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* June 30, 1999 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 tridiagonal matrix T. Eigenvalues and 27* eigenvectors can be selected by specifying either a range of values 28* or a range of 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', DSTEBZ and ZSTEIN are called 79* 80* UPLO (input) CHARACTER*1 81* = 'U': Upper triangle of A is stored; 82* = 'L': Lower triangle of A is stored. 83* 84* N (input) INTEGER 85* The order of the matrix A. N >= 0. 86* 87* A (input/output) COMPLEX*16 array, dimension (LDA, N) 88* On entry, the Hermitian matrix A. If UPLO = 'U', the 89* leading N-by-N upper triangular part of A contains the 90* upper triangular part of the matrix A. If UPLO = 'L', 91* the leading N-by-N lower triangular part of A contains 92* the lower triangular part of the matrix A. 93* On exit, the lower triangle (if UPLO='L') or the upper 94* triangle (if UPLO='U') of A, including the diagonal, is 95* destroyed. 96* 97* LDA (input) INTEGER 98* The leading dimension of the array A. LDA >= max(1,N). 99* 100* VL (input) DOUBLE PRECISION 101* VU (input) DOUBLE PRECISION 102* If RANGE='V', the lower and upper bounds of the interval to 103* be searched for eigenvalues. VL < VU. 104* Not referenced if RANGE = 'A' or 'I'. 105* 106* IL (input) INTEGER 107* IU (input) INTEGER 108* If RANGE='I', the indices (in ascending order) of the 109* smallest and largest eigenvalues to be returned. 110* 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 111* Not referenced if RANGE = 'A' or 'V'. 112* 113* ABSTOL (input) DOUBLE PRECISION 114* The absolute error tolerance for the eigenvalues. 115* An approximate eigenvalue is accepted as converged 116* when it is determined to lie in an interval [a,b] 117* of width less than or equal to 118* 119* ABSTOL + EPS * max( |a|,|b| ) , 120* 121* where EPS is the machine precision. If ABSTOL is less than 122* or equal to zero, then EPS*|T| will be used in its place, 123* where |T| is the 1-norm of the tridiagonal matrix obtained 124* by reducing A to tridiagonal form. 125* 126* See "Computing Small Singular Values of Bidiagonal Matrices 127* with Guaranteed High Relative Accuracy," by Demmel and 128* Kahan, LAPACK Working Note #3. 129* 130* If high relative accuracy is important, set ABSTOL to 131* DLAMCH( 'Safe minimum' ). Doing so will guarantee that 132* eigenvalues are computed to high relative accuracy when 133* possible in future releases. The current code does not 134* make any guarantees about high relative accuracy, but 135* furutre releases will. See J. Barlow and J. Demmel, 136* "Computing Accurate Eigensystems of Scaled Diagonally 137* Dominant Matrices", LAPACK Working Note #7, for a discussion 138* of which matrices define their eigenvalues to high relative 139* accuracy. 140* 141* M (output) INTEGER 142* The total number of eigenvalues found. 0 <= M <= N. 143* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 144* 145* W (output) DOUBLE PRECISION array, dimension (N) 146* The first M elements contain the selected eigenvalues in 147* ascending order. 148* 149* Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M)) 150* If JOBZ = 'V', then if INFO = 0, the first M columns of Z 151* contain the orthonormal eigenvectors of the matrix A 152* corresponding to the selected eigenvalues, with the i-th 153* column of Z holding the eigenvector associated with W(i). 154* If JOBZ = 'N', then Z is not referenced. 155* Note: the user must ensure that at least max(1,M) columns are 156* supplied in the array Z; if RANGE = 'V', the exact value of M 157* is not known in advance and an upper bound must be used. 158* 159* LDZ (input) INTEGER 160* The leading dimension of the array Z. LDZ >= 1, and if 161* JOBZ = 'V', LDZ >= max(1,N). 162* 163* ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) ) 164* The support of the eigenvectors in Z, i.e., the indices 165* indicating the nonzero elements in Z. The i-th eigenvector 166* is nonzero only in elements ISUPPZ( 2*i-1 ) through 167* ISUPPZ( 2*i ). 168* 169* WORK (workspace/output) COMPLEX*16 array, dimension (LWORK) 170* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 171* 172* LWORK (input) INTEGER 173* The length of the array WORK. LWORK >= max(1,2*N). 174* For optimal efficiency, LWORK >= (NB+1)*N, 175* where NB is the max of the blocksize for ZHETRD and for 176* ZUNMTR as returned by ILAENV. 177* 178* If LWORK = -1, then a workspace query is assumed; the routine 179* only calculates the optimal size of the WORK array, returns 180* this value as the first entry of the WORK array, and no error 181* message related to LWORK is issued by XERBLA. 182* 183* RWORK (workspace/output) DOUBLE PRECISION array, dimension (LRWORK) 184* On exit, if INFO = 0, RWORK(1) returns the optimal 185* (and minimal) LRWORK. 186* 187* LRWORK (input) INTEGER 188* The length of the array RWORK. LRWORK >= max(1,24*N). 189* 190* If LRWORK = -1, then a workspace query is assumed; the routine 191* only calculates the optimal size of the RWORK array, returns 192* this value as the first entry of the RWORK array, and no error 193* message related to LRWORK is issued by XERBLA. 194* 195* IWORK (workspace/output) INTEGER array, dimension (LIWORK) 196* On exit, if INFO = 0, IWORK(1) returns the optimal 197* (and minimal) LIWORK. 198* 199* LIWORK (input) INTEGER 200* The dimension of the array IWORK. LIWORK >= max(1,10*N). 201* 202* If LIWORK = -1, then a workspace query is assumed; the 203* routine only calculates the optimal size of the IWORK array, 204* returns this value as the first entry of the IWORK array, and 205* no error message related to LIWORK is issued by XERBLA. 206* 207* INFO (output) INTEGER 208* = 0: successful exit 209* < 0: if INFO = -i, the i-th argument had an illegal value 210* > 0: Internal error 211* 212* Further Details 213* =============== 214* 215* Based on contributions by 216* Inderjit Dhillon, IBM Almaden, USA 217* Osni Marques, LBNL/NERSC, USA 218* Ken Stanley, Computer Science Division, University of 219* California at Berkeley, USA 220* 221* ===================================================================== 222* 223* .. Parameters .. 224 DOUBLE PRECISION ZERO, ONE 225 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 226* .. 227* .. Local Scalars .. 228 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ 229 CHARACTER ORDER 230 INTEGER I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP, 231 $ INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK, 232 $ INDTAU, INDWK, INDWKN, ISCALE, ITMP1, J, JJ, 233 $ LIWMIN, LLWORK, LLWRKN, LRWMIN, LWKOPT, LWMIN, 234 $ NB, NSPLIT 235 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, 236 $ SIGMA, SMLNUM, TMP1, VLL, VUU 237* .. 238* .. External Functions .. 239 LOGICAL LSAME 240 INTEGER ILAENV 241 DOUBLE PRECISION DLAMCH, ZLANSY 242 EXTERNAL LSAME, ILAENV, DLAMCH, ZLANSY 243* .. 244* .. External Subroutines .. 245 EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL, 246 $ ZHETRD, ZSTEGR, ZSTEIN, ZSWAP, ZUNMTR 247* .. 248* .. Intrinsic Functions .. 249 INTRINSIC DBLE, MAX, MIN, SQRT 250* .. 251* .. Executable Statements .. 252* 253* Test the input parameters. 254* 255 IEEEOK = ILAENV( 10, 'ZHEEVR', 'N', 1, 2, 3, 4 ) 256* 257 LOWER = LSAME( UPLO, 'L' ) 258 WANTZ = LSAME( JOBZ, 'V' ) 259 ALLEIG = LSAME( RANGE, 'A' ) 260 VALEIG = LSAME( RANGE, 'V' ) 261 INDEIG = LSAME( RANGE, 'I' ) 262* 263 LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LRWORK.EQ.-1 ) .OR. 264 $ ( LIWORK.EQ.-1 ) ) 265* 266 LRWMIN = MAX( 1, 24*N ) 267 LIWMIN = MAX( 1, 10*N ) 268 LWMIN = MAX( 1, 2*N ) 269* 270 INFO = 0 271 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 272 INFO = -1 273 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 274 INFO = -2 275 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN 276 INFO = -3 277 ELSE IF( N.LT.0 ) THEN 278 INFO = -4 279 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 280 INFO = -6 281 ELSE 282 IF( VALEIG ) THEN 283 IF( N.GT.0 .AND. VU.LE.VL ) 284 $ INFO = -8 285 ELSE IF( INDEIG ) THEN 286 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 287 INFO = -9 288 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 289 INFO = -10 290 END IF 291 END IF 292 END IF 293 IF( INFO.EQ.0 ) THEN 294 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 295 INFO = -15 296 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 297 INFO = -18 298 ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN 299 INFO = -20 300 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 301 INFO = -22 302 END IF 303 END IF 304* 305 IF( INFO.EQ.0 ) THEN 306 NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 ) 307 NB = MAX( NB, ILAENV( 1, 'ZUNMTR', UPLO, N, -1, -1, -1 ) ) 308 LWKOPT = MAX( ( NB+1 )*N, LWMIN ) 309 WORK( 1 ) = LWKOPT 310 RWORK( 1 ) = LRWMIN 311 IWORK( 1 ) = LIWMIN 312 END IF 313* 314 IF( INFO.NE.0 ) THEN 315 CALL XERBLA( 'ZHEEVR', -INFO ) 316 RETURN 317 ELSE IF( LQUERY ) THEN 318 RETURN 319 END IF 320* 321* Quick return if possible 322* 323 M = 0 324 IF( N.EQ.0 ) THEN 325 WORK( 1 ) = 1 326 RETURN 327 END IF 328* 329 IF( N.EQ.1 ) THEN 330 WORK( 1 ) = 7 331 IF( ALLEIG .OR. INDEIG ) THEN 332 M = 1 333 W( 1 ) = DBLE( A( 1, 1 ) ) 334 ELSE 335 IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) ) 336 $ THEN 337 M = 1 338 W( 1 ) = DBLE( A( 1, 1 ) ) 339 END IF 340 END IF 341 IF( WANTZ ) 342 $ Z( 1, 1 ) = ONE 343 RETURN 344 END IF 345* 346* Get machine constants. 347* 348 SAFMIN = DLAMCH( 'Safe minimum' ) 349 EPS = DLAMCH( 'Precision' ) 350 SMLNUM = SAFMIN / EPS 351 BIGNUM = ONE / SMLNUM 352 RMIN = SQRT( SMLNUM ) 353 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 354* 355* Scale matrix to allowable range, if necessary. 356* 357 ISCALE = 0 358 ABSTLL = ABSTOL 359 VLL = VL 360 VUU = VU 361 ANRM = ZLANSY( 'M', UPLO, N, A, LDA, RWORK ) 362 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN 363 ISCALE = 1 364 SIGMA = RMIN / ANRM 365 ELSE IF( ANRM.GT.RMAX ) THEN 366 ISCALE = 1 367 SIGMA = RMAX / ANRM 368 END IF 369 IF( ISCALE.EQ.1 ) THEN 370 IF( LOWER ) THEN 371 DO 10 J = 1, N 372 CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 ) 373 10 CONTINUE 374 ELSE 375 DO 20 J = 1, N 376 CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 ) 377 20 CONTINUE 378 END IF 379 IF( ABSTOL.GT.0 ) 380 $ ABSTLL = ABSTOL*SIGMA 381 IF( VALEIG ) THEN 382 VLL = VL*SIGMA 383 VUU = VU*SIGMA 384 END IF 385 END IF 386* 387* Call ZHETRD to reduce Hermitian matrix to tridiagonal form. 388* 389 INDTAU = 1 390 INDWK = INDTAU + N 391* 392 INDRE = 1 393 INDRD = INDRE + N 394 INDREE = INDRD + N 395 INDRDD = INDREE + N 396 INDRWK = INDRDD + N 397 LLWORK = LWORK - INDWK + 1 398 CALL ZHETRD( UPLO, N, A, LDA, RWORK( INDRD ), RWORK( INDRE ), 399 $ WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO ) 400* 401* If all eigenvalues are desired 402* then call DSTERF or ZSTEGR and ZUNMTR. 403* 404 IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND. 405 $ IEEEOK.EQ.1 ) THEN 406 IF( .NOT.WANTZ ) THEN 407 CALL DCOPY( N, RWORK( INDRD ), 1, W, 1 ) 408 CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 ) 409 CALL DSTERF( N, W, RWORK( INDREE ), INFO ) 410 ELSE 411 CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 ) 412 CALL DCOPY( N, RWORK( INDRD ), 1, RWORK( INDRDD ), 1 ) 413* 414 CALL ZSTEGR( JOBZ, RANGE, N, RWORK( INDRDD ), 415 $ RWORK( INDREE ), VL, VU, IL, IU, ABSTOL, M, W, 416 $ Z, LDZ, ISUPPZ, RWORK( INDRWK ), LWORK, IWORK, 417 $ LIWORK, INFO ) 418* 419* 420* 421* Apply unitary matrix used in reduction to tridiagonal 422* form to eigenvectors returned by ZSTEIN. 423* 424 IF( WANTZ .AND. INFO.EQ.0 ) THEN 425 INDWKN = INDWK 426 LLWRKN = LWORK - INDWKN + 1 427 CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, 428 $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ), 429 $ LLWRKN, IINFO ) 430 END IF 431 END IF 432* 433* 434 IF( INFO.EQ.0 ) THEN 435 M = N 436 GO TO 30 437 END IF 438 INFO = 0 439 END IF 440* 441* Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN. 442* Also call DSTEBZ and ZSTEIN if CSTEGR fails. 443* 444 IF( WANTZ ) THEN 445 ORDER = 'B' 446 ELSE 447 ORDER = 'E' 448 END IF 449 INDIFL = 1 450 INDIBL = INDIFL + N 451 INDISP = INDIBL + N 452 INDIWO = INDISP + N 453 CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, 454 $ RWORK( INDRD ), RWORK( INDRE ), M, NSPLIT, W, 455 $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ), 456 $ IWORK( INDIWO ), INFO ) 457* 458 IF( WANTZ ) THEN 459 CALL ZSTEIN( N, RWORK( INDRD ), RWORK( INDRE ), M, W, 460 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, 461 $ RWORK( INDRWK ), IWORK( INDIWO ), IWORK( INDIFL ), 462 $ INFO ) 463* 464* Apply unitary matrix used in reduction to tridiagonal 465* form to eigenvectors returned by ZSTEIN. 466* 467 INDWKN = INDWK 468 LLWRKN = LWORK - INDWKN + 1 469 CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z, 470 $ LDZ, WORK( INDWKN ), LLWRKN, IINFO ) 471 END IF 472* 473* If matrix was scaled, then rescale eigenvalues appropriately. 474* 475 30 CONTINUE 476 IF( ISCALE.EQ.1 ) THEN 477 IF( INFO.EQ.0 ) THEN 478 IMAX = M 479 ELSE 480 IMAX = INFO - 1 481 END IF 482 CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) 483 END IF 484* 485* If eigenvalues are not in order, then sort them, along with 486* eigenvectors. 487* 488 IF( WANTZ ) THEN 489 DO 50 J = 1, M - 1 490 I = 0 491 TMP1 = W( J ) 492 DO 40 JJ = J + 1, M 493 IF( W( JJ ).LT.TMP1 ) THEN 494 I = JJ 495 TMP1 = W( JJ ) 496 END IF 497 40 CONTINUE 498* 499 IF( I.NE.0 ) THEN 500 ITMP1 = IWORK( INDIBL+I-1 ) 501 W( I ) = W( J ) 502 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) 503 W( J ) = TMP1 504 IWORK( INDIBL+J-1 ) = ITMP1 505 CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 506 END IF 507 50 CONTINUE 508 END IF 509* 510* Set WORK(1) to optimal workspace size. 511* 512 WORK( 1 ) = LWKOPT 513 RWORK( 1 ) = LRWMIN 514 IWORK( 1 ) = LIWMIN 515* 516 RETURN 517* 518* End of ZHEEVR 519* 520 END 521