1*> \brief \b SSTEMR 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download SSTEMR + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sstemr.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sstemr.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sstemr.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE SSTEMR( 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* REAL VL, VU 30* .. 31* .. Array Arguments .. 32* INTEGER ISUPPZ( * ), IWORK( * ) 33* REAL D( * ), E( * ), W( * ), WORK( * ) 34* REAL Z( LDZ, * ) 35* .. 36* 37* 38*> \par Purpose: 39* ============= 40*> 41*> \verbatim 42*> 43*> SSTEMR 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.SSTEMR 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 SLARRE, 292*> if INFO = 2X, internal error in SLARRV. 293*> Here, the digit X = ABS( IINFO ) < 10, where IINFO is 294*> the nonzero error code returned by SLARRE or 295*> SLARRV, 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 realOTHERcomputational 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 SSTEMR( 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 REAL VL, VU 331* .. 332* .. Array Arguments .. 333 INTEGER ISUPPZ( * ), IWORK( * ) 334 REAL D( * ), E( * ), W( * ), WORK( * ) 335 REAL Z( LDZ, * ) 336* .. 337* 338* ===================================================================== 339* 340* .. Parameters .. 341 REAL ZERO, ONE, FOUR, MINRGP 342 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, 343 $ FOUR = 4.0E0, 344 $ MINRGP = 3.0E-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 REAL 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 REAL SLAMCH, SLANST 361 EXTERNAL LSAME, SLAMCH, SLANST 362* .. 363* .. External Subroutines .. 364 EXTERNAL SCOPY, SLAE2, SLAEV2, SLARRC, SLARRE, SLARRJ, 365 $ SLARRR, SLARRV, SLASRT, SSCAL, SSWAP, XERBLA 366* .. 367* .. Intrinsic Functions .. 368 INTRINSIC MAX, MIN, SQRT 369* .. 370* .. Executable Statements .. 371* 372* Test the input parameters. 373* 374 WANTZ = LSAME( JOBZ, 'V' ) 375 ALLEIG = LSAME( RANGE, 'A' ) 376 VALEIG = LSAME( RANGE, 'V' ) 377 INDEIG = LSAME( RANGE, 'I' ) 378* 379 LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) ) 380 ZQUERY = ( NZC.EQ.-1 ) 381 382* SSTEMR needs WORK of size 6*N, IWORK of size 3*N. 383* In addition, SLARRE needs WORK of size 6*N, IWORK of size 5*N. 384* Furthermore, SLARRV needs WORK of size 12*N, IWORK of size 7*N. 385 IF( WANTZ ) THEN 386 LWMIN = 18*N 387 LIWMIN = 10*N 388 ELSE 389* need less workspace if only the eigenvalues are wanted 390 LWMIN = 12*N 391 LIWMIN = 8*N 392 ENDIF 393 394 WL = ZERO 395 WU = ZERO 396 IIL = 0 397 IIU = 0 398 NSPLIT = 0 399 400 IF( VALEIG ) THEN 401* We do not reference VL, VU in the cases RANGE = 'I','A' 402* The interval (WL, WU] contains all the wanted eigenvalues. 403* It is either given by the user or computed in SLARRE. 404 WL = VL 405 WU = VU 406 ELSEIF( INDEIG ) THEN 407* We do not reference IL, IU in the cases RANGE = 'V','A' 408 IIL = IL 409 IIU = IU 410 ENDIF 411* 412 INFO = 0 413 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 414 INFO = -1 415 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 416 INFO = -2 417 ELSE IF( N.LT.0 ) THEN 418 INFO = -3 419 ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN 420 INFO = -7 421 ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN 422 INFO = -8 423 ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN 424 INFO = -9 425 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 426 INFO = -13 427 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 428 INFO = -17 429 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 430 INFO = -19 431 END IF 432* 433* Get machine constants. 434* 435 SAFMIN = SLAMCH( 'Safe minimum' ) 436 EPS = SLAMCH( 'Precision' ) 437 SMLNUM = SAFMIN / EPS 438 BIGNUM = ONE / SMLNUM 439 RMIN = SQRT( SMLNUM ) 440 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 441* 442 IF( INFO.EQ.0 ) THEN 443 WORK( 1 ) = LWMIN 444 IWORK( 1 ) = LIWMIN 445* 446 IF( WANTZ .AND. ALLEIG ) THEN 447 NZCMIN = N 448 ELSE IF( WANTZ .AND. VALEIG ) THEN 449 CALL SLARRC( 'T', N, VL, VU, D, E, SAFMIN, 450 $ NZCMIN, ITMP, ITMP2, INFO ) 451 ELSE IF( WANTZ .AND. INDEIG ) THEN 452 NZCMIN = IIU-IIL+1 453 ELSE 454* WANTZ .EQ. FALSE. 455 NZCMIN = 0 456 ENDIF 457 IF( ZQUERY .AND. INFO.EQ.0 ) THEN 458 Z( 1,1 ) = NZCMIN 459 ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN 460 INFO = -14 461 END IF 462 END IF 463 464 IF( INFO.NE.0 ) THEN 465* 466 CALL XERBLA( 'SSTEMR', -INFO ) 467* 468 RETURN 469 ELSE IF( LQUERY .OR. ZQUERY ) THEN 470 RETURN 471 END IF 472* 473* Handle N = 0, 1, and 2 cases immediately 474* 475 M = 0 476 IF( N.EQ.0 ) 477 $ RETURN 478* 479 IF( N.EQ.1 ) THEN 480 IF( ALLEIG .OR. INDEIG ) THEN 481 M = 1 482 W( 1 ) = D( 1 ) 483 ELSE 484 IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN 485 M = 1 486 W( 1 ) = D( 1 ) 487 END IF 488 END IF 489 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 490 Z( 1, 1 ) = ONE 491 ISUPPZ(1) = 1 492 ISUPPZ(2) = 1 493 END IF 494 RETURN 495 END IF 496* 497 IF( N.EQ.2 ) THEN 498 IF( .NOT.WANTZ ) THEN 499 CALL SLAE2( D(1), E(1), D(2), R1, R2 ) 500 ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 501 CALL SLAEV2( D(1), E(1), D(2), R1, R2, CS, SN ) 502 END IF 503 IF( ALLEIG.OR. 504 $ (VALEIG.AND.(R2.GT.WL).AND. 505 $ (R2.LE.WU)).OR. 506 $ (INDEIG.AND.(IIL.EQ.1)) ) THEN 507 M = M+1 508 W( M ) = R2 509 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 510 Z( 1, M ) = -SN 511 Z( 2, M ) = CS 512* Note: At most one of SN and CS can be zero. 513 IF (SN.NE.ZERO) THEN 514 IF (CS.NE.ZERO) THEN 515 ISUPPZ(2*M-1) = 1 516 ISUPPZ(2*M) = 2 517 ELSE 518 ISUPPZ(2*M-1) = 1 519 ISUPPZ(2*M) = 1 520 END IF 521 ELSE 522 ISUPPZ(2*M-1) = 2 523 ISUPPZ(2*M) = 2 524 END IF 525 ENDIF 526 ENDIF 527 IF( ALLEIG.OR. 528 $ (VALEIG.AND.(R1.GT.WL).AND. 529 $ (R1.LE.WU)).OR. 530 $ (INDEIG.AND.(IIU.EQ.2)) ) THEN 531 M = M+1 532 W( M ) = R1 533 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN 534 Z( 1, M ) = CS 535 Z( 2, M ) = SN 536* Note: At most one of SN and CS can be zero. 537 IF (SN.NE.ZERO) THEN 538 IF (CS.NE.ZERO) THEN 539 ISUPPZ(2*M-1) = 1 540 ISUPPZ(2*M) = 2 541 ELSE 542 ISUPPZ(2*M-1) = 1 543 ISUPPZ(2*M) = 1 544 END IF 545 ELSE 546 ISUPPZ(2*M-1) = 2 547 ISUPPZ(2*M) = 2 548 END IF 549 ENDIF 550 ENDIF 551 ELSE 552 553* Continue with general N 554 555 INDGRS = 1 556 INDERR = 2*N + 1 557 INDGP = 3*N + 1 558 INDD = 4*N + 1 559 INDE2 = 5*N + 1 560 INDWRK = 6*N + 1 561* 562 IINSPL = 1 563 IINDBL = N + 1 564 IINDW = 2*N + 1 565 IINDWK = 3*N + 1 566* 567* Scale matrix to allowable range, if necessary. 568* The allowable range is related to the PIVMIN parameter; see the 569* comments in SLARRD. The preference for scaling small values 570* up is heuristic; we expect users' matrices not to be close to the 571* RMAX threshold. 572* 573 SCALE = ONE 574 TNRM = SLANST( 'M', N, D, E ) 575 IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN 576 SCALE = RMIN / TNRM 577 ELSE IF( TNRM.GT.RMAX ) THEN 578 SCALE = RMAX / TNRM 579 END IF 580 IF( SCALE.NE.ONE ) THEN 581 CALL SSCAL( N, SCALE, D, 1 ) 582 CALL SSCAL( N-1, SCALE, E, 1 ) 583 TNRM = TNRM*SCALE 584 IF( VALEIG ) THEN 585* If eigenvalues in interval have to be found, 586* scale (WL, WU] accordingly 587 WL = WL*SCALE 588 WU = WU*SCALE 589 ENDIF 590 END IF 591* 592* Compute the desired eigenvalues of the tridiagonal after splitting 593* into smaller subblocks if the corresponding off-diagonal elements 594* are small 595* THRESH is the splitting parameter for SLARRE 596* A negative THRESH forces the old splitting criterion based on the 597* size of the off-diagonal. A positive THRESH switches to splitting 598* which preserves relative accuracy. 599* 600 IF( TRYRAC ) THEN 601* Test whether the matrix warrants the more expensive relative approach. 602 CALL SLARRR( N, D, E, IINFO ) 603 ELSE 604* The user does not care about relative accurately eigenvalues 605 IINFO = -1 606 ENDIF 607* Set the splitting criterion 608 IF (IINFO.EQ.0) THEN 609 THRESH = EPS 610 ELSE 611 THRESH = -EPS 612* relative accuracy is desired but T does not guarantee it 613 TRYRAC = .FALSE. 614 ENDIF 615* 616 IF( TRYRAC ) THEN 617* Copy original diagonal, needed to guarantee relative accuracy 618 CALL SCOPY(N,D,1,WORK(INDD),1) 619 ENDIF 620* Store the squares of the offdiagonal values of T 621 DO 5 J = 1, N-1 622 WORK( INDE2+J-1 ) = E(J)**2 623 5 CONTINUE 624 625* Set the tolerance parameters for bisection 626 IF( .NOT.WANTZ ) THEN 627* SLARRE computes the eigenvalues to full precision. 628 RTOL1 = FOUR * EPS 629 RTOL2 = FOUR * EPS 630 ELSE 631* SLARRE computes the eigenvalues to less than full precision. 632* SLARRV will refine the eigenvalue approximations, and we can 633* need less accurate initial bisection in SLARRE. 634* Note: these settings do only affect the subset case and SLARRE 635 RTOL1 = MAX( SQRT(EPS)*5.0E-2, FOUR * EPS ) 636 RTOL2 = MAX( SQRT(EPS)*5.0E-3, FOUR * EPS ) 637 ENDIF 638 CALL SLARRE( RANGE, N, WL, WU, IIL, IIU, D, E, 639 $ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT, 640 $ IWORK( IINSPL ), M, W, WORK( INDERR ), 641 $ WORK( INDGP ), IWORK( IINDBL ), 642 $ IWORK( IINDW ), WORK( INDGRS ), PIVMIN, 643 $ WORK( INDWRK ), IWORK( IINDWK ), IINFO ) 644 IF( IINFO.NE.0 ) THEN 645 INFO = 10 + ABS( IINFO ) 646 RETURN 647 END IF 648* Note that if RANGE .NE. 'V', SLARRE computes bounds on the desired 649* part of the spectrum. All desired eigenvalues are contained in 650* (WL,WU] 651 652 653 IF( WANTZ ) THEN 654* 655* Compute the desired eigenvectors corresponding to the computed 656* eigenvalues 657* 658 CALL SLARRV( N, WL, WU, D, E, 659 $ PIVMIN, IWORK( IINSPL ), M, 660 $ 1, M, MINRGP, RTOL1, RTOL2, 661 $ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ), 662 $ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ, 663 $ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO ) 664 IF( IINFO.NE.0 ) THEN 665 INFO = 20 + ABS( IINFO ) 666 RETURN 667 END IF 668 ELSE 669* SLARRE computes eigenvalues of the (shifted) root representation 670* SLARRV returns the eigenvalues of the unshifted matrix. 671* However, if the eigenvectors are not desired by the user, we need 672* to apply the corresponding shifts from SLARRE to obtain the 673* eigenvalues of the original matrix. 674 DO 20 J = 1, M 675 ITMP = IWORK( IINDBL+J-1 ) 676 W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) ) 677 20 CONTINUE 678 END IF 679* 680 681 IF ( TRYRAC ) THEN 682* Refine computed eigenvalues so that they are relatively accurate 683* with respect to the original matrix T. 684 IBEGIN = 1 685 WBEGIN = 1 686 DO 39 JBLK = 1, IWORK( IINDBL+M-1 ) 687 IEND = IWORK( IINSPL+JBLK-1 ) 688 IN = IEND - IBEGIN + 1 689 WEND = WBEGIN - 1 690* check if any eigenvalues have to be refined in this block 691 36 CONTINUE 692 IF( WEND.LT.M ) THEN 693 IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN 694 WEND = WEND + 1 695 GO TO 36 696 END IF 697 END IF 698 IF( WEND.LT.WBEGIN ) THEN 699 IBEGIN = IEND + 1 700 GO TO 39 701 END IF 702 703 OFFSET = IWORK(IINDW+WBEGIN-1)-1 704 IFIRST = IWORK(IINDW+WBEGIN-1) 705 ILAST = IWORK(IINDW+WEND-1) 706 RTOL2 = FOUR * EPS 707 CALL SLARRJ( IN, 708 $ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1), 709 $ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN), 710 $ WORK( INDERR+WBEGIN-1 ), 711 $ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN, 712 $ TNRM, IINFO ) 713 IBEGIN = IEND + 1 714 WBEGIN = WEND + 1 715 39 CONTINUE 716 ENDIF 717* 718* If matrix was scaled, then rescale eigenvalues appropriately. 719* 720 IF( SCALE.NE.ONE ) THEN 721 CALL SSCAL( M, ONE / SCALE, W, 1 ) 722 END IF 723 END IF 724* 725* If eigenvalues are not in increasing order, then sort them, 726* possibly along with eigenvectors. 727* 728 IF( NSPLIT.GT.1 .OR. N.EQ.2 ) THEN 729 IF( .NOT. WANTZ ) THEN 730 CALL SLASRT( 'I', M, W, IINFO ) 731 IF( IINFO.NE.0 ) THEN 732 INFO = 3 733 RETURN 734 END IF 735 ELSE 736 DO 60 J = 1, M - 1 737 I = 0 738 TMP = W( J ) 739 DO 50 JJ = J + 1, M 740 IF( W( JJ ).LT.TMP ) THEN 741 I = JJ 742 TMP = W( JJ ) 743 END IF 744 50 CONTINUE 745 IF( I.NE.0 ) THEN 746 W( I ) = W( J ) 747 W( J ) = TMP 748 IF( WANTZ ) THEN 749 CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 750 ITMP = ISUPPZ( 2*I-1 ) 751 ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 ) 752 ISUPPZ( 2*J-1 ) = ITMP 753 ITMP = ISUPPZ( 2*I ) 754 ISUPPZ( 2*I ) = ISUPPZ( 2*J ) 755 ISUPPZ( 2*J ) = ITMP 756 END IF 757 END IF 758 60 CONTINUE 759 END IF 760 ENDIF 761* 762* 763 WORK( 1 ) = LWMIN 764 IWORK( 1 ) = LIWMIN 765 RETURN 766* 767* End of SSTEMR 768* 769 END 770