1*> \brief <b> CHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE 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 CHEEVR + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cheevr.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cheevr.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cheevr.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE CHEEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, 22* ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, 23* RWORK, LRWORK, IWORK, LIWORK, INFO ) 24* 25* .. Scalar Arguments .. 26* CHARACTER JOBZ, RANGE, UPLO 27* INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK, 28* $ M, N 29* REAL ABSTOL, VL, VU 30* .. 31* .. Array Arguments .. 32* INTEGER ISUPPZ( * ), IWORK( * ) 33* REAL RWORK( * ), W( * ) 34* COMPLEX A( LDA, * ), WORK( * ), Z( LDZ, * ) 35* .. 36* 37* 38*> \par Purpose: 39* ============= 40*> 41*> \verbatim 42*> 43*> CHEEVR computes selected eigenvalues and, optionally, eigenvectors 44*> of a complex Hermitian matrix A. Eigenvalues and eigenvectors can 45*> be selected by specifying either a range of values or a range of 46*> indices for the desired eigenvalues. 47*> 48*> CHEEVR first reduces the matrix A to tridiagonal form T with a call 49*> to CHETRD. Then, whenever possible, CHEEVR calls CSTEMR to compute 50*> the eigenspectrum using Relatively Robust Representations. CSTEMR 51*> computes eigenvalues by the dqds algorithm, while orthogonal 52*> eigenvectors are computed from various "good" L D L^T representations 53*> (also known as Relatively Robust Representations). Gram-Schmidt 54*> orthogonalization is avoided as far as possible. More specifically, 55*> the various steps of the algorithm are as follows. 56*> 57*> For each unreduced block (submatrix) of T, 58*> (a) Compute T - sigma I = L D L^T, so that L and D 59*> define all the wanted eigenvalues to high relative accuracy. 60*> This means that small relative changes in the entries of D and L 61*> cause only small relative changes in the eigenvalues and 62*> eigenvectors. The standard (unfactored) representation of the 63*> tridiagonal matrix T does not have this property in general. 64*> (b) Compute the eigenvalues to suitable accuracy. 65*> If the eigenvectors are desired, the algorithm attains full 66*> accuracy of the computed eigenvalues only right before 67*> the corresponding vectors have to be computed, see steps c) and d). 68*> (c) For each cluster of close eigenvalues, select a new 69*> shift close to the cluster, find a new factorization, and refine 70*> the shifted eigenvalues to suitable accuracy. 71*> (d) For each eigenvalue with a large enough relative separation compute 72*> the corresponding eigenvector by forming a rank revealing twisted 73*> factorization. Go back to (c) for any clusters that remain. 74*> 75*> The desired accuracy of the output can be specified by the input 76*> parameter ABSTOL. 77*> 78*> For more details, see DSTEMR's documentation and: 79*> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations 80*> to compute orthogonal eigenvectors of symmetric tridiagonal matrices," 81*> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. 82*> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and 83*> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, 84*> 2004. Also LAPACK Working Note 154. 85*> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric 86*> tridiagonal eigenvalue/eigenvector problem", 87*> Computer Science Division Technical Report No. UCB/CSD-97-971, 88*> UC Berkeley, May 1997. 89*> 90*> 91*> Note 1 : CHEEVR calls CSTEMR when the full spectrum is requested 92*> on machines which conform to the ieee-754 floating point standard. 93*> CHEEVR calls SSTEBZ and CSTEIN on non-ieee machines and 94*> when partial spectrum requests are made. 95*> 96*> Normal execution of CSTEMR may create NaNs and infinities and 97*> hence may abort due to a floating point exception in environments 98*> which do not handle NaNs and infinities in the ieee standard default 99*> manner. 100*> \endverbatim 101* 102* Arguments: 103* ========== 104* 105*> \param[in] JOBZ 106*> \verbatim 107*> JOBZ is CHARACTER*1 108*> = 'N': Compute eigenvalues only; 109*> = 'V': Compute eigenvalues and eigenvectors. 110*> \endverbatim 111*> 112*> \param[in] RANGE 113*> \verbatim 114*> RANGE is CHARACTER*1 115*> = 'A': all eigenvalues will be found. 116*> = 'V': all eigenvalues in the half-open interval (VL,VU] 117*> will be found. 118*> = 'I': the IL-th through IU-th eigenvalues will be found. 119*> For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and 120*> CSTEIN are called 121*> \endverbatim 122*> 123*> \param[in] UPLO 124*> \verbatim 125*> UPLO is CHARACTER*1 126*> = 'U': Upper triangle of A is stored; 127*> = 'L': Lower triangle of A is stored. 128*> \endverbatim 129*> 130*> \param[in] N 131*> \verbatim 132*> N is INTEGER 133*> The order of the matrix A. N >= 0. 134*> \endverbatim 135*> 136*> \param[in,out] A 137*> \verbatim 138*> A is COMPLEX array, dimension (LDA, N) 139*> On entry, the Hermitian matrix A. If UPLO = 'U', the 140*> leading N-by-N upper triangular part of A contains the 141*> upper triangular part of the matrix A. If UPLO = 'L', 142*> the leading N-by-N lower triangular part of A contains 143*> the lower triangular part of the matrix A. 144*> On exit, the lower triangle (if UPLO='L') or the upper 145*> triangle (if UPLO='U') of A, including the diagonal, is 146*> destroyed. 147*> \endverbatim 148*> 149*> \param[in] LDA 150*> \verbatim 151*> LDA is INTEGER 152*> The leading dimension of the array A. LDA >= max(1,N). 153*> \endverbatim 154*> 155*> \param[in] VL 156*> \verbatim 157*> VL is REAL 158*> \endverbatim 159*> 160*> \param[in] VU 161*> \verbatim 162*> VU is REAL 163*> If RANGE='V', the lower and upper bounds of the interval to 164*> be searched for eigenvalues. VL < VU. 165*> Not referenced if RANGE = 'A' or 'I'. 166*> \endverbatim 167*> 168*> \param[in] IL 169*> \verbatim 170*> IL is INTEGER 171*> \endverbatim 172*> 173*> \param[in] IU 174*> \verbatim 175*> IU is INTEGER 176*> If RANGE='I', the indices (in ascending order) of the 177*> smallest and largest eigenvalues to be returned. 178*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 179*> Not referenced if RANGE = 'A' or 'V'. 180*> \endverbatim 181*> 182*> \param[in] ABSTOL 183*> \verbatim 184*> ABSTOL is REAL 185*> The absolute error tolerance for the eigenvalues. 186*> An approximate eigenvalue is accepted as converged 187*> when it is determined to lie in an interval [a,b] 188*> of width less than or equal to 189*> 190*> ABSTOL + EPS * max( |a|,|b| ) , 191*> 192*> where EPS is the machine precision. If ABSTOL is less than 193*> or equal to zero, then EPS*|T| will be used in its place, 194*> where |T| is the 1-norm of the tridiagonal matrix obtained 195*> by reducing A to tridiagonal form. 196*> 197*> See "Computing Small Singular Values of Bidiagonal Matrices 198*> with Guaranteed High Relative Accuracy," by Demmel and 199*> Kahan, LAPACK Working Note #3. 200*> 201*> If high relative accuracy is important, set ABSTOL to 202*> SLAMCH( 'Safe minimum' ). Doing so will guarantee that 203*> eigenvalues are computed to high relative accuracy when 204*> possible in future releases. The current code does not 205*> make any guarantees about high relative accuracy, but 206*> furutre releases will. See J. Barlow and J. Demmel, 207*> "Computing Accurate Eigensystems of Scaled Diagonally 208*> Dominant Matrices", LAPACK Working Note #7, for a discussion 209*> of which matrices define their eigenvalues to high relative 210*> accuracy. 211*> \endverbatim 212*> 213*> \param[out] M 214*> \verbatim 215*> M is INTEGER 216*> The total number of eigenvalues found. 0 <= M <= N. 217*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 218*> \endverbatim 219*> 220*> \param[out] W 221*> \verbatim 222*> W is REAL array, dimension (N) 223*> The first M elements contain the selected eigenvalues in 224*> ascending order. 225*> \endverbatim 226*> 227*> \param[out] Z 228*> \verbatim 229*> Z is COMPLEX array, dimension (LDZ, max(1,M)) 230*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z 231*> contain the orthonormal eigenvectors of the matrix A 232*> corresponding to the selected eigenvalues, with the i-th 233*> column of Z holding the eigenvector associated with W(i). 234*> If JOBZ = 'N', then Z is not referenced. 235*> Note: the user must ensure that at least max(1,M) columns are 236*> supplied in the array Z; if RANGE = 'V', the exact value of M 237*> is not known in advance and an upper bound must be used. 238*> \endverbatim 239*> 240*> \param[in] LDZ 241*> \verbatim 242*> LDZ is INTEGER 243*> The leading dimension of the array Z. LDZ >= 1, and if 244*> JOBZ = 'V', LDZ >= max(1,N). 245*> \endverbatim 246*> 247*> \param[out] ISUPPZ 248*> \verbatim 249*> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) 250*> The support of the eigenvectors in Z, i.e., the indices 251*> indicating the nonzero elements in Z. The i-th eigenvector 252*> is nonzero only in elements ISUPPZ( 2*i-1 ) through 253*> ISUPPZ( 2*i ). 254*> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 255*> \endverbatim 256*> 257*> \param[out] WORK 258*> \verbatim 259*> WORK is COMPLEX array, dimension (MAX(1,LWORK)) 260*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 261*> \endverbatim 262*> 263*> \param[in] LWORK 264*> \verbatim 265*> LWORK is INTEGER 266*> The length of the array WORK. LWORK >= max(1,2*N). 267*> For optimal efficiency, LWORK >= (NB+1)*N, 268*> where NB is the max of the blocksize for CHETRD and for 269*> CUNMTR as returned by ILAENV. 270*> 271*> If LWORK = -1, then a workspace query is assumed; the routine 272*> only calculates the optimal sizes of the WORK, RWORK and 273*> IWORK arrays, returns these values as the first entries of 274*> the WORK, RWORK and IWORK arrays, and no error message 275*> related to LWORK or LRWORK or LIWORK is issued by XERBLA. 276*> \endverbatim 277*> 278*> \param[out] RWORK 279*> \verbatim 280*> RWORK is REAL array, dimension (MAX(1,LRWORK)) 281*> On exit, if INFO = 0, RWORK(1) returns the optimal 282*> (and minimal) LRWORK. 283*> \endverbatim 284*> 285*> \param[in] LRWORK 286*> \verbatim 287*> LRWORK is INTEGER 288*> The length of the array RWORK. LRWORK >= max(1,24*N). 289*> 290*> If LRWORK = -1, then a workspace query is assumed; the 291*> routine only calculates the optimal sizes of the WORK, RWORK 292*> and IWORK arrays, returns these values as the first entries 293*> of the WORK, RWORK and IWORK arrays, and no error message 294*> related to LWORK or LRWORK or LIWORK is issued by XERBLA. 295*> \endverbatim 296*> 297*> \param[out] IWORK 298*> \verbatim 299*> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) 300*> On exit, if INFO = 0, IWORK(1) returns the optimal 301*> (and minimal) LIWORK. 302*> \endverbatim 303*> 304*> \param[in] LIWORK 305*> \verbatim 306*> LIWORK is INTEGER 307*> The dimension of the array IWORK. LIWORK >= max(1,10*N). 308*> 309*> If LIWORK = -1, then a workspace query is assumed; the 310*> routine only calculates the optimal sizes of the WORK, RWORK 311*> and IWORK arrays, returns these values as the first entries 312*> of the WORK, RWORK and IWORK arrays, and no error message 313*> related to LWORK or LRWORK or LIWORK is issued by XERBLA. 314*> \endverbatim 315*> 316*> \param[out] INFO 317*> \verbatim 318*> INFO is INTEGER 319*> = 0: successful exit 320*> < 0: if INFO = -i, the i-th argument had an illegal value 321*> > 0: Internal error 322*> \endverbatim 323* 324* Authors: 325* ======== 326* 327*> \author Univ. of Tennessee 328*> \author Univ. of California Berkeley 329*> \author Univ. of Colorado Denver 330*> \author NAG Ltd. 331* 332*> \date September 2012 333* 334*> \ingroup complexHEeigen 335* 336*> \par Contributors: 337* ================== 338*> 339*> Inderjit Dhillon, IBM Almaden, USA \n 340*> Osni Marques, LBNL/NERSC, USA \n 341*> Ken Stanley, Computer Science Division, University of 342*> California at Berkeley, USA \n 343*> Jason Riedy, Computer Science Division, University of 344*> California at Berkeley, USA \n 345*> 346* ===================================================================== 347 SUBROUTINE CHEEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, 348 $ ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, 349 $ RWORK, LRWORK, IWORK, LIWORK, INFO ) 350* 351* -- LAPACK driver routine (version 3.4.2) -- 352* -- LAPACK is a software package provided by Univ. of Tennessee, -- 353* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 354* September 2012 355* 356* .. Scalar Arguments .. 357 CHARACTER JOBZ, RANGE, UPLO 358 INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK, 359 $ M, N 360 REAL ABSTOL, VL, VU 361* .. 362* .. Array Arguments .. 363 INTEGER ISUPPZ( * ), IWORK( * ) 364 REAL RWORK( * ), W( * ) 365 COMPLEX A( LDA, * ), WORK( * ), Z( LDZ, * ) 366* .. 367* 368* ===================================================================== 369* 370* .. Parameters .. 371 REAL ZERO, ONE, TWO 372 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 ) 373* .. 374* .. Local Scalars .. 375 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG, 376 $ WANTZ, TRYRAC 377 CHARACTER ORDER 378 INTEGER I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP, 379 $ INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK, 380 $ INDTAU, INDWK, INDWKN, ISCALE, ITMP1, J, JJ, 381 $ LIWMIN, LLWORK, LLRWORK, LLWRKN, LRWMIN, 382 $ LWKOPT, LWMIN, NB, NSPLIT 383 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, 384 $ SIGMA, SMLNUM, TMP1, VLL, VUU 385* .. 386* .. External Functions .. 387 LOGICAL LSAME 388 INTEGER ILAENV 389 REAL CLANSY, SLAMCH 390 EXTERNAL LSAME, ILAENV, CLANSY, SLAMCH 391* .. 392* .. External Subroutines .. 393 EXTERNAL CHETRD, CSSCAL, CSTEMR, CSTEIN, CSWAP, CUNMTR, 394 $ SCOPY, SSCAL, SSTEBZ, SSTERF, XERBLA 395* .. 396* .. Intrinsic Functions .. 397 INTRINSIC MAX, MIN, REAL, SQRT 398* .. 399* .. Executable Statements .. 400* 401* Test the input parameters. 402* 403 IEEEOK = ILAENV( 10, 'CHEEVR', 'N', 1, 2, 3, 4 ) 404* 405 LOWER = LSAME( UPLO, 'L' ) 406 WANTZ = LSAME( JOBZ, 'V' ) 407 ALLEIG = LSAME( RANGE, 'A' ) 408 VALEIG = LSAME( RANGE, 'V' ) 409 INDEIG = LSAME( RANGE, 'I' ) 410* 411 LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LRWORK.EQ.-1 ) .OR. 412 $ ( LIWORK.EQ.-1 ) ) 413* 414 LRWMIN = MAX( 1, 24*N ) 415 LIWMIN = MAX( 1, 10*N ) 416 LWMIN = MAX( 1, 2*N ) 417* 418 INFO = 0 419 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 420 INFO = -1 421 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 422 INFO = -2 423 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN 424 INFO = -3 425 ELSE IF( N.LT.0 ) THEN 426 INFO = -4 427 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 428 INFO = -6 429 ELSE 430 IF( VALEIG ) THEN 431 IF( N.GT.0 .AND. VU.LE.VL ) 432 $ INFO = -8 433 ELSE IF( INDEIG ) THEN 434 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 435 INFO = -9 436 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 437 INFO = -10 438 END IF 439 END IF 440 END IF 441 IF( INFO.EQ.0 ) THEN 442 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 443 INFO = -15 444 END IF 445 END IF 446* 447 IF( INFO.EQ.0 ) THEN 448 NB = ILAENV( 1, 'CHETRD', UPLO, N, -1, -1, -1 ) 449 NB = MAX( NB, ILAENV( 1, 'CUNMTR', UPLO, N, -1, -1, -1 ) ) 450 LWKOPT = MAX( ( NB+1 )*N, LWMIN ) 451 WORK( 1 ) = LWKOPT 452 RWORK( 1 ) = LRWMIN 453 IWORK( 1 ) = LIWMIN 454* 455 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 456 INFO = -18 457 ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN 458 INFO = -20 459 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 460 INFO = -22 461 END IF 462 END IF 463* 464 IF( INFO.NE.0 ) THEN 465 CALL XERBLA( 'CHEEVR', -INFO ) 466 RETURN 467 ELSE IF( LQUERY ) THEN 468 RETURN 469 END IF 470* 471* Quick return if possible 472* 473 M = 0 474 IF( N.EQ.0 ) THEN 475 WORK( 1 ) = 1 476 RETURN 477 END IF 478* 479 IF( N.EQ.1 ) THEN 480 WORK( 1 ) = 2 481 IF( ALLEIG .OR. INDEIG ) THEN 482 M = 1 483 W( 1 ) = REAL( A( 1, 1 ) ) 484 ELSE 485 IF( VL.LT.REAL( A( 1, 1 ) ) .AND. VU.GE.REAL( A( 1, 1 ) ) ) 486 $ THEN 487 M = 1 488 W( 1 ) = REAL( A( 1, 1 ) ) 489 END IF 490 END IF 491 IF( WANTZ ) THEN 492 Z( 1, 1 ) = ONE 493 ISUPPZ( 1 ) = 1 494 ISUPPZ( 2 ) = 1 495 END IF 496 RETURN 497 END IF 498* 499* Get machine constants. 500* 501 SAFMIN = SLAMCH( 'Safe minimum' ) 502 EPS = SLAMCH( 'Precision' ) 503 SMLNUM = SAFMIN / EPS 504 BIGNUM = ONE / SMLNUM 505 RMIN = SQRT( SMLNUM ) 506 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 507* 508* Scale matrix to allowable range, if necessary. 509* 510 ISCALE = 0 511 ABSTLL = ABSTOL 512 IF (VALEIG) THEN 513 VLL = VL 514 VUU = VU 515 END IF 516 ANRM = CLANSY( 'M', UPLO, N, A, LDA, RWORK ) 517 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN 518 ISCALE = 1 519 SIGMA = RMIN / ANRM 520 ELSE IF( ANRM.GT.RMAX ) THEN 521 ISCALE = 1 522 SIGMA = RMAX / ANRM 523 END IF 524 IF( ISCALE.EQ.1 ) THEN 525 IF( LOWER ) THEN 526 DO 10 J = 1, N 527 CALL CSSCAL( N-J+1, SIGMA, A( J, J ), 1 ) 528 10 CONTINUE 529 ELSE 530 DO 20 J = 1, N 531 CALL CSSCAL( J, SIGMA, A( 1, J ), 1 ) 532 20 CONTINUE 533 END IF 534 IF( ABSTOL.GT.0 ) 535 $ ABSTLL = ABSTOL*SIGMA 536 IF( VALEIG ) THEN 537 VLL = VL*SIGMA 538 VUU = VU*SIGMA 539 END IF 540 END IF 541 542* Initialize indices into workspaces. Note: The IWORK indices are 543* used only if SSTERF or CSTEMR fail. 544 545* WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the 546* elementary reflectors used in CHETRD. 547 INDTAU = 1 548* INDWK is the starting offset of the remaining complex workspace, 549* and LLWORK is the remaining complex workspace size. 550 INDWK = INDTAU + N 551 LLWORK = LWORK - INDWK + 1 552 553* RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal 554* entries. 555 INDRD = 1 556* RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the 557* tridiagonal matrix from CHETRD. 558 INDRE = INDRD + N 559* RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over 560* -written by CSTEMR (the SSTERF path copies the diagonal to W). 561 INDRDD = INDRE + N 562* RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over 563* -written while computing the eigenvalues in SSTERF and CSTEMR. 564 INDREE = INDRDD + N 565* INDRWK is the starting offset of the left-over real workspace, and 566* LLRWORK is the remaining workspace size. 567 INDRWK = INDREE + N 568 LLRWORK = LRWORK - INDRWK + 1 569 570* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and 571* stores the block indices of each of the M<=N eigenvalues. 572 INDIBL = 1 573* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and 574* stores the starting and finishing indices of each block. 575 INDISP = INDIBL + N 576* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors 577* that corresponding to eigenvectors that fail to converge in 578* SSTEIN. This information is discarded; if any fail, the driver 579* returns INFO > 0. 580 INDIFL = INDISP + N 581* INDIWO is the offset of the remaining integer workspace. 582 INDIWO = INDIFL + N 583 584* 585* Call CHETRD to reduce Hermitian matrix to tridiagonal form. 586* 587 CALL CHETRD( UPLO, N, A, LDA, RWORK( INDRD ), RWORK( INDRE ), 588 $ WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO ) 589* 590* If all eigenvalues are desired 591* then call SSTERF or CSTEMR and CUNMTR. 592* 593 TEST = .FALSE. 594 IF( INDEIG ) THEN 595 IF( IL.EQ.1 .AND. IU.EQ.N ) THEN 596 TEST = .TRUE. 597 END IF 598 END IF 599 IF( ( ALLEIG.OR.TEST ) .AND. ( IEEEOK.EQ.1 ) ) THEN 600 IF( .NOT.WANTZ ) THEN 601 CALL SCOPY( N, RWORK( INDRD ), 1, W, 1 ) 602 CALL SCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 ) 603 CALL SSTERF( N, W, RWORK( INDREE ), INFO ) 604 ELSE 605 CALL SCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 ) 606 CALL SCOPY( N, RWORK( INDRD ), 1, RWORK( INDRDD ), 1 ) 607* 608 IF (ABSTOL .LE. TWO*N*EPS) THEN 609 TRYRAC = .TRUE. 610 ELSE 611 TRYRAC = .FALSE. 612 END IF 613 CALL CSTEMR( JOBZ, 'A', N, RWORK( INDRDD ), 614 $ RWORK( INDREE ), VL, VU, IL, IU, M, W, 615 $ Z, LDZ, N, ISUPPZ, TRYRAC, 616 $ RWORK( INDRWK ), LLRWORK, 617 $ IWORK, LIWORK, INFO ) 618* 619* Apply unitary matrix used in reduction to tridiagonal 620* form to eigenvectors returned by CSTEIN. 621* 622 IF( WANTZ .AND. INFO.EQ.0 ) THEN 623 INDWKN = INDWK 624 LLWRKN = LWORK - INDWKN + 1 625 CALL CUNMTR( 'L', UPLO, 'N', N, M, A, LDA, 626 $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ), 627 $ LLWRKN, IINFO ) 628 END IF 629 END IF 630* 631* 632 IF( INFO.EQ.0 ) THEN 633 M = N 634 GO TO 30 635 END IF 636 INFO = 0 637 END IF 638* 639* Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN. 640* Also call SSTEBZ and CSTEIN if CSTEMR fails. 641* 642 IF( WANTZ ) THEN 643 ORDER = 'B' 644 ELSE 645 ORDER = 'E' 646 END IF 647 648 CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, 649 $ RWORK( INDRD ), RWORK( INDRE ), M, NSPLIT, W, 650 $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ), 651 $ IWORK( INDIWO ), INFO ) 652* 653 IF( WANTZ ) THEN 654 CALL CSTEIN( N, RWORK( INDRD ), RWORK( INDRE ), M, W, 655 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, 656 $ RWORK( INDRWK ), IWORK( INDIWO ), IWORK( INDIFL ), 657 $ INFO ) 658* 659* Apply unitary matrix used in reduction to tridiagonal 660* form to eigenvectors returned by CSTEIN. 661* 662 INDWKN = INDWK 663 LLWRKN = LWORK - INDWKN + 1 664 CALL CUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z, 665 $ LDZ, WORK( INDWKN ), LLWRKN, IINFO ) 666 END IF 667* 668* If matrix was scaled, then rescale eigenvalues appropriately. 669* 670 30 CONTINUE 671 IF( ISCALE.EQ.1 ) THEN 672 IF( INFO.EQ.0 ) THEN 673 IMAX = M 674 ELSE 675 IMAX = INFO - 1 676 END IF 677 CALL SSCAL( IMAX, ONE / SIGMA, W, 1 ) 678 END IF 679* 680* If eigenvalues are not in order, then sort them, along with 681* eigenvectors. 682* 683 IF( WANTZ ) THEN 684 DO 50 J = 1, M - 1 685 I = 0 686 TMP1 = W( J ) 687 DO 40 JJ = J + 1, M 688 IF( W( JJ ).LT.TMP1 ) THEN 689 I = JJ 690 TMP1 = W( JJ ) 691 END IF 692 40 CONTINUE 693* 694 IF( I.NE.0 ) THEN 695 ITMP1 = IWORK( INDIBL+I-1 ) 696 W( I ) = W( J ) 697 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) 698 W( J ) = TMP1 699 IWORK( INDIBL+J-1 ) = ITMP1 700 CALL CSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 701 END IF 702 50 CONTINUE 703 END IF 704* 705* Set WORK(1) to optimal workspace size. 706* 707 WORK( 1 ) = LWKOPT 708 RWORK( 1 ) = LRWMIN 709 IWORK( 1 ) = LIWMIN 710* 711 RETURN 712* 713* End of CHEEVR 714* 715 END 716