1*> \brief <b> ZHEEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices</b> 2* 3* @precisions fortran z -> s d c 4* 5* =========== DOCUMENTATION =========== 6* 7* Online html documentation available at 8* http://www.netlib.org/lapack/explore-html/ 9* 10*> \htmlonly 11*> Download ZHEEVR_2STAGE + dependencies 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zheevr_2stage.f"> 13*> [TGZ]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zheevr_2stage.f"> 15*> [ZIP]</a> 16*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zheevr_2stage.f"> 17*> [TXT]</a> 18*> \endhtmlonly 19* 20* Definition: 21* =========== 22* 23* SUBROUTINE ZHEEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, 24* IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, 25* WORK, LWORK, RWORK, LRWORK, IWORK, 26* LIWORK, INFO ) 27* 28* IMPLICIT NONE 29* 30* .. Scalar Arguments .. 31* CHARACTER JOBZ, RANGE, UPLO 32* INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK, 33* $ M, N 34* DOUBLE PRECISION ABSTOL, VL, VU 35* .. 36* .. Array Arguments .. 37* INTEGER ISUPPZ( * ), IWORK( * ) 38* DOUBLE PRECISION RWORK( * ), W( * ) 39* COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * ) 40* .. 41* 42* 43*> \par Purpose: 44* ============= 45*> 46*> \verbatim 47*> 48*> ZHEEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors 49*> of a complex Hermitian matrix A using the 2stage technique for 50*> the reduction to tridiagonal. Eigenvalues and eigenvectors can 51*> be selected by specifying either a range of values or a range of 52*> indices for the desired eigenvalues. 53*> 54*> ZHEEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call 55*> to ZHETRD. Then, whenever possible, ZHEEVR_2STAGE calls ZSTEMR to compute 56*> eigenspectrum using Relatively Robust Representations. ZSTEMR 57*> computes eigenvalues by the dqds algorithm, while orthogonal 58*> eigenvectors are computed from various "good" L D L^T representations 59*> (also known as Relatively Robust Representations). Gram-Schmidt 60*> orthogonalization is avoided as far as possible. More specifically, 61*> the various steps of the algorithm are as follows. 62*> 63*> For each unreduced block (submatrix) of T, 64*> (a) Compute T - sigma I = L D L^T, so that L and D 65*> define all the wanted eigenvalues to high relative accuracy. 66*> This means that small relative changes in the entries of D and L 67*> cause only small relative changes in the eigenvalues and 68*> eigenvectors. The standard (unfactored) representation of the 69*> tridiagonal matrix T does not have this property in general. 70*> (b) Compute the eigenvalues to suitable accuracy. 71*> If the eigenvectors are desired, the algorithm attains full 72*> accuracy of the computed eigenvalues only right before 73*> the corresponding vectors have to be computed, see steps c) and d). 74*> (c) For each cluster of close eigenvalues, select a new 75*> shift close to the cluster, find a new factorization, and refine 76*> the shifted eigenvalues to suitable accuracy. 77*> (d) For each eigenvalue with a large enough relative separation compute 78*> the corresponding eigenvector by forming a rank revealing twisted 79*> factorization. Go back to (c) for any clusters that remain. 80*> 81*> The desired accuracy of the output can be specified by the input 82*> parameter ABSTOL. 83*> 84*> For more details, see DSTEMR's documentation and: 85*> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations 86*> to compute orthogonal eigenvectors of symmetric tridiagonal matrices," 87*> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. 88*> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and 89*> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, 90*> 2004. Also LAPACK Working Note 154. 91*> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric 92*> tridiagonal eigenvalue/eigenvector problem", 93*> Computer Science Division Technical Report No. UCB/CSD-97-971, 94*> UC Berkeley, May 1997. 95*> 96*> 97*> Note 1 : ZHEEVR_2STAGE calls ZSTEMR when the full spectrum is requested 98*> on machines which conform to the ieee-754 floating point standard. 99*> ZHEEVR_2STAGE calls DSTEBZ and ZSTEIN on non-ieee machines and 100*> when partial spectrum requests are made. 101*> 102*> Normal execution of ZSTEMR may create NaNs and infinities and 103*> hence may abort due to a floating point exception in environments 104*> which do not handle NaNs and infinities in the ieee standard default 105*> manner. 106*> \endverbatim 107* 108* Arguments: 109* ========== 110* 111*> \param[in] JOBZ 112*> \verbatim 113*> JOBZ is CHARACTER*1 114*> = 'N': Compute eigenvalues only; 115*> = 'V': Compute eigenvalues and eigenvectors. 116*> Not available in this release. 117*> \endverbatim 118*> 119*> \param[in] RANGE 120*> \verbatim 121*> RANGE is CHARACTER*1 122*> = 'A': all eigenvalues will be found. 123*> = 'V': all eigenvalues in the half-open interval (VL,VU] 124*> will be found. 125*> = 'I': the IL-th through IU-th eigenvalues will be found. 126*> For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and 127*> ZSTEIN are called 128*> \endverbatim 129*> 130*> \param[in] UPLO 131*> \verbatim 132*> UPLO is CHARACTER*1 133*> = 'U': Upper triangle of A is stored; 134*> = 'L': Lower triangle of A is stored. 135*> \endverbatim 136*> 137*> \param[in] N 138*> \verbatim 139*> N is INTEGER 140*> The order of the matrix A. N >= 0. 141*> \endverbatim 142*> 143*> \param[in,out] A 144*> \verbatim 145*> A is COMPLEX*16 array, dimension (LDA, N) 146*> On entry, the Hermitian matrix A. If UPLO = 'U', the 147*> leading N-by-N upper triangular part of A contains the 148*> upper triangular part of the matrix A. If UPLO = 'L', 149*> the leading N-by-N lower triangular part of A contains 150*> the lower triangular part of the matrix A. 151*> On exit, the lower triangle (if UPLO='L') or the upper 152*> triangle (if UPLO='U') of A, including the diagonal, is 153*> destroyed. 154*> \endverbatim 155*> 156*> \param[in] LDA 157*> \verbatim 158*> LDA is INTEGER 159*> The leading dimension of the array A. LDA >= max(1,N). 160*> \endverbatim 161*> 162*> \param[in] VL 163*> \verbatim 164*> VL is DOUBLE PRECISION 165*> If RANGE='V', the lower bound of the interval to 166*> be searched for eigenvalues. VL < VU. 167*> Not referenced if RANGE = 'A' or 'I'. 168*> \endverbatim 169*> 170*> \param[in] VU 171*> \verbatim 172*> VU is DOUBLE PRECISION 173*> If RANGE='V', the upper bound of the interval to 174*> be searched for eigenvalues. VL < VU. 175*> Not referenced if RANGE = 'A' or 'I'. 176*> \endverbatim 177*> 178*> \param[in] IL 179*> \verbatim 180*> IL is INTEGER 181*> If RANGE='I', the index of the 182*> smallest eigenvalue to be returned. 183*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 184*> Not referenced if RANGE = 'A' or 'V'. 185*> \endverbatim 186*> 187*> \param[in] IU 188*> \verbatim 189*> IU is INTEGER 190*> If RANGE='I', the index of the 191*> largest eigenvalue to be returned. 192*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 193*> Not referenced if RANGE = 'A' or 'V'. 194*> \endverbatim 195*> 196*> \param[in] ABSTOL 197*> \verbatim 198*> ABSTOL is DOUBLE PRECISION 199*> The absolute error tolerance for the eigenvalues. 200*> An approximate eigenvalue is accepted as converged 201*> when it is determined to lie in an interval [a,b] 202*> of width less than or equal to 203*> 204*> ABSTOL + EPS * max( |a|,|b| ) , 205*> 206*> where EPS is the machine precision. If ABSTOL is less than 207*> or equal to zero, then EPS*|T| will be used in its place, 208*> where |T| is the 1-norm of the tridiagonal matrix obtained 209*> by reducing A to tridiagonal form. 210*> 211*> See "Computing Small Singular Values of Bidiagonal Matrices 212*> with Guaranteed High Relative Accuracy," by Demmel and 213*> Kahan, LAPACK Working Note #3. 214*> 215*> If high relative accuracy is important, set ABSTOL to 216*> DLAMCH( 'Safe minimum' ). Doing so will guarantee that 217*> eigenvalues are computed to high relative accuracy when 218*> possible in future releases. The current code does not 219*> make any guarantees about high relative accuracy, but 220*> future releases will. See J. Barlow and J. Demmel, 221*> "Computing Accurate Eigensystems of Scaled Diagonally 222*> Dominant Matrices", LAPACK Working Note #7, for a discussion 223*> of which matrices define their eigenvalues to high relative 224*> accuracy. 225*> \endverbatim 226*> 227*> \param[out] M 228*> \verbatim 229*> M is INTEGER 230*> The total number of eigenvalues found. 0 <= M <= N. 231*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 232*> \endverbatim 233*> 234*> \param[out] W 235*> \verbatim 236*> W is DOUBLE PRECISION array, dimension (N) 237*> The first M elements contain the selected eigenvalues in 238*> ascending order. 239*> \endverbatim 240*> 241*> \param[out] Z 242*> \verbatim 243*> Z is COMPLEX*16 array, dimension (LDZ, max(1,M)) 244*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z 245*> contain the orthonormal eigenvectors of the matrix A 246*> corresponding to the selected eigenvalues, with the i-th 247*> column of Z holding the eigenvector associated with W(i). 248*> If JOBZ = 'N', then Z is not referenced. 249*> Note: the user must ensure that at least max(1,M) columns are 250*> supplied in the array Z; if RANGE = 'V', the exact value of M 251*> is not known in advance and an upper bound must be used. 252*> \endverbatim 253*> 254*> \param[in] LDZ 255*> \verbatim 256*> LDZ is INTEGER 257*> The leading dimension of the array Z. LDZ >= 1, and if 258*> JOBZ = 'V', LDZ >= max(1,N). 259*> \endverbatim 260*> 261*> \param[out] ISUPPZ 262*> \verbatim 263*> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) 264*> The support of the eigenvectors in Z, i.e., the indices 265*> indicating the nonzero elements in Z. The i-th eigenvector 266*> is nonzero only in elements ISUPPZ( 2*i-1 ) through 267*> ISUPPZ( 2*i ). This is an output of ZSTEMR (tridiagonal 268*> matrix). The support of the eigenvectors of A is typically 269*> 1:N because of the unitary transformations applied by ZUNMTR. 270*> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 271*> \endverbatim 272*> 273*> \param[out] WORK 274*> \verbatim 275*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) 276*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 277*> \endverbatim 278*> 279*> \param[in] LWORK 280*> \verbatim 281*> LWORK is INTEGER 282*> The dimension of the array WORK. 283*> If JOBZ = 'N' and N > 1, LWORK must be queried. 284*> LWORK = MAX(1, 26*N, dimension) where 285*> dimension = max(stage1,stage2) + (KD+1)*N + N 286*> = N*KD + N*max(KD+1,FACTOPTNB) 287*> + max(2*KD*KD, KD*NTHREADS) 288*> + (KD+1)*N + N 289*> where KD is the blocking size of the reduction, 290*> FACTOPTNB is the blocking used by the QR or LQ 291*> algorithm, usually FACTOPTNB=128 is a good choice 292*> NTHREADS is the number of threads used when 293*> openMP compilation is enabled, otherwise =1. 294*> If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available 295*> 296*> If LWORK = -1, then a workspace query is assumed; the routine 297*> only calculates the optimal sizes of the WORK, RWORK and 298*> IWORK arrays, returns these values as the first entries of 299*> the WORK, RWORK and IWORK arrays, and no error message 300*> related to LWORK or LRWORK or LIWORK is issued by XERBLA. 301*> \endverbatim 302*> 303*> \param[out] RWORK 304*> \verbatim 305*> RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) 306*> On exit, if INFO = 0, RWORK(1) returns the optimal 307*> (and minimal) LRWORK. 308*> \endverbatim 309*> 310*> \param[in] LRWORK 311*> \verbatim 312*> LRWORK is INTEGER 313*> The length of the array RWORK. LRWORK >= max(1,24*N). 314*> 315*> If LRWORK = -1, then a workspace query is assumed; the 316*> routine only calculates the optimal sizes of the WORK, RWORK 317*> and IWORK arrays, returns these values as the first entries 318*> of the WORK, RWORK and IWORK arrays, and no error message 319*> related to LWORK or LRWORK or LIWORK is issued by XERBLA. 320*> \endverbatim 321*> 322*> \param[out] IWORK 323*> \verbatim 324*> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) 325*> On exit, if INFO = 0, IWORK(1) returns the optimal 326*> (and minimal) LIWORK. 327*> \endverbatim 328*> 329*> \param[in] LIWORK 330*> \verbatim 331*> LIWORK is INTEGER 332*> The dimension of the array IWORK. LIWORK >= max(1,10*N). 333*> 334*> If LIWORK = -1, then a workspace query is assumed; the 335*> routine only calculates the optimal sizes of the WORK, RWORK 336*> and IWORK arrays, returns these values as the first entries 337*> of the WORK, RWORK and IWORK arrays, and no error message 338*> related to LWORK or LRWORK or LIWORK is issued by XERBLA. 339*> \endverbatim 340*> 341*> \param[out] INFO 342*> \verbatim 343*> INFO is INTEGER 344*> = 0: successful exit 345*> < 0: if INFO = -i, the i-th argument had an illegal value 346*> > 0: Internal error 347*> \endverbatim 348* 349* Authors: 350* ======== 351* 352*> \author Univ. of Tennessee 353*> \author Univ. of California Berkeley 354*> \author Univ. of Colorado Denver 355*> \author NAG Ltd. 356* 357*> \date June 2016 358* 359*> \ingroup complex16HEeigen 360* 361*> \par Contributors: 362* ================== 363*> 364*> Inderjit Dhillon, IBM Almaden, USA \n 365*> Osni Marques, LBNL/NERSC, USA \n 366*> Ken Stanley, Computer Science Division, University of 367*> California at Berkeley, USA \n 368*> Jason Riedy, Computer Science Division, University of 369*> California at Berkeley, USA \n 370*> 371*> \par Further Details: 372* ===================== 373*> 374*> \verbatim 375*> 376*> All details about the 2stage techniques are available in: 377*> 378*> Azzam Haidar, Hatem Ltaief, and Jack Dongarra. 379*> Parallel reduction to condensed forms for symmetric eigenvalue problems 380*> using aggregated fine-grained and memory-aware kernels. In Proceedings 381*> of 2011 International Conference for High Performance Computing, 382*> Networking, Storage and Analysis (SC '11), New York, NY, USA, 383*> Article 8 , 11 pages. 384*> http://doi.acm.org/10.1145/2063384.2063394 385*> 386*> A. Haidar, J. Kurzak, P. Luszczek, 2013. 387*> An improved parallel singular value algorithm and its implementation 388*> for multicore hardware, In Proceedings of 2013 International Conference 389*> for High Performance Computing, Networking, Storage and Analysis (SC '13). 390*> Denver, Colorado, USA, 2013. 391*> Article 90, 12 pages. 392*> http://doi.acm.org/10.1145/2503210.2503292 393*> 394*> A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. 395*> A novel hybrid CPU-GPU generalized eigensolver for electronic structure 396*> calculations based on fine-grained memory aware tasks. 397*> International Journal of High Performance Computing Applications. 398*> Volume 28 Issue 2, Pages 196-209, May 2014. 399*> http://hpc.sagepub.com/content/28/2/196 400*> 401*> \endverbatim 402* 403* ===================================================================== 404 SUBROUTINE ZHEEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, 405 $ IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, 406 $ WORK, LWORK, RWORK, LRWORK, IWORK, 407 $ LIWORK, INFO ) 408* 409 IMPLICIT NONE 410* 411* -- LAPACK driver routine (version 3.8.0) -- 412* -- LAPACK is a software package provided by Univ. of Tennessee, -- 413* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 414* June 2016 415* 416* .. Scalar Arguments .. 417 CHARACTER JOBZ, RANGE, UPLO 418 INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK, 419 $ M, N 420 DOUBLE PRECISION ABSTOL, VL, VU 421* .. 422* .. Array Arguments .. 423 INTEGER ISUPPZ( * ), IWORK( * ) 424 DOUBLE PRECISION RWORK( * ), W( * ) 425 COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * ) 426* .. 427* 428* ===================================================================== 429* 430* .. Parameters .. 431 DOUBLE PRECISION ZERO, ONE, TWO 432 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 ) 433* .. 434* .. Local Scalars .. 435 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG, 436 $ WANTZ, TRYRAC 437 CHARACTER ORDER 438 INTEGER I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP, 439 $ INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK, 440 $ INDTAU, INDWK, INDWKN, ISCALE, ITMP1, J, JJ, 441 $ LIWMIN, LLWORK, LLRWORK, LLWRKN, LRWMIN, 442 $ LWMIN, NSPLIT, LHTRD, LWTRD, KD, IB, INDHOUS 443 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, 444 $ SIGMA, SMLNUM, TMP1, VLL, VUU 445* .. 446* .. External Functions .. 447 LOGICAL LSAME 448 INTEGER ILAENV, ILAENV2STAGE 449 DOUBLE PRECISION DLAMCH, ZLANSY 450 EXTERNAL LSAME, DLAMCH, ZLANSY, ILAENV, ILAENV2STAGE 451* .. 452* .. External Subroutines .. 453 EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL, 454 $ ZHETRD_2STAGE, ZSTEMR, ZSTEIN, ZSWAP, ZUNMTR 455* .. 456* .. Intrinsic Functions .. 457 INTRINSIC DBLE, MAX, MIN, SQRT 458* .. 459* .. Executable Statements .. 460* 461* Test the input parameters. 462* 463 IEEEOK = ILAENV( 10, 'ZHEEVR', 'N', 1, 2, 3, 4 ) 464* 465 LOWER = LSAME( UPLO, 'L' ) 466 WANTZ = LSAME( JOBZ, 'V' ) 467 ALLEIG = LSAME( RANGE, 'A' ) 468 VALEIG = LSAME( RANGE, 'V' ) 469 INDEIG = LSAME( RANGE, 'I' ) 470* 471 LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LRWORK.EQ.-1 ) .OR. 472 $ ( LIWORK.EQ.-1 ) ) 473* 474 KD = ILAENV2STAGE( 1, 'ZHETRD_2STAGE', JOBZ, N, -1, -1, -1 ) 475 IB = ILAENV2STAGE( 2, 'ZHETRD_2STAGE', JOBZ, N, KD, -1, -1 ) 476 LHTRD = ILAENV2STAGE( 3, 'ZHETRD_2STAGE', JOBZ, N, KD, IB, -1 ) 477 LWTRD = ILAENV2STAGE( 4, 'ZHETRD_2STAGE', JOBZ, N, KD, IB, -1 ) 478 LWMIN = N + LHTRD + LWTRD 479 LRWMIN = MAX( 1, 24*N ) 480 LIWMIN = MAX( 1, 10*N ) 481* 482 INFO = 0 483 IF( .NOT.( LSAME( JOBZ, 'N' ) ) ) THEN 484 INFO = -1 485 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 486 INFO = -2 487 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN 488 INFO = -3 489 ELSE IF( N.LT.0 ) THEN 490 INFO = -4 491 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 492 INFO = -6 493 ELSE 494 IF( VALEIG ) THEN 495 IF( N.GT.0 .AND. VU.LE.VL ) 496 $ INFO = -8 497 ELSE IF( INDEIG ) THEN 498 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 499 INFO = -9 500 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 501 INFO = -10 502 END IF 503 END IF 504 END IF 505 IF( INFO.EQ.0 ) THEN 506 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 507 INFO = -15 508 END IF 509 END IF 510* 511 IF( INFO.EQ.0 ) THEN 512 WORK( 1 ) = LWMIN 513 RWORK( 1 ) = LRWMIN 514 IWORK( 1 ) = LIWMIN 515* 516 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 517 INFO = -18 518 ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN 519 INFO = -20 520 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 521 INFO = -22 522 END IF 523 END IF 524* 525 IF( INFO.NE.0 ) THEN 526 CALL XERBLA( 'ZHEEVR_2STAGE', -INFO ) 527 RETURN 528 ELSE IF( LQUERY ) THEN 529 RETURN 530 END IF 531* 532* Quick return if possible 533* 534 M = 0 535 IF( N.EQ.0 ) THEN 536 WORK( 1 ) = 1 537 RETURN 538 END IF 539* 540 IF( N.EQ.1 ) THEN 541 WORK( 1 ) = 2 542 IF( ALLEIG .OR. INDEIG ) THEN 543 M = 1 544 W( 1 ) = DBLE( A( 1, 1 ) ) 545 ELSE 546 IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) ) 547 $ THEN 548 M = 1 549 W( 1 ) = DBLE( A( 1, 1 ) ) 550 END IF 551 END IF 552 IF( WANTZ ) THEN 553 Z( 1, 1 ) = ONE 554 ISUPPZ( 1 ) = 1 555 ISUPPZ( 2 ) = 1 556 END IF 557 RETURN 558 END IF 559* 560* Get machine constants. 561* 562 SAFMIN = DLAMCH( 'Safe minimum' ) 563 EPS = DLAMCH( 'Precision' ) 564 SMLNUM = SAFMIN / EPS 565 BIGNUM = ONE / SMLNUM 566 RMIN = SQRT( SMLNUM ) 567 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 568* 569* Scale matrix to allowable range, if necessary. 570* 571 ISCALE = 0 572 ABSTLL = ABSTOL 573 IF (VALEIG) THEN 574 VLL = VL 575 VUU = VU 576 END IF 577 ANRM = ZLANSY( 'M', UPLO, N, A, LDA, RWORK ) 578 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN 579 ISCALE = 1 580 SIGMA = RMIN / ANRM 581 ELSE IF( ANRM.GT.RMAX ) THEN 582 ISCALE = 1 583 SIGMA = RMAX / ANRM 584 END IF 585 IF( ISCALE.EQ.1 ) THEN 586 IF( LOWER ) THEN 587 DO 10 J = 1, N 588 CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 ) 589 10 CONTINUE 590 ELSE 591 DO 20 J = 1, N 592 CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 ) 593 20 CONTINUE 594 END IF 595 IF( ABSTOL.GT.0 ) 596 $ ABSTLL = ABSTOL*SIGMA 597 IF( VALEIG ) THEN 598 VLL = VL*SIGMA 599 VUU = VU*SIGMA 600 END IF 601 END IF 602 603* Initialize indices into workspaces. Note: The IWORK indices are 604* used only if DSTERF or ZSTEMR fail. 605 606* WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the 607* elementary reflectors used in ZHETRD. 608 INDTAU = 1 609* INDWK is the starting offset of the remaining complex workspace, 610* and LLWORK is the remaining complex workspace size. 611 INDHOUS = INDTAU + N 612 INDWK = INDHOUS + LHTRD 613 LLWORK = LWORK - INDWK + 1 614 615* RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal 616* entries. 617 INDRD = 1 618* RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the 619* tridiagonal matrix from ZHETRD. 620 INDRE = INDRD + N 621* RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over 622* -written by ZSTEMR (the DSTERF path copies the diagonal to W). 623 INDRDD = INDRE + N 624* RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over 625* -written while computing the eigenvalues in DSTERF and ZSTEMR. 626 INDREE = INDRDD + N 627* INDRWK is the starting offset of the left-over real workspace, and 628* LLRWORK is the remaining workspace size. 629 INDRWK = INDREE + N 630 LLRWORK = LRWORK - INDRWK + 1 631 632* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and 633* stores the block indices of each of the M<=N eigenvalues. 634 INDIBL = 1 635* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and 636* stores the starting and finishing indices of each block. 637 INDISP = INDIBL + N 638* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors 639* that corresponding to eigenvectors that fail to converge in 640* ZSTEIN. This information is discarded; if any fail, the driver 641* returns INFO > 0. 642 INDIFL = INDISP + N 643* INDIWO is the offset of the remaining integer workspace. 644 INDIWO = INDIFL + N 645 646* 647* Call ZHETRD_2STAGE to reduce Hermitian matrix to tridiagonal form. 648* 649 CALL ZHETRD_2STAGE( JOBZ, UPLO, N, A, LDA, RWORK( INDRD ), 650 $ RWORK( INDRE ), WORK( INDTAU ), 651 $ WORK( INDHOUS ), LHTRD, 652 $ WORK( INDWK ), LLWORK, IINFO ) 653* 654* If all eigenvalues are desired 655* then call DSTERF or ZSTEMR and ZUNMTR. 656* 657 TEST = .FALSE. 658 IF( INDEIG ) THEN 659 IF( IL.EQ.1 .AND. IU.EQ.N ) THEN 660 TEST = .TRUE. 661 END IF 662 END IF 663 IF( ( ALLEIG.OR.TEST ) .AND. ( IEEEOK.EQ.1 ) ) THEN 664 IF( .NOT.WANTZ ) THEN 665 CALL DCOPY( N, RWORK( INDRD ), 1, W, 1 ) 666 CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 ) 667 CALL DSTERF( N, W, RWORK( INDREE ), INFO ) 668 ELSE 669 CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 ) 670 CALL DCOPY( N, RWORK( INDRD ), 1, RWORK( INDRDD ), 1 ) 671* 672 IF (ABSTOL .LE. TWO*N*EPS) THEN 673 TRYRAC = .TRUE. 674 ELSE 675 TRYRAC = .FALSE. 676 END IF 677 CALL ZSTEMR( JOBZ, 'A', N, RWORK( INDRDD ), 678 $ RWORK( INDREE ), VL, VU, IL, IU, M, W, 679 $ Z, LDZ, N, ISUPPZ, TRYRAC, 680 $ RWORK( INDRWK ), LLRWORK, 681 $ IWORK, LIWORK, INFO ) 682* 683* Apply unitary matrix used in reduction to tridiagonal 684* form to eigenvectors returned by ZSTEMR. 685* 686 IF( WANTZ .AND. INFO.EQ.0 ) THEN 687 INDWKN = INDWK 688 LLWRKN = LWORK - INDWKN + 1 689 CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, 690 $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ), 691 $ LLWRKN, IINFO ) 692 END IF 693 END IF 694* 695* 696 IF( INFO.EQ.0 ) THEN 697 M = N 698 GO TO 30 699 END IF 700 INFO = 0 701 END IF 702* 703* Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN. 704* Also call DSTEBZ and ZSTEIN if ZSTEMR fails. 705* 706 IF( WANTZ ) THEN 707 ORDER = 'B' 708 ELSE 709 ORDER = 'E' 710 END IF 711 712 CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, 713 $ RWORK( INDRD ), RWORK( INDRE ), M, NSPLIT, W, 714 $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ), 715 $ IWORK( INDIWO ), INFO ) 716* 717 IF( WANTZ ) THEN 718 CALL ZSTEIN( N, RWORK( INDRD ), RWORK( INDRE ), M, W, 719 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, 720 $ RWORK( INDRWK ), IWORK( INDIWO ), IWORK( INDIFL ), 721 $ INFO ) 722* 723* Apply unitary matrix used in reduction to tridiagonal 724* form to eigenvectors returned by ZSTEIN. 725* 726 INDWKN = INDWK 727 LLWRKN = LWORK - INDWKN + 1 728 CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z, 729 $ LDZ, WORK( INDWKN ), LLWRKN, IINFO ) 730 END IF 731* 732* If matrix was scaled, then rescale eigenvalues appropriately. 733* 734 30 CONTINUE 735 IF( ISCALE.EQ.1 ) THEN 736 IF( INFO.EQ.0 ) THEN 737 IMAX = M 738 ELSE 739 IMAX = INFO - 1 740 END IF 741 CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) 742 END IF 743* 744* If eigenvalues are not in order, then sort them, along with 745* eigenvectors. 746* 747 IF( WANTZ ) THEN 748 DO 50 J = 1, M - 1 749 I = 0 750 TMP1 = W( J ) 751 DO 40 JJ = J + 1, M 752 IF( W( JJ ).LT.TMP1 ) THEN 753 I = JJ 754 TMP1 = W( JJ ) 755 END IF 756 40 CONTINUE 757* 758 IF( I.NE.0 ) THEN 759 ITMP1 = IWORK( INDIBL+I-1 ) 760 W( I ) = W( J ) 761 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) 762 W( J ) = TMP1 763 IWORK( INDIBL+J-1 ) = ITMP1 764 CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 765 END IF 766 50 CONTINUE 767 END IF 768* 769* Set WORK(1) to optimal workspace size. 770* 771 WORK( 1 ) = LWMIN 772 RWORK( 1 ) = LRWMIN 773 IWORK( 1 ) = LIWMIN 774* 775 RETURN 776* 777* End of ZHEEVR_2STAGE 778* 779 END 780