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