 $ (VALEIG.AND.(R1.GT.WL).AND. 531 $ (R1.LE.WU)).OR. 532 $ (INDEIG.AND.(IIU.EQ.2)) ) THEN 533 M = M+1 534 W( M ) = R1 535 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 536 Z( 1, M ) = CS 537 Z( 2, M ) = SN 538* Note: At most one of SN and CS can be zero. 539 IF (SN.NE.ZERO) THEN 540 IF (CS.NE.ZERO) THEN 541 ISUPPZ(2*M-1) = 1 542 ISUPPZ(2*M) = 2 543 ELSE 544 ISUPPZ(2*M-1) = 1 545 ISUPPZ(2*M) = 1 546 END IF 547 ELSE 548 ISUPPZ(2*M-1) = 2 549 ISUPPZ(2*M) = 2 550 END IF 551 ENDIF 552 ENDIF 553 554 ELSE 555 556* Continue with general N 557 558 INDGRS = 1 559 INDERR = 2*N + 1 560 INDGP = 3*N + 1 561 INDD = 4*N + 1 562 INDE2 = 5*N + 1 563 INDWRK = 6*N + 1 564* 565 IINSPL = 1 566 IINDBL = N + 1 567 IINDW = 2*N + 1 568 IINDWK = 3*N + 1 569* 570* Scale matrix to allowable range, if necessary. 571* The allowable range is related to the PIVMIN parameter; see the 572* comments in DLARRD. The preference for scaling small values 573* up is heuristic; we expect users' matrices not to be close to the 574* RMAX threshold. 575* 576 SCALE = ONE 577 TNRM = DLANST( 'M', N, D, E ) 578 IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN 579 SCALE = RMIN / TNRM 580 ELSE IF( TNRM.GT.RMAX ) THEN 581 SCALE = RMAX / TNRM 582 END IF 583 IF( SCALE.NE.ONE ) THEN 584 CALL DSCAL( N, SCALE, D, 1 ) 585 CALL DSCAL( N-1, SCALE, E, 1 ) 586 TNRM = TNRM*SCALE 587 IF( VALEIG ) THEN 588* If eigenvalues in interval have to be found, 589* scale (WL, WU] accordingly 590 WL = WL*SCALE 591 WU = WU*SCALE 592 ENDIF 593 END IF 594* 595* Compute the desired eigenvalues of the tridiagonal after splitting 596* into smaller subblocks if the corresponding off-diagonal elements 597* are small 598* THRESH is the splitting parameter for DLARRE 599* A negative THRESH forces the old splitting criterion based on the 600* size of the off-diagonal. A positive THRESH switches to splitting 601* which preserves relative accuracy. 602* 603 IF( TRYRAC ) THEN 604* Test whether the matrix warrants the more expensive relative approach. 605 CALL DLARRR( N, D, E, IINFO ) 606 ELSE 607* The user does not care about relative accurately eigenvalues 608 IINFO = -1 609 ENDIF 610* Set the splitting criterion 611 IF (IINFO.EQ.0) THEN 612 THRESH = EPS 613 ELSE 614 THRESH = -EPS 615* relative accuracy is desired but T does not guarantee it 616 TRYRAC = .FALSE. 617 ENDIF 618* 619 IF( TRYRAC ) THEN 620* Copy original diagonal, needed to guarantee relative accuracy 621 CALL DCOPY(N,D,1,WORK(INDD),1) 622 ENDIF 623* Store the squares of the offdiagonal values of T 624 DO 5 J = 1, N-1 625 WORK( INDE2+J-1 ) = E(J)**2 626 5 CONTINUE 627 628* Set the tolerance parameters for bisection 629 IF( .NOT.WANTZ ) THEN 630* DLARRE computes the eigenvalues to full precision. 631 RTOL1 = FOUR * EPS 632 RTOL2 = FOUR * EPS 633 ELSE 634* DLARRE computes the eigenvalues to less than full precision. 635* DLARRV will refine the eigenvalue approximations, and we can 636* need less accurate initial bisection in DLARRE. 637* Note: these settings do only affect the subset case and DLARRE 638 RTOL1 = SQRT(EPS) 639 RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS ) 640 ENDIF 641 CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E, 642 $ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT, 643 $ IWORK( IINSPL ), M, W, WORK( INDERR ), 644 $ WORK( INDGP ), IWORK( IINDBL ), 645 $ IWORK( IINDW ), WORK( INDGRS ), PIVMIN, 646 $ WORK( INDWRK ), IWORK( IINDWK ), IINFO ) 647 IF( IINFO.NE.0 ) THEN 648 INFO = 10 + ABS( IINFO ) 649 RETURN 650 END IF 651* Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired 652* part of the spectrum. All desired eigenvalues are contained in 653* (WL,WU] 654 655 656 IF( WANTZ ) THEN 657* 658* Compute the desired eigenvectors corresponding to the computed 659* eigenvalues 660* 661 CALL DLARRV( N, WL, WU, D, E, 662 $ PIVMIN, IWORK( IINSPL ), M, 663 $ 1, M, MINRGP, RTOL1, RTOL2, 664 $ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ), 665 $ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ, 666 $ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO ) 667 IF( IINFO.NE.0 ) THEN 668 INFO = 20 + ABS( IINFO ) 669 RETURN 670 END IF 671 ELSE 672* DLARRE computes eigenvalues of the (shifted) root representation 673* DLARRV returns the eigenvalues of the unshifted matrix. 674* However, if the eigenvectors are not desired by the user, we need 675* to apply the corresponding shifts from DLARRE to obtain the 676* eigenvalues of the original matrix. 677 DO 20 J = 1, M 678 ITMP = IWORK( IINDBL+J-1 ) 679 W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) ) 680 20 CONTINUE 681 END IF 682* 683 684 IF ( TRYRAC ) THEN 685* Refine computed eigenvalues so that they are relatively accurate 686* with respect to the original matrix T. 687 IBEGIN = 1 688 WBEGIN = 1 689 DO 39 JBLK = 1, IWORK( IINDBL+M-1 ) 690 IEND = IWORK( IINSPL+JBLK-1 ) 691 IN = IEND - IBEGIN + 1 692 WEND = WBEGIN - 1 693* check if any eigenvalues have to be refined in this block 694 36 CONTINUE 695 IF( WEND.LT.M ) THEN 696 IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN 697 WEND = WEND + 1 698 GO TO 36 699 END IF 700 END IF 701 IF( WEND.LT.WBEGIN ) THEN 702 IBEGIN = IEND + 1 703 GO TO 39 704 END IF 705 706 OFFSET = IWORK(IINDW+WBEGIN-1)-1 707 IFIRST = IWORK(IINDW+WBEGIN-1) 708 ILAST = IWORK(IINDW+WEND-1) 709 RTOL2 = FOUR * EPS 710 CALL DLARRJ( IN, 711 $ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1), 712 $ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN), 713 $ WORK( INDERR+WBEGIN-1 ), 714 $ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN, 715 $ TNRM, IINFO ) 716 IBEGIN = IEND + 1 717 WBEGIN = WEND + 1 718 39 CONTINUE 719 ENDIF 720* 721* If matrix was scaled, then rescale eigenvalues appropriately. 722* 723 IF( SCALE.NE.ONE ) THEN 724 CALL DSCAL( M, ONE / SCALE, W, 1 ) 725 END IF 726 727 END IF 728 729* 730* If eigenvalues are not in increasing order, then sort them, 731* possibly along with eigenvectors. 732* 733 IF( NSPLIT.GT.1 .OR. N.EQ.2 ) THEN 734 IF( .NOT. WANTZ ) THEN 735 CALL DLASRT( 'I', M, W, IINFO ) 736 IF( IINFO.NE.0 ) THEN 737 INFO = 3 738 RETURN 739 END IF 740 ELSE 741 DO 60 J = 1, M - 1 742 I = 0 743 TMP = W( J ) 744 DO 50 JJ = J + 1, M 745 IF( W( JJ ).LT.TMP ) THEN 746 I = JJ 747 TMP = W( JJ ) 748 END IF 749 50 CONTINUE 750 IF( I.NE.0 ) THEN 751 W( I ) = W( J ) 752 W( J ) = TMP 753 IF( WANTZ ) THEN 754 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 755 ITMP = ISUPPZ( 2*I-1 ) 756 ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 ) 757 ISUPPZ( 2*J-1 ) = ITMP 758 ITMP = ISUPPZ( 2*I ) 759 ISUPPZ( 2*I ) = ISUPPZ( 2*J ) 760 ISUPPZ( 2*J ) = ITMP 761 END IF 762 END IF 763 60 CONTINUE 764 END IF 765 ENDIF 766* 767* 768 WORK( 1 ) = LWMIN 769 IWORK( 1 ) = LIWMIN 770 RETURN 771* 772* End of DSTEMR 773* 774 END 775