1 SUBROUTINE ZHEEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, 2 $ ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, 3 $ RWORK, LRWORK, IWORK, LIWORK, INFO ) 4* 5* -- LAPACK driver routine (version 3.0) -- 6* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., 7* Courant Institute, Argonne National Lab, and Rice University 8* March 20, 2000 9* 10* .. Scalar Arguments .. 11 CHARACTER JOBZ, RANGE, UPLO 12 INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK, 13 $ M, N 14 DOUBLE PRECISION ABSTOL, VL, VU 15* .. 16* .. Array Arguments .. 17 INTEGER ISUPPZ( * ), IWORK( * ) 18 DOUBLE PRECISION RWORK( * ), W( * ) 19 COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * ) 20* .. 21* 22* Purpose 23* ======= 24* 25* ZHEEVR computes selected eigenvalues and, optionally, eigenvectors 26* of a complex Hermitian matrix T. Eigenvalues and eigenvectors can 27* be selected by specifying either a range of values or a range of 28* indices for the desired eigenvalues. 29* 30* Whenever possible, ZHEEVR calls ZSTEGR to compute the 31* eigenspectrum using Relatively Robust Representations. ZSTEGR 32* computes eigenvalues by the dqds algorithm, while orthogonal 33* eigenvectors are computed from various "good" L D L^T representations 34* (also known as Relatively Robust Representations). Gram-Schmidt 35* orthogonalization is avoided as far as possible. More specifically, 36* the various steps of the algorithm are as follows. For the i-th 37* unreduced block of T, 38* (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T 39* is a relatively robust representation, 40* (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high 41* relative accuracy by the dqds algorithm, 42* (c) If there is a cluster of close eigenvalues, "choose" sigma_i 43* close to the cluster, and go to step (a), 44* (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, 45* compute the corresponding eigenvector by forming a 46* rank-revealing twisted factorization. 47* The desired accuracy of the output can be specified by the input 48* parameter ABSTOL. 49* 50* For more details, see "A new O(n^2) algorithm for the symmetric 51* tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, 52* Computer Science Division Technical Report No. UCB//CSD-97-971, 53* UC Berkeley, May 1997. 54* 55* 56* Note 1 : ZHEEVR calls ZSTEGR when the full spectrum is requested 57* on machines which conform to the ieee-754 floating point standard. 58* ZHEEVR calls DSTEBZ and ZSTEIN on non-ieee machines and 59* when partial spectrum requests are made. 60* 61* Normal execution of ZSTEGR may create NaNs and infinities and 62* hence may abort due to a floating point exception in environments 63* which do not handle NaNs and infinities in the ieee standard default 64* manner. 65* 66* Arguments 67* ========= 68* 69* JOBZ (input) CHARACTER*1 70* = 'N': Compute eigenvalues only; 71* = 'V': Compute eigenvalues and eigenvectors. 72* 73* RANGE (input) CHARACTER*1 74* = 'A': all eigenvalues will be found. 75* = 'V': all eigenvalues in the half-open interval (VL,VU] 76* will be found. 77* = 'I': the IL-th through IU-th eigenvalues will be found. 78********** For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and 79********** ZSTEIN are called 80* 81* UPLO (input) CHARACTER*1 82* = 'U': Upper triangle of A is stored; 83* = 'L': Lower triangle of A is stored. 84* 85* N (input) INTEGER 86* The order of the matrix A. N >= 0. 87* 88* A (input/output) COMPLEX*16 array, dimension (LDA, N) 89* On entry, the Hermitian matrix A. If UPLO = 'U', the 90* leading N-by-N upper triangular part of A contains the 91* upper triangular part of the matrix A. If UPLO = 'L', 92* the leading N-by-N lower triangular part of A contains 93* the lower triangular part of the matrix A. 94* On exit, the lower triangle (if UPLO='L') or the upper 95* triangle (if UPLO='U') of A, including the diagonal, is 96* destroyed. 97* 98* LDA (input) INTEGER 99* The leading dimension of the array A. LDA >= max(1,N). 100* 101* VL (input) DOUBLE PRECISION 102* VU (input) DOUBLE PRECISION 103* If RANGE='V', the lower and upper bounds of the interval to 104* be searched for eigenvalues. VL < VU. 105* Not referenced if RANGE = 'A' or 'I'. 106* 107* IL (input) INTEGER 108* IU (input) INTEGER 109* If RANGE='I', the indices (in ascending order) of the 110* smallest and largest eigenvalues to be returned. 111* 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 112* Not referenced if RANGE = 'A' or 'V'. 113* 114* ABSTOL (input) DOUBLE PRECISION 115* The absolute error tolerance for the eigenvalues. 116* An approximate eigenvalue is accepted as converged 117* when it is determined to lie in an interval [a,b] 118* of width less than or equal to 119* 120* ABSTOL + EPS * max( |a|,|b| ) , 121* 122* where EPS is the machine precision. If ABSTOL is less than 123* or equal to zero, then EPS*|T| will be used in its place, 124* where |T| is the 1-norm of the tridiagonal matrix obtained 125* by reducing A to tridiagonal form. 126* 127* See "Computing Small Singular Values of Bidiagonal Matrices 128* with Guaranteed High Relative Accuracy," by Demmel and 129* Kahan, LAPACK Working Note #3. 130* 131* If high relative accuracy is important, set ABSTOL to 132* DLAMCH( 'Safe minimum' ). Doing so will guarantee that 133* eigenvalues are computed to high relative accuracy when 134* possible in future releases. The current code does not 135* make any guarantees about high relative accuracy, but 136* furutre releases will. See J. Barlow and J. Demmel, 137* "Computing Accurate Eigensystems of Scaled Diagonally 138* Dominant Matrices", LAPACK Working Note #7, for a discussion 139* of which matrices define their eigenvalues to high relative 140* accuracy. 141* 142* M (output) INTEGER 143* The total number of eigenvalues found. 0 <= M <= N. 144* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 145* 146* W (output) DOUBLE PRECISION array, dimension (N) 147* The first M elements contain the selected eigenvalues in 148* ascending order. 149* 150* Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M)) 151* If JOBZ = 'V', then if INFO = 0, the first M columns of Z 152* contain the orthonormal eigenvectors of the matrix A 153* corresponding to the selected eigenvalues, with the i-th 154* column of Z holding the eigenvector associated with W(i). 155* If JOBZ = 'N', then Z is not referenced. 156* Note: the user must ensure that at least max(1,M) columns are 157* supplied in the array Z; if RANGE = 'V', the exact value of M 158* is not known in advance and an upper bound must be used. 159* 160* LDZ (input) INTEGER 161* The leading dimension of the array Z. LDZ >= 1, and if 162* JOBZ = 'V', LDZ >= max(1,N). 163* 164* ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) ) 165* The support of the eigenvectors in Z, i.e., the indices 166* indicating the nonzero elements in Z. The i-th eigenvector 167* is nonzero only in elements ISUPPZ( 2*i-1 ) through 168* ISUPPZ( 2*i ). 169********** Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 170* 171* WORK (workspace/output) COMPLEX*16 array, dimension (LWORK) 172* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 173* 174* LWORK (input) INTEGER 175* The length of the array WORK. LWORK >= max(1,2*N). 176* For optimal efficiency, LWORK >= (NB+1)*N, 177* where NB is the max of the blocksize for ZHETRD and for 178* ZUNMTR as returned by ILAENV. 179* 180* If LWORK = -1, then a workspace query is assumed; the routine 181* only calculates the optimal size of the WORK array, returns 182* this value as the first entry of the WORK array, and no error 183* message related to LWORK is issued by XERBLA. 184* 185* RWORK (workspace/output) DOUBLE PRECISION array, dimension (LRWORK) 186* On exit, if INFO = 0, RWORK(1) returns the optimal 187* (and minimal) LRWORK. 188* 189* LRWORK (input) INTEGER 190* The length of the array RWORK. LRWORK >= max(1,24*N). 191* 192* If LRWORK = -1, then a workspace query is assumed; the routine 193* only calculates the optimal size of the RWORK array, returns 194* this value as the first entry of the RWORK array, and no error 195* message related to LRWORK is issued by XERBLA. 196* 197* IWORK (workspace/output) INTEGER array, dimension (LIWORK) 198* On exit, if INFO = 0, IWORK(1) returns the optimal 199* (and minimal) LIWORK. 200* 201* LIWORK (input) INTEGER 202* The dimension of the array IWORK. LIWORK >= max(1,10*N). 203* 204* If LIWORK = -1, then a workspace query is assumed; the 205* routine only calculates the optimal size of the IWORK array, 206* returns this value as the first entry of the IWORK array, and 207* no error message related to LIWORK is issued by XERBLA. 208* 209* INFO (output) INTEGER 210* = 0: successful exit 211* < 0: if INFO = -i, the i-th argument had an illegal value 212* > 0: Internal error 213* 214* Further Details 215* =============== 216* 217* Based on contributions by 218* Inderjit Dhillon, IBM Almaden, USA 219* Osni Marques, LBNL/NERSC, USA 220* Ken Stanley, Computer Science Division, University of 221* California at Berkeley, USA 222* 223* ===================================================================== 224* 225* .. Parameters .. 226 DOUBLE PRECISION ZERO, ONE 227 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 228* .. 229* .. Local Scalars .. 230 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ 231 CHARACTER ORDER 232 INTEGER I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP, 233 $ INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK, 234 $ INDTAU, INDWK, INDWKN, ISCALE, ITMP1, J, JJ, 235 $ LIWMIN, LLWORK, LLWRKN, LRWMIN, LWKOPT, LWMIN, 236 $ NB, NSPLIT 237 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, 238 $ SIGMA, SMLNUM, TMP1, VLL, VUU 239* .. 240* .. External Functions .. 241 LOGICAL LSAME 242 INTEGER ILAENV 243 DOUBLE PRECISION DLAMCH, ZLANSY 244 EXTERNAL LSAME, ILAENV, DLAMCH, ZLANSY 245* .. 246* .. External Subroutines .. 247 EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL, 248 $ ZHETRD, ZSTEGR, ZSTEIN, ZSWAP, ZUNMTR 249* .. 250* .. Intrinsic Functions .. 251 INTRINSIC DBLE, MAX, MIN, SQRT 252* .. 253* .. Executable Statements .. 254* 255* Test the input parameters. 256* 257 IEEEOK = ILAENV( 10, 'ZHEEVR', 'N', 1, 2, 3, 4 ) 258* 259 LOWER = LSAME( UPLO, 'L' ) 260 WANTZ = LSAME( JOBZ, 'V' ) 261 ALLEIG = LSAME( RANGE, 'A' ) 262 VALEIG = LSAME( RANGE, 'V' ) 263 INDEIG = LSAME( RANGE, 'I' ) 264* 265 LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LRWORK.EQ.-1 ) .OR. 266 $ ( LIWORK.EQ.-1 ) ) 267* 268 LRWMIN = MAX( 1, 24*N ) 269 LIWMIN = MAX( 1, 10*N ) 270 LWMIN = MAX( 1, 2*N ) 271* 272 INFO = 0 273 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 274 INFO = -1 275 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 276 INFO = -2 277 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN 278 INFO = -3 279 ELSE IF( N.LT.0 ) THEN 280 INFO = -4 281 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 282 INFO = -6 283 ELSE 284 IF( VALEIG ) THEN 285 IF( N.GT.0 .AND. VU.LE.VL ) 286 $ INFO = -8 287 ELSE IF( INDEIG ) THEN 288 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 289 INFO = -9 290 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 291 INFO = -10 292 END IF 293 END IF 294 END IF 295 IF( INFO.EQ.0 ) THEN 296 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 297 INFO = -15 298 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 299 INFO = -18 300 ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN 301 INFO = -20 302 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 303 INFO = -22 304 END IF 305 END IF 306* 307 IF( INFO.EQ.0 ) THEN 308 NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 ) 309 NB = MAX( NB, ILAENV( 1, 'ZUNMTR', UPLO, N, -1, -1, -1 ) ) 310 LWKOPT = MAX( ( NB+1 )*N, LWMIN ) 311 WORK( 1 ) = LWKOPT 312 RWORK( 1 ) = LRWMIN 313 IWORK( 1 ) = LIWMIN 314 END IF 315* 316 IF( INFO.NE.0 ) THEN 317 CALL XERBLA( 'ZHEEVR', -INFO ) 318 RETURN 319 ELSE IF( LQUERY ) THEN 320 RETURN 321 END IF 322* 323* Quick return if possible 324* 325 M = 0 326 IF( N.EQ.0 ) THEN 327 WORK( 1 ) = 1 328 RETURN 329 END IF 330* 331 IF( N.EQ.1 ) THEN 332 WORK( 1 ) = 7 333 IF( ALLEIG .OR. INDEIG ) THEN 334 M = 1 335 W( 1 ) = DBLE( A( 1, 1 ) ) 336 ELSE 337 IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) ) 338 $ THEN 339 M = 1 340 W( 1 ) = DBLE( A( 1, 1 ) ) 341 END IF 342 END IF 343 IF( WANTZ ) 344 $ Z( 1, 1 ) = ONE 345 RETURN 346 END IF 347* 348* Get machine constants. 349* 350 SAFMIN = DLAMCH( 'Safe minimum' ) 351 EPS = DLAMCH( 'Precision' ) 352 SMLNUM = SAFMIN / EPS 353 BIGNUM = ONE / SMLNUM 354 RMIN = SQRT( SMLNUM ) 355 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 356* 357* Scale matrix to allowable range, if necessary. 358* 359 ISCALE = 0 360 ABSTLL = ABSTOL 361 VLL = VL 362 VUU = VU 363 ANRM = ZLANSY( 'M', UPLO, N, A, LDA, RWORK ) 364 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN 365 ISCALE = 1 366 SIGMA = RMIN / ANRM 367 ELSE IF( ANRM.GT.RMAX ) THEN 368 ISCALE = 1 369 SIGMA = RMAX / ANRM 370 END IF 371 IF( ISCALE.EQ.1 ) THEN 372 IF( LOWER ) THEN 373 DO 10 J = 1, N 374 CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 ) 375 10 CONTINUE 376 ELSE 377 DO 20 J = 1, N 378 CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 ) 379 20 CONTINUE 380 END IF 381 IF( ABSTOL.GT.0 ) 382 $ ABSTLL = ABSTOL*SIGMA 383 IF( VALEIG ) THEN 384 VLL = VL*SIGMA 385 VUU = VU*SIGMA 386 END IF 387 END IF 388* 389* Call ZHETRD to reduce Hermitian matrix to tridiagonal form. 390* 391 INDTAU = 1 392 INDWK = INDTAU + N 393* 394 INDRE = 1 395 INDRD = INDRE + N 396 INDREE = INDRD + N 397 INDRDD = INDREE + N 398 INDRWK = INDRDD + N 399 LLWORK = LWORK - INDWK + 1 400 CALL ZHETRD( UPLO, N, A, LDA, RWORK( INDRD ), RWORK( INDRE ), 401 $ WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO ) 402* 403* If all eigenvalues are desired 404* then call DSTERF or ZSTEGR and ZUNMTR. 405* 406 IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND. 407 $ IEEEOK.EQ.1 ) THEN 408 IF( .NOT.WANTZ ) THEN 409 CALL DCOPY( N, RWORK( INDRD ), 1, W, 1 ) 410 CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 ) 411 CALL DSTERF( N, W, RWORK( INDREE ), INFO ) 412 ELSE 413 CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 ) 414 CALL DCOPY( N, RWORK( INDRD ), 1, RWORK( INDRDD ), 1 ) 415* 416 CALL ZSTEGR( JOBZ, 'A', N, RWORK( INDRDD ), 417 $ RWORK( INDREE ), VL, VU, IL, IU, ABSTOL, M, W, 418 $ Z, LDZ, ISUPPZ, RWORK( INDRWK ), LWORK, IWORK, 419 $ LIWORK, INFO ) 420* 421* 422* 423* Apply unitary matrix used in reduction to tridiagonal 424* form to eigenvectors returned by ZSTEIN. 425* 426 IF( WANTZ .AND. INFO.EQ.0 ) THEN 427 INDWKN = INDWK 428 LLWRKN = LWORK - INDWKN + 1 429 CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, 430 $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ), 431 $ LLWRKN, IINFO ) 432 END IF 433 END IF 434* 435* 436 IF( INFO.EQ.0 ) THEN 437 M = N 438 GO TO 30 439 END IF 440 INFO = 0 441 END IF 442* 443* Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN. 444* Also call DSTEBZ and ZSTEIN if CSTEGR fails. 445* 446 IF( WANTZ ) THEN 447 ORDER = 'B' 448 ELSE 449 ORDER = 'E' 450 END IF 451 INDIFL = 1 452 INDIBL = INDIFL + N 453 INDISP = INDIBL + N 454 INDIWO = INDISP + N 455 CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, 456 $ RWORK( INDRD ), RWORK( INDRE ), M, NSPLIT, W, 457 $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ), 458 $ IWORK( INDIWO ), INFO ) 459* 460 IF( WANTZ ) THEN 461 CALL ZSTEIN( N, RWORK( INDRD ), RWORK( INDRE ), M, W, 462 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, 463 $ RWORK( INDRWK ), IWORK( INDIWO ), IWORK( INDIFL ), 464 $ INFO ) 465* 466* Apply unitary matrix used in reduction to tridiagonal 467* form to eigenvectors returned by ZSTEIN. 468* 469 INDWKN = INDWK 470 LLWRKN = LWORK - INDWKN + 1 471 CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z, 472 $ LDZ, WORK( INDWKN ), LLWRKN, IINFO ) 473 END IF 474* 475* If matrix was scaled, then rescale eigenvalues appropriately. 476* 477 30 CONTINUE 478 IF( ISCALE.EQ.1 ) THEN 479 IF( INFO.EQ.0 ) THEN 480 IMAX = M 481 ELSE 482 IMAX = INFO - 1 483 END IF 484 CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) 485 END IF 486* 487* If eigenvalues are not in order, then sort them, along with 488* eigenvectors. 489* 490 IF( WANTZ ) THEN 491 DO 50 J = 1, M - 1 492 I = 0 493 TMP1 = W( J ) 494 DO 40 JJ = J + 1, M 495 IF( W( JJ ).LT.TMP1 ) THEN 496 I = JJ 497 TMP1 = W( JJ ) 498 END IF 499 40 CONTINUE 500* 501 IF( I.NE.0 ) THEN 502 ITMP1 = IWORK( INDIBL+I-1 ) 503 W( I ) = W( J ) 504 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) 505 W( J ) = TMP1 506 IWORK( INDIBL+J-1 ) = ITMP1 507 CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 508 END IF 509 50 CONTINUE 510 END IF 511* 512* Set WORK(1) to optimal workspace size. 513* 514 WORK( 1 ) = LWKOPT 515 RWORK( 1 ) = LRWMIN 516 IWORK( 1 ) = LIWMIN 517* 518 RETURN 519* 520* End of ZHEEVR 521* 522 END 523