1*> \brief <b> DSTEVR 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 DSTEVR + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstevr.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstevr.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstevr.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE DSTEVR( 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* DOUBLE PRECISION ABSTOL, VL, VU 29* .. 30* .. Array Arguments .. 31* INTEGER ISUPPZ( * ), IWORK( * ) 32* DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) 33* .. 34* 35* 36*> \par Purpose: 37* ============= 38*> 39*> \verbatim 40*> 41*> DSTEVR 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, DSTEVR calls DSTEMR to compute the 47*> eigenspectrum using Relatively Robust Representations. DSTEMR 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 : DSTEVR calls DSTEMR when the full spectrum is requested 73*> on machines which conform to the ieee-754 floating point standard. 74*> DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and 75*> when partial spectrum requests are made. 76*> 77*> Normal execution of DSTEMR 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, DSTEBZ and 101*> DSTEIN 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 131*> \endverbatim 132*> 133*> \param[in] VU 134*> \verbatim 135*> VU is DOUBLE PRECISION 136*> If RANGE='V', the lower and upper bounds of the interval to 137*> be searched for eigenvalues. VL < VU. 138*> Not referenced if RANGE = 'A' or 'I'. 139*> \endverbatim 140*> 141*> \param[in] IL 142*> \verbatim 143*> IL is INTEGER 144*> \endverbatim 145*> 146*> \param[in] IU 147*> \verbatim 148*> IU is INTEGER 149*> If RANGE='I', the indices (in ascending order) of the 150*> smallest and largest eigenvalues to be returned. 151*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 152*> Not referenced if RANGE = 'A' or 'V'. 153*> \endverbatim 154*> 155*> \param[in] ABSTOL 156*> \verbatim 157*> ABSTOL is DOUBLE PRECISION 158*> The absolute error tolerance for the eigenvalues. 159*> An approximate eigenvalue is accepted as converged 160*> when it is determined to lie in an interval [a,b] 161*> of width less than or equal to 162*> 163*> ABSTOL + EPS * max( |a|,|b| ) , 164*> 165*> where EPS is the machine precision. If ABSTOL is less than 166*> or equal to zero, then EPS*|T| will be used in its place, 167*> where |T| is the 1-norm of the tridiagonal matrix obtained 168*> by reducing A to tridiagonal form. 169*> 170*> See "Computing Small Singular Values of Bidiagonal Matrices 171*> with Guaranteed High Relative Accuracy," by Demmel and 172*> Kahan, LAPACK Working Note #3. 173*> 174*> If high relative accuracy is important, set ABSTOL to 175*> DLAMCH( 'Safe minimum' ). Doing so will guarantee that 176*> eigenvalues are computed to high relative accuracy when 177*> possible in future releases. The current code does not 178*> make any guarantees about high relative accuracy, but 179*> future releases will. See J. Barlow and J. Demmel, 180*> "Computing Accurate Eigensystems of Scaled Diagonally 181*> Dominant Matrices", LAPACK Working Note #7, for a discussion 182*> of which matrices define their eigenvalues to high relative 183*> accuracy. 184*> \endverbatim 185*> 186*> \param[out] M 187*> \verbatim 188*> M is INTEGER 189*> The total number of eigenvalues found. 0 <= M <= N. 190*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 191*> \endverbatim 192*> 193*> \param[out] W 194*> \verbatim 195*> W is DOUBLE PRECISION array, dimension (N) 196*> The first M elements contain the selected eigenvalues in 197*> ascending order. 198*> \endverbatim 199*> 200*> \param[out] Z 201*> \verbatim 202*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) ) 203*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z 204*> contain the orthonormal eigenvectors of the matrix A 205*> corresponding to the selected eigenvalues, with the i-th 206*> column of Z holding the eigenvector associated with W(i). 207*> Note: the user must ensure that at least max(1,M) columns are 208*> supplied in the array Z; if RANGE = 'V', the exact value of M 209*> is not known in advance and an upper bound must be used. 210*> \endverbatim 211*> 212*> \param[in] LDZ 213*> \verbatim 214*> LDZ is INTEGER 215*> The leading dimension of the array Z. LDZ >= 1, and if 216*> JOBZ = 'V', LDZ >= max(1,N). 217*> \endverbatim 218*> 219*> \param[out] ISUPPZ 220*> \verbatim 221*> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) 222*> The support of the eigenvectors in Z, i.e., the indices 223*> indicating the nonzero elements in Z. The i-th eigenvector 224*> is nonzero only in elements ISUPPZ( 2*i-1 ) through 225*> ISUPPZ( 2*i ). 226*> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 227*> \endverbatim 228*> 229*> \param[out] WORK 230*> \verbatim 231*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) 232*> On exit, if INFO = 0, WORK(1) returns the optimal (and 233*> minimal) LWORK. 234*> \endverbatim 235*> 236*> \param[in] LWORK 237*> \verbatim 238*> LWORK is INTEGER 239*> The dimension of the array WORK. LWORK >= max(1,20*N). 240*> 241*> If LWORK = -1, then a workspace query is assumed; the routine 242*> only calculates the optimal sizes of the WORK and IWORK 243*> arrays, returns these values as the first entries of the WORK 244*> and IWORK arrays, and no error message related to LWORK or 245*> LIWORK is issued by XERBLA. 246*> \endverbatim 247*> 248*> \param[out] IWORK 249*> \verbatim 250*> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) 251*> On exit, if INFO = 0, IWORK(1) returns the optimal (and 252*> minimal) LIWORK. 253*> \endverbatim 254*> 255*> \param[in] LIWORK 256*> \verbatim 257*> LIWORK is INTEGER 258*> The dimension of the array IWORK. LIWORK >= max(1,10*N). 259*> 260*> If LIWORK = -1, then a workspace query is assumed; the 261*> routine only calculates the optimal sizes of the WORK and 262*> IWORK arrays, returns these values as the first entries of 263*> the WORK and IWORK arrays, and no error message related to 264*> LWORK or LIWORK is issued by XERBLA. 265*> \endverbatim 266*> 267*> \param[out] INFO 268*> \verbatim 269*> INFO is INTEGER 270*> = 0: successful exit 271*> < 0: if INFO = -i, the i-th argument had an illegal value 272*> > 0: Internal error 273*> \endverbatim 274* 275* Authors: 276* ======== 277* 278*> \author Univ. of Tennessee 279*> \author Univ. of California Berkeley 280*> \author Univ. of Colorado Denver 281*> \author NAG Ltd. 282* 283*> \date November 2015 284* 285*> \ingroup doubleOTHEReigen 286* 287*> \par Contributors: 288* ================== 289*> 290*> Inderjit Dhillon, IBM Almaden, USA \n 291*> Osni Marques, LBNL/NERSC, USA \n 292*> Ken Stanley, Computer Science Division, University of 293*> California at Berkeley, USA \n 294*> 295* ===================================================================== 296 SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, 297 $ M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, 298 $ LIWORK, INFO ) 299* 300* -- LAPACK driver routine (version 3.6.0) -- 301* -- LAPACK is a software package provided by Univ. of Tennessee, -- 302* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 303* November 2015 304* 305* .. Scalar Arguments .. 306 CHARACTER JOBZ, RANGE 307 INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N 308 DOUBLE PRECISION ABSTOL, VL, VU 309* .. 310* .. Array Arguments .. 311 INTEGER ISUPPZ( * ), IWORK( * ) 312 DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) 313* .. 314* 315* ===================================================================== 316* 317* .. Parameters .. 318 DOUBLE PRECISION ZERO, ONE, TWO 319 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 ) 320* .. 321* .. Local Scalars .. 322 LOGICAL ALLEIG, INDEIG, TEST, LQUERY, VALEIG, WANTZ, 323 $ TRYRAC 324 CHARACTER ORDER 325 INTEGER I, IEEEOK, IMAX, INDIBL, INDIFL, INDISP, 326 $ INDIWO, ISCALE, ITMP1, J, JJ, LIWMIN, LWMIN, 327 $ NSPLIT 328 DOUBLE PRECISION BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, 329 $ TMP1, TNRM, VLL, VUU 330* .. 331* .. External Functions .. 332 LOGICAL LSAME 333 INTEGER ILAENV 334 DOUBLE PRECISION DLAMCH, DLANST 335 EXTERNAL LSAME, ILAENV, DLAMCH, DLANST 336* .. 337* .. External Subroutines .. 338 EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTEMR, DSTEIN, DSTERF, 339 $ DSWAP, XERBLA 340* .. 341* .. Intrinsic Functions .. 342 INTRINSIC MAX, MIN, SQRT 343* .. 344* .. Executable Statements .. 345* 346* 347* Test the input parameters. 348* 349 IEEEOK = ILAENV( 10, 'DSTEVR', 'N', 1, 2, 3, 4 ) 350* 351 WANTZ = LSAME( JOBZ, 'V' ) 352 ALLEIG = LSAME( RANGE, 'A' ) 353 VALEIG = LSAME( RANGE, 'V' ) 354 INDEIG = LSAME( RANGE, 'I' ) 355* 356 LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) ) 357 LWMIN = MAX( 1, 20*N ) 358 LIWMIN = MAX( 1, 10*N ) 359* 360* 361 INFO = 0 362 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 363 INFO = -1 364 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 365 INFO = -2 366 ELSE IF( N.LT.0 ) THEN 367 INFO = -3 368 ELSE 369 IF( VALEIG ) THEN 370 IF( N.GT.0 .AND. VU.LE.VL ) 371 $ INFO = -7 372 ELSE IF( INDEIG ) THEN 373 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 374 INFO = -8 375 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 376 INFO = -9 377 END IF 378 END IF 379 END IF 380 IF( INFO.EQ.0 ) THEN 381 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 382 INFO = -14 383 END IF 384 END IF 385* 386 IF( INFO.EQ.0 ) THEN 387 WORK( 1 ) = LWMIN 388 IWORK( 1 ) = LIWMIN 389* 390 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 391 INFO = -17 392 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 393 INFO = -19 394 END IF 395 END IF 396* 397 IF( INFO.NE.0 ) THEN 398 CALL XERBLA( 'DSTEVR', -INFO ) 399 RETURN 400 ELSE IF( LQUERY ) THEN 401 RETURN 402 END IF 403* 404* Quick return if possible 405* 406 M = 0 407 IF( N.EQ.0 ) 408 $ RETURN 409* 410 IF( N.EQ.1 ) THEN 411 IF( ALLEIG .OR. INDEIG ) THEN 412 M = 1 413 W( 1 ) = D( 1 ) 414 ELSE 415 IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN 416 M = 1 417 W( 1 ) = D( 1 ) 418 END IF 419 END IF 420 IF( WANTZ ) 421 $ Z( 1, 1 ) = ONE 422 RETURN 423 END IF 424* 425* Get machine constants. 426* 427 SAFMIN = DLAMCH( 'Safe minimum' ) 428 EPS = DLAMCH( 'Precision' ) 429 SMLNUM = SAFMIN / EPS 430 BIGNUM = ONE / SMLNUM 431 RMIN = SQRT( SMLNUM ) 432 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 433* 434* 435* Scale matrix to allowable range, if necessary. 436* 437 ISCALE = 0 438 IF( VALEIG ) THEN 439 VLL = VL 440 VUU = VU 441 END IF 442* 443 TNRM = DLANST( 'M', N, D, E ) 444 IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN 445 ISCALE = 1 446 SIGMA = RMIN / TNRM 447 ELSE IF( TNRM.GT.RMAX ) THEN 448 ISCALE = 1 449 SIGMA = RMAX / TNRM 450 END IF 451 IF( ISCALE.EQ.1 ) THEN 452 CALL DSCAL( N, SIGMA, D, 1 ) 453 CALL DSCAL( N-1, SIGMA, E( 1 ), 1 ) 454 IF( VALEIG ) THEN 455 VLL = VL*SIGMA 456 VUU = VU*SIGMA 457 END IF 458 END IF 459 460* Initialize indices into workspaces. Note: These indices are used only 461* if DSTERF or DSTEMR fail. 462 463* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and 464* stores the block indices of each of the M<=N eigenvalues. 465 INDIBL = 1 466* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and 467* stores the starting and finishing indices of each block. 468 INDISP = INDIBL + N 469* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors 470* that corresponding to eigenvectors that fail to converge in 471* DSTEIN. This information is discarded; if any fail, the driver 472* returns INFO > 0. 473 INDIFL = INDISP + N 474* INDIWO is the offset of the remaining integer workspace. 475 INDIWO = INDISP + N 476* 477* If all eigenvalues are desired, then 478* call DSTERF or DSTEMR. If this fails for some eigenvalue, then 479* try DSTEBZ. 480* 481* 482 TEST = .FALSE. 483 IF( INDEIG ) THEN 484 IF( IL.EQ.1 .AND. IU.EQ.N ) THEN 485 TEST = .TRUE. 486 END IF 487 END IF 488 IF( ( ALLEIG .OR. TEST ) .AND. IEEEOK.EQ.1 ) THEN 489 CALL DCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 ) 490 IF( .NOT.WANTZ ) THEN 491 CALL DCOPY( N, D, 1, W, 1 ) 492 CALL DSTERF( N, W, WORK, INFO ) 493 ELSE 494 CALL DCOPY( N, D, 1, WORK( N+1 ), 1 ) 495 IF (ABSTOL .LE. TWO*N*EPS) THEN 496 TRYRAC = .TRUE. 497 ELSE 498 TRYRAC = .FALSE. 499 END IF 500 CALL DSTEMR( JOBZ, 'A', N, WORK( N+1 ), WORK, VL, VU, IL, 501 $ IU, M, W, Z, LDZ, N, ISUPPZ, TRYRAC, 502 $ WORK( 2*N+1 ), LWORK-2*N, IWORK, LIWORK, INFO ) 503* 504 END IF 505 IF( INFO.EQ.0 ) THEN 506 M = N 507 GO TO 10 508 END IF 509 INFO = 0 510 END IF 511* 512* Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN. 513* 514 IF( WANTZ ) THEN 515 ORDER = 'B' 516 ELSE 517 ORDER = 'E' 518 END IF 519 520 CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M, 521 $ NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK, 522 $ IWORK( INDIWO ), INFO ) 523* 524 IF( WANTZ ) THEN 525 CALL DSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ), 526 $ Z, LDZ, WORK, IWORK( INDIWO ), IWORK( INDIFL ), 527 $ INFO ) 528 END IF 529* 530* If matrix was scaled, then rescale eigenvalues appropriately. 531* 532 10 CONTINUE 533 IF( ISCALE.EQ.1 ) THEN 534 IF( INFO.EQ.0 ) THEN 535 IMAX = M 536 ELSE 537 IMAX = INFO - 1 538 END IF 539 CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) 540 END IF 541* 542* If eigenvalues are not in order, then sort them, along with 543* eigenvectors. 544* 545 IF( WANTZ ) THEN 546 DO 30 J = 1, M - 1 547 I = 0 548 TMP1 = W( J ) 549 DO 20 JJ = J + 1, M 550 IF( W( JJ ).LT.TMP1 ) THEN 551 I = JJ 552 TMP1 = W( JJ ) 553 END IF 554 20 CONTINUE 555* 556 IF( I.NE.0 ) THEN 557 ITMP1 = IWORK( I ) 558 W( I ) = W( J ) 559 IWORK( I ) = IWORK( J ) 560 W( J ) = TMP1 561 IWORK( J ) = ITMP1 562 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 563 END IF 564 30 CONTINUE 565 END IF 566* 567* Causes problems with tests 19 & 20: 568* IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002 569* 570* 571 WORK( 1 ) = LWMIN 572 IWORK( 1 ) = LIWMIN 573 RETURN 574* 575* End of DSTEVR 576* 577 END 578