1*> \brief <b> SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b> 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download SSTEVR + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sstevr.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sstevr.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sstevr.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE SSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, 22* M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, 23* LIWORK, INFO ) 24* 25* .. Scalar Arguments .. 26* CHARACTER JOBZ, RANGE 27* INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N 28* REAL ABSTOL, VL, VU 29* .. 30* .. Array Arguments .. 31* INTEGER ISUPPZ( * ), IWORK( * ) 32* REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) 33* .. 34* 35* 36*> \par Purpose: 37* ============= 38*> 39*> \verbatim 40*> 41*> SSTEVR computes selected eigenvalues and, optionally, eigenvectors 42*> of a real symmetric tridiagonal matrix T. Eigenvalues and 43*> eigenvectors can be selected by specifying either a range of values 44*> or a range of indices for the desired eigenvalues. 45*> 46*> Whenever possible, SSTEVR calls SSTEMR to compute the 47*> eigenspectrum using Relatively Robust Representations. SSTEMR 48*> computes eigenvalues by the dqds algorithm, while orthogonal 49*> eigenvectors are computed from various "good" L D L^T representations 50*> (also known as Relatively Robust Representations). Gram-Schmidt 51*> orthogonalization is avoided as far as possible. More specifically, 52*> the various steps of the algorithm are as follows. For the i-th 53*> unreduced block of T, 54*> (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T 55*> is a relatively robust representation, 56*> (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high 57*> relative accuracy by the dqds algorithm, 58*> (c) If there is a cluster of close eigenvalues, "choose" sigma_i 59*> close to the cluster, and go to step (a), 60*> (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, 61*> compute the corresponding eigenvector by forming a 62*> rank-revealing twisted factorization. 63*> The desired accuracy of the output can be specified by the input 64*> parameter ABSTOL. 65*> 66*> For more details, see "A new O(n^2) algorithm for the symmetric 67*> tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, 68*> Computer Science Division Technical Report No. UCB//CSD-97-971, 69*> UC Berkeley, May 1997. 70*> 71*> 72*> Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested 73*> on machines which conform to the ieee-754 floating point standard. 74*> SSTEVR calls SSTEBZ and SSTEIN on non-ieee machines and 75*> when partial spectrum requests are made. 76*> 77*> Normal execution of SSTEMR may create NaNs and infinities and 78*> hence may abort due to a floating point exception in environments 79*> which do not handle NaNs and infinities in the ieee standard default 80*> manner. 81*> \endverbatim 82* 83* Arguments: 84* ========== 85* 86*> \param[in] JOBZ 87*> \verbatim 88*> JOBZ is CHARACTER*1 89*> = 'N': Compute eigenvalues only; 90*> = 'V': Compute eigenvalues and eigenvectors. 91*> \endverbatim 92*> 93*> \param[in] RANGE 94*> \verbatim 95*> RANGE is CHARACTER*1 96*> = 'A': all eigenvalues will be found. 97*> = 'V': all eigenvalues in the half-open interval (VL,VU] 98*> will be found. 99*> = 'I': the IL-th through IU-th eigenvalues will be found. 100*> For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and 101*> SSTEIN are called 102*> \endverbatim 103*> 104*> \param[in] N 105*> \verbatim 106*> N is INTEGER 107*> The order of the matrix. N >= 0. 108*> \endverbatim 109*> 110*> \param[in,out] D 111*> \verbatim 112*> D is REAL array, dimension (N) 113*> On entry, the n diagonal elements of the tridiagonal matrix 114*> A. 115*> On exit, D may be multiplied by a constant factor chosen 116*> to avoid over/underflow in computing the eigenvalues. 117*> \endverbatim 118*> 119*> \param[in,out] E 120*> \verbatim 121*> E is REAL array, dimension (max(1,N-1)) 122*> On entry, the (n-1) subdiagonal elements of the tridiagonal 123*> matrix A in elements 1 to N-1 of E. 124*> On exit, E may be multiplied by a constant factor chosen 125*> to avoid over/underflow in computing the eigenvalues. 126*> \endverbatim 127*> 128*> \param[in] VL 129*> \verbatim 130*> VL is REAL 131*> If RANGE='V', the lower bound of the interval to 132*> be searched for eigenvalues. VL < VU. 133*> Not referenced if RANGE = 'A' or 'I'. 134*> \endverbatim 135*> 136*> \param[in] VU 137*> \verbatim 138*> VU is REAL 139*> If RANGE='V', the upper bound of the interval to 140*> be searched for eigenvalues. VL < VU. 141*> Not referenced if RANGE = 'A' or 'I'. 142*> \endverbatim 143*> 144*> \param[in] IL 145*> \verbatim 146*> IL is INTEGER 147*> If RANGE='I', the index of the 148*> smallest eigenvalue to be returned. 149*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 150*> Not referenced if RANGE = 'A' or 'V'. 151*> \endverbatim 152*> 153*> \param[in] IU 154*> \verbatim 155*> IU is INTEGER 156*> If RANGE='I', the index of the 157*> largest eigenvalue to be returned. 158*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 159*> Not referenced if RANGE = 'A' or 'V'. 160*> \endverbatim 161*> 162*> \param[in] ABSTOL 163*> \verbatim 164*> ABSTOL is REAL 165*> The absolute error tolerance for the eigenvalues. 166*> An approximate eigenvalue is accepted as converged 167*> when it is determined to lie in an interval [a,b] 168*> of width less than or equal to 169*> 170*> ABSTOL + EPS * max( |a|,|b| ) , 171*> 172*> where EPS is the machine precision. If ABSTOL is less than 173*> or equal to zero, then EPS*|T| will be used in its place, 174*> where |T| is the 1-norm of the tridiagonal matrix obtained 175*> by reducing A to tridiagonal form. 176*> 177*> See "Computing Small Singular Values of Bidiagonal Matrices 178*> with Guaranteed High Relative Accuracy," by Demmel and 179*> Kahan, LAPACK Working Note #3. 180*> 181*> If high relative accuracy is important, set ABSTOL to 182*> SLAMCH( 'Safe minimum' ). Doing so will guarantee that 183*> eigenvalues are computed to high relative accuracy when 184*> possible in future releases. The current code does not 185*> make any guarantees about high relative accuracy, but 186*> future releases will. See J. Barlow and J. Demmel, 187*> "Computing Accurate Eigensystems of Scaled Diagonally 188*> Dominant Matrices", LAPACK Working Note #7, for a discussion 189*> of which matrices define their eigenvalues to high relative 190*> accuracy. 191*> \endverbatim 192*> 193*> \param[out] M 194*> \verbatim 195*> M is INTEGER 196*> The total number of eigenvalues found. 0 <= M <= N. 197*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 198*> \endverbatim 199*> 200*> \param[out] W 201*> \verbatim 202*> W is REAL array, dimension (N) 203*> The first M elements contain the selected eigenvalues in 204*> ascending order. 205*> \endverbatim 206*> 207*> \param[out] Z 208*> \verbatim 209*> Z is REAL array, dimension (LDZ, max(1,M) ) 210*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z 211*> contain the orthonormal eigenvectors of the matrix A 212*> corresponding to the selected eigenvalues, with the i-th 213*> column of Z holding the eigenvector associated with W(i). 214*> Note: the user must ensure that at least max(1,M) columns are 215*> supplied in the array Z; if RANGE = 'V', the exact value of M 216*> is not known in advance and an upper bound must be used. 217*> \endverbatim 218*> 219*> \param[in] LDZ 220*> \verbatim 221*> LDZ is INTEGER 222*> The leading dimension of the array Z. LDZ >= 1, and if 223*> JOBZ = 'V', LDZ >= max(1,N). 224*> \endverbatim 225*> 226*> \param[out] ISUPPZ 227*> \verbatim 228*> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) 229*> The support of the eigenvectors in Z, i.e., the indices 230*> indicating the nonzero elements in Z. The i-th eigenvector 231*> is nonzero only in elements ISUPPZ( 2*i-1 ) through 232*> ISUPPZ( 2*i ). 233*> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 234*> \endverbatim 235*> 236*> \param[out] WORK 237*> \verbatim 238*> WORK is REAL array, dimension (MAX(1,LWORK)) 239*> On exit, if INFO = 0, WORK(1) returns the optimal (and 240*> minimal) LWORK. 241*> \endverbatim 242*> 243*> \param[in] LWORK 244*> \verbatim 245*> LWORK is INTEGER 246*> The dimension of the array WORK. LWORK >= 20*N. 247*> 248*> If LWORK = -1, then a workspace query is assumed; the routine 249*> only calculates the optimal sizes of the WORK and IWORK 250*> arrays, returns these values as the first entries of the WORK 251*> and IWORK arrays, and no error message related to LWORK or 252*> LIWORK is issued by XERBLA. 253*> \endverbatim 254*> 255*> \param[out] IWORK 256*> \verbatim 257*> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) 258*> On exit, if INFO = 0, IWORK(1) returns the optimal (and 259*> minimal) LIWORK. 260*> \endverbatim 261*> 262*> \param[in] LIWORK 263*> \verbatim 264*> LIWORK is INTEGER 265*> The dimension of the array IWORK. LIWORK >= 10*N. 266*> 267*> If LIWORK = -1, then a workspace query is assumed; the 268*> routine only calculates the optimal sizes of the WORK and 269*> IWORK arrays, returns these values as the first entries of 270*> the WORK and IWORK arrays, and no error message related to 271*> LWORK or LIWORK is issued by XERBLA. 272*> \endverbatim 273*> 274*> \param[out] INFO 275*> \verbatim 276*> INFO is INTEGER 277*> = 0: successful exit 278*> < 0: if INFO = -i, the i-th argument had an illegal value 279*> > 0: Internal error 280*> \endverbatim 281* 282* Authors: 283* ======== 284* 285*> \author Univ. of Tennessee 286*> \author Univ. of California Berkeley 287*> \author Univ. of Colorado Denver 288*> \author NAG Ltd. 289* 290*> \ingroup realOTHEReigen 291* 292*> \par Contributors: 293* ================== 294*> 295*> Inderjit Dhillon, IBM Almaden, USA \n 296*> Osni Marques, LBNL/NERSC, USA \n 297*> Ken Stanley, Computer Science Division, University of 298*> California at Berkeley, USA \n 299*> Jason Riedy, Computer Science Division, University of 300*> California at Berkeley, USA \n 301*> 302* ===================================================================== 303 SUBROUTINE SSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, 304 $ M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, 305 $ LIWORK, INFO ) 306* 307* -- LAPACK driver routine -- 308* -- LAPACK is a software package provided by Univ. of Tennessee, -- 309* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 310* 311* .. Scalar Arguments .. 312 CHARACTER JOBZ, RANGE 313 INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N 314 REAL ABSTOL, VL, VU 315* .. 316* .. Array Arguments .. 317 INTEGER ISUPPZ( * ), IWORK( * ) 318 REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) 319* .. 320* 321* ===================================================================== 322* 323* .. Parameters .. 324 REAL ZERO, ONE, TWO 325 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 ) 326* .. 327* .. Local Scalars .. 328 LOGICAL ALLEIG, INDEIG, TEST, LQUERY, VALEIG, WANTZ, 329 $ TRYRAC 330 CHARACTER ORDER 331 INTEGER I, IEEEOK, IMAX, INDIBL, INDIFL, INDISP, 332 $ INDIWO, ISCALE, J, JJ, LIWMIN, LWMIN, NSPLIT 333 REAL BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, 334 $ TMP1, TNRM, VLL, VUU 335* .. 336* .. External Functions .. 337 LOGICAL LSAME 338 INTEGER ILAENV 339 REAL SLAMCH, SLANST 340 EXTERNAL LSAME, ILAENV, SLAMCH, SLANST 341* .. 342* .. External Subroutines .. 343 EXTERNAL SCOPY, SSCAL, SSTEBZ, SSTEMR, SSTEIN, SSTERF, 344 $ SSWAP, XERBLA 345* .. 346* .. Intrinsic Functions .. 347 INTRINSIC MAX, MIN, SQRT 348* .. 349* .. Executable Statements .. 350* 351* 352* Test the input parameters. 353* 354 IEEEOK = ILAENV( 10, 'SSTEVR', 'N', 1, 2, 3, 4 ) 355* 356 WANTZ = LSAME( JOBZ, 'V' ) 357 ALLEIG = LSAME( RANGE, 'A' ) 358 VALEIG = LSAME( RANGE, 'V' ) 359 INDEIG = LSAME( RANGE, 'I' ) 360* 361 LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) ) 362 LWMIN = MAX( 1, 20*N ) 363 LIWMIN = MAX(1, 10*N ) 364* 365* 366 INFO = 0 367 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 368 INFO = -1 369 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 370 INFO = -2 371 ELSE IF( N.LT.0 ) THEN 372 INFO = -3 373 ELSE 374 IF( VALEIG ) THEN 375 IF( N.GT.0 .AND. VU.LE.VL ) 376 $ INFO = -7 377 ELSE IF( INDEIG ) THEN 378 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 379 INFO = -8 380 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 381 INFO = -9 382 END IF 383 END IF 384 END IF 385 IF( INFO.EQ.0 ) THEN 386 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 387 INFO = -14 388 END IF 389 END IF 390* 391 IF( INFO.EQ.0 ) THEN 392 WORK( 1 ) = LWMIN 393 IWORK( 1 ) = LIWMIN 394* 395 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 396 INFO = -17 397 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 398 INFO = -19 399 END IF 400 END IF 401* 402 IF( INFO.NE.0 ) THEN 403 CALL XERBLA( 'SSTEVR', -INFO ) 404 RETURN 405 ELSE IF( LQUERY ) THEN 406 RETURN 407 END IF 408* 409* Quick return if possible 410* 411 M = 0 412 IF( N.EQ.0 ) 413 $ RETURN 414* 415 IF( N.EQ.1 ) THEN 416 IF( ALLEIG .OR. INDEIG ) THEN 417 M = 1 418 W( 1 ) = D( 1 ) 419 ELSE 420 IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN 421 M = 1 422 W( 1 ) = D( 1 ) 423 END IF 424 END IF 425 IF( WANTZ ) 426 $ Z( 1, 1 ) = ONE 427 RETURN 428 END IF 429* 430* Get machine constants. 431* 432 SAFMIN = SLAMCH( 'Safe minimum' ) 433 EPS = SLAMCH( 'Precision' ) 434 SMLNUM = SAFMIN / EPS 435 BIGNUM = ONE / SMLNUM 436 RMIN = SQRT( SMLNUM ) 437 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 438* 439* 440* Scale matrix to allowable range, if necessary. 441* 442 ISCALE = 0 443 IF( VALEIG ) THEN 444 VLL = VL 445 VUU = VU 446 END IF 447* 448 TNRM = SLANST( 'M', N, D, E ) 449 IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN 450 ISCALE = 1 451 SIGMA = RMIN / TNRM 452 ELSE IF( TNRM.GT.RMAX ) THEN 453 ISCALE = 1 454 SIGMA = RMAX / TNRM 455 END IF 456 IF( ISCALE.EQ.1 ) THEN 457 CALL SSCAL( N, SIGMA, D, 1 ) 458 CALL SSCAL( N-1, SIGMA, E( 1 ), 1 ) 459 IF( VALEIG ) THEN 460 VLL = VL*SIGMA 461 VUU = VU*SIGMA 462 END IF 463 END IF 464 465* Initialize indices into workspaces. Note: These indices are used only 466* if SSTERF or SSTEMR fail. 467 468* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and 469* stores the block indices of each of the M<=N eigenvalues. 470 INDIBL = 1 471* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and 472* stores the starting and finishing indices of each block. 473 INDISP = INDIBL + N 474* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors 475* that corresponding to eigenvectors that fail to converge in 476* SSTEIN. This information is discarded; if any fail, the driver 477* returns INFO > 0. 478 INDIFL = INDISP + N 479* INDIWO is the offset of the remaining integer workspace. 480 INDIWO = INDISP + N 481* 482* If all eigenvalues are desired, then 483* call SSTERF or SSTEMR. If this fails for some eigenvalue, then 484* try SSTEBZ. 485* 486* 487 TEST = .FALSE. 488 IF( INDEIG ) THEN 489 IF( IL.EQ.1 .AND. IU.EQ.N ) THEN 490 TEST = .TRUE. 491 END IF 492 END IF 493 IF( ( ALLEIG .OR. TEST ) .AND. IEEEOK.EQ.1 ) THEN 494 CALL SCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 ) 495 IF( .NOT.WANTZ ) THEN 496 CALL SCOPY( N, D, 1, W, 1 ) 497 CALL SSTERF( N, W, WORK, INFO ) 498 ELSE 499 CALL SCOPY( N, D, 1, WORK( N+1 ), 1 ) 500 IF (ABSTOL .LE. TWO*N*EPS) THEN 501 TRYRAC = .TRUE. 502 ELSE 503 TRYRAC = .FALSE. 504 END IF 505 CALL SSTEMR( JOBZ, 'A', N, WORK( N+1 ), WORK, VL, VU, IL, 506 $ IU, M, W, Z, LDZ, N, ISUPPZ, TRYRAC, 507 $ WORK( 2*N+1 ), LWORK-2*N, IWORK, LIWORK, INFO ) 508* 509 END IF 510 IF( INFO.EQ.0 ) THEN 511 M = N 512 GO TO 10 513 END IF 514 INFO = 0 515 END IF 516* 517* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. 518* 519 IF( WANTZ ) THEN 520 ORDER = 'B' 521 ELSE 522 ORDER = 'E' 523 END IF 524 525 CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M, 526 $ NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK, 527 $ IWORK( INDIWO ), INFO ) 528* 529 IF( WANTZ ) THEN 530 CALL SSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ), 531 $ Z, LDZ, WORK, IWORK( INDIWO ), IWORK( INDIFL ), 532 $ INFO ) 533 END IF 534* 535* If matrix was scaled, then rescale eigenvalues appropriately. 536* 537 10 CONTINUE 538 IF( ISCALE.EQ.1 ) THEN 539 IF( INFO.EQ.0 ) THEN 540 IMAX = M 541 ELSE 542 IMAX = INFO - 1 543 END IF 544 CALL SSCAL( IMAX, ONE / SIGMA, W, 1 ) 545 END IF 546* 547* If eigenvalues are not in order, then sort them, along with 548* eigenvectors. 549* 550 IF( WANTZ ) THEN 551 DO 30 J = 1, M - 1 552 I = 0 553 TMP1 = W( J ) 554 DO 20 JJ = J + 1, M 555 IF( W( JJ ).LT.TMP1 ) THEN 556 I = JJ 557 TMP1 = W( JJ ) 558 END IF 559 20 CONTINUE 560* 561 IF( I.NE.0 ) THEN 562 W( I ) = W( J ) 563 W( J ) = TMP1 564 CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 565 END IF 566 30 CONTINUE 567 END IF 568* 569* Causes problems with tests 19 & 20: 570* IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002 571* 572* 573 WORK( 1 ) = LWMIN 574 IWORK( 1 ) = LIWMIN 575 RETURN 576* 577* End of SSTEVR 578* 579 END 580