1*> \brief <b> SSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b> 2* 3* @generated from dsyevr_2stage.f, fortran d -> s, Sat Nov 5 23:50:10 2016 4* 5* =========== DOCUMENTATION =========== 6* 7* Online html documentation available at 8* http://www.netlib.org/lapack/explore-html/ 9* 10*> \htmlonly 11*> Download SSYEVR_2STAGE + dependencies 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ssyevr_2stage.f"> 13*> [TGZ]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ssyevr_2stage.f"> 15*> [ZIP]</a> 16*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ssyevr_2stage.f"> 17*> [TXT]</a> 18*> \endhtmlonly 19* 20* Definition: 21* =========== 22* 23* SUBROUTINE SSYEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, 24* IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, 25* LWORK, IWORK, LIWORK, INFO ) 26* 27* IMPLICIT NONE 28* 29* .. Scalar Arguments .. 30* CHARACTER JOBZ, RANGE, UPLO 31* INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N 32* REAL ABSTOL, VL, VU 33* .. 34* .. Array Arguments .. 35* INTEGER ISUPPZ( * ), IWORK( * ) 36* REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * ) 37* .. 38* 39* 40*> \par Purpose: 41* ============= 42*> 43*> \verbatim 44*> 45*> SSYEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors 46*> of a real symmetric matrix A using the 2stage technique for 47*> the reduction to tridiagonal. Eigenvalues and eigenvectors can be 48*> selected by specifying either a range of values or a range of 49*> indices for the desired eigenvalues. 50*> 51*> SSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call 52*> to SSYTRD. Then, whenever possible, SSYEVR_2STAGE calls SSTEMR to compute 53*> the eigenspectrum using Relatively Robust Representations. SSTEMR 54*> computes eigenvalues by the dqds algorithm, while orthogonal 55*> eigenvectors are computed from various "good" L D L^T representations 56*> (also known as Relatively Robust Representations). Gram-Schmidt 57*> orthogonalization is avoided as far as possible. More specifically, 58*> the various steps of the algorithm are as follows. 59*> 60*> For each unreduced block (submatrix) of T, 61*> (a) Compute T - sigma I = L D L^T, so that L and D 62*> define all the wanted eigenvalues to high relative accuracy. 63*> This means that small relative changes in the entries of D and L 64*> cause only small relative changes in the eigenvalues and 65*> eigenvectors. The standard (unfactored) representation of the 66*> tridiagonal matrix T does not have this property in general. 67*> (b) Compute the eigenvalues to suitable accuracy. 68*> If the eigenvectors are desired, the algorithm attains full 69*> accuracy of the computed eigenvalues only right before 70*> the corresponding vectors have to be computed, see steps c) and d). 71*> (c) For each cluster of close eigenvalues, select a new 72*> shift close to the cluster, find a new factorization, and refine 73*> the shifted eigenvalues to suitable accuracy. 74*> (d) For each eigenvalue with a large enough relative separation compute 75*> the corresponding eigenvector by forming a rank revealing twisted 76*> factorization. Go back to (c) for any clusters that remain. 77*> 78*> The desired accuracy of the output can be specified by the input 79*> parameter ABSTOL. 80*> 81*> For more details, see SSTEMR's documentation and: 82*> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations 83*> to compute orthogonal eigenvectors of symmetric tridiagonal matrices," 84*> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. 85*> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and 86*> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, 87*> 2004. Also LAPACK Working Note 154. 88*> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric 89*> tridiagonal eigenvalue/eigenvector problem", 90*> Computer Science Division Technical Report No. UCB/CSD-97-971, 91*> UC Berkeley, May 1997. 92*> 93*> 94*> Note 1 : SSYEVR_2STAGE calls SSTEMR when the full spectrum is requested 95*> on machines which conform to the ieee-754 floating point standard. 96*> SSYEVR_2STAGE calls SSTEBZ and SSTEIN on non-ieee machines and 97*> when partial spectrum requests are made. 98*> 99*> Normal execution of SSTEMR may create NaNs and infinities and 100*> hence may abort due to a floating point exception in environments 101*> which do not handle NaNs and infinities in the ieee standard default 102*> manner. 103*> \endverbatim 104* 105* Arguments: 106* ========== 107* 108*> \param[in] JOBZ 109*> \verbatim 110*> JOBZ is CHARACTER*1 111*> = 'N': Compute eigenvalues only; 112*> = 'V': Compute eigenvalues and eigenvectors. 113*> Not available in this release. 114*> \endverbatim 115*> 116*> \param[in] RANGE 117*> \verbatim 118*> RANGE is CHARACTER*1 119*> = 'A': all eigenvalues will be found. 120*> = 'V': all eigenvalues in the half-open interval (VL,VU] 121*> will be found. 122*> = 'I': the IL-th through IU-th eigenvalues will be found. 123*> For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and 124*> SSTEIN are called 125*> \endverbatim 126*> 127*> \param[in] UPLO 128*> \verbatim 129*> UPLO is CHARACTER*1 130*> = 'U': Upper triangle of A is stored; 131*> = 'L': Lower triangle of A is stored. 132*> \endverbatim 133*> 134*> \param[in] N 135*> \verbatim 136*> N is INTEGER 137*> The order of the matrix A. N >= 0. 138*> \endverbatim 139*> 140*> \param[in,out] A 141*> \verbatim 142*> A is REAL array, dimension (LDA, N) 143*> On entry, the symmetric matrix A. If UPLO = 'U', the 144*> leading N-by-N upper triangular part of A contains the 145*> upper triangular part of the matrix A. If UPLO = 'L', 146*> the leading N-by-N lower triangular part of A contains 147*> the lower triangular part of the matrix A. 148*> On exit, the lower triangle (if UPLO='L') or the upper 149*> triangle (if UPLO='U') of A, including the diagonal, is 150*> destroyed. 151*> \endverbatim 152*> 153*> \param[in] LDA 154*> \verbatim 155*> LDA is INTEGER 156*> The leading dimension of the array A. LDA >= max(1,N). 157*> \endverbatim 158*> 159*> \param[in] VL 160*> \verbatim 161*> VL is REAL 162*> If RANGE='V', the lower bound of the interval to 163*> be searched for eigenvalues. VL < VU. 164*> Not referenced if RANGE = 'A' or 'I'. 165*> \endverbatim 166*> 167*> \param[in] VU 168*> \verbatim 169*> VU is REAL 170*> If RANGE='V', the upper bound of the interval to 171*> be searched for eigenvalues. VL < VU. 172*> Not referenced if RANGE = 'A' or 'I'. 173*> \endverbatim 174*> 175*> \param[in] IL 176*> \verbatim 177*> IL is INTEGER 178*> If RANGE='I', the index of the 179*> smallest eigenvalue to be returned. 180*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 181*> Not referenced if RANGE = 'A' or 'V'. 182*> \endverbatim 183*> 184*> \param[in] IU 185*> \verbatim 186*> IU is INTEGER 187*> If RANGE='I', the index of the 188*> largest eigenvalue to be returned. 189*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 190*> Not referenced if RANGE = 'A' or 'V'. 191*> \endverbatim 192*> 193*> \param[in] ABSTOL 194*> \verbatim 195*> ABSTOL is REAL 196*> The absolute error tolerance for the eigenvalues. 197*> An approximate eigenvalue is accepted as converged 198*> when it is determined to lie in an interval [a,b] 199*> of width less than or equal to 200*> 201*> ABSTOL + EPS * max( |a|,|b| ) , 202*> 203*> where EPS is the machine precision. If ABSTOL is less than 204*> or equal to zero, then EPS*|T| will be used in its place, 205*> where |T| is the 1-norm of the tridiagonal matrix obtained 206*> by reducing A to tridiagonal form. 207*> 208*> See "Computing Small Singular Values of Bidiagonal Matrices 209*> with Guaranteed High Relative Accuracy," by Demmel and 210*> Kahan, LAPACK Working Note #3. 211*> 212*> If high relative accuracy is important, set ABSTOL to 213*> SLAMCH( 'Safe minimum' ). Doing so will guarantee that 214*> eigenvalues are computed to high relative accuracy when 215*> possible in future releases. The current code does not 216*> make any guarantees about high relative accuracy, but 217*> future releases will. See J. Barlow and J. Demmel, 218*> "Computing Accurate Eigensystems of Scaled Diagonally 219*> Dominant Matrices", LAPACK Working Note #7, for a discussion 220*> of which matrices define their eigenvalues to high relative 221*> accuracy. 222*> \endverbatim 223*> 224*> \param[out] M 225*> \verbatim 226*> M is INTEGER 227*> The total number of eigenvalues found. 0 <= M <= N. 228*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 229*> \endverbatim 230*> 231*> \param[out] W 232*> \verbatim 233*> W is REAL array, dimension (N) 234*> The first M elements contain the selected eigenvalues in 235*> ascending order. 236*> \endverbatim 237*> 238*> \param[out] Z 239*> \verbatim 240*> Z is REAL array, dimension (LDZ, max(1,M)) 241*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z 242*> contain the orthonormal eigenvectors of the matrix A 243*> corresponding to the selected eigenvalues, with the i-th 244*> column of Z holding the eigenvector associated with W(i). 245*> If JOBZ = 'N', then Z is not referenced. 246*> Note: the user must ensure that at least max(1,M) columns are 247*> supplied in the array Z; if RANGE = 'V', the exact value of M 248*> is not known in advance and an upper bound must be used. 249*> Supplying N columns is always safe. 250*> \endverbatim 251*> 252*> \param[in] LDZ 253*> \verbatim 254*> LDZ is INTEGER 255*> The leading dimension of the array Z. LDZ >= 1, and if 256*> JOBZ = 'V', LDZ >= max(1,N). 257*> \endverbatim 258*> 259*> \param[out] ISUPPZ 260*> \verbatim 261*> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) 262*> The support of the eigenvectors in Z, i.e., the indices 263*> indicating the nonzero elements in Z. The i-th eigenvector 264*> is nonzero only in elements ISUPPZ( 2*i-1 ) through 265*> ISUPPZ( 2*i ). This is an output of SSTEMR (tridiagonal 266*> matrix). The support of the eigenvectors of A is typically 267*> 1:N because of the orthogonal transformations applied by SORMTR. 268*> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 269*> \endverbatim 270*> 271*> \param[out] WORK 272*> \verbatim 273*> WORK is REAL array, dimension (MAX(1,LWORK)) 274*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 275*> \endverbatim 276*> 277*> \param[in] LWORK 278*> \verbatim 279*> LWORK is INTEGER 280*> The dimension of the array WORK. 281*> If JOBZ = 'N' and N > 1, LWORK must be queried. 282*> LWORK = MAX(1, 26*N, dimension) where 283*> dimension = max(stage1,stage2) + (KD+1)*N + 5*N 284*> = N*KD + N*max(KD+1,FACTOPTNB) 285*> + max(2*KD*KD, KD*NTHREADS) 286*> + (KD+1)*N + 5*N 287*> where KD is the blocking size of the reduction, 288*> FACTOPTNB is the blocking used by the QR or LQ 289*> algorithm, usually FACTOPTNB=128 is a good choice 290*> NTHREADS is the number of threads used when 291*> openMP compilation is enabled, otherwise =1. 292*> If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available 293*> 294*> If LWORK = -1, then a workspace query is assumed; the routine 295*> only calculates the optimal size of the WORK array, returns 296*> this value as the first entry of the WORK array, and no error 297*> message related to LWORK is issued by XERBLA. 298*> \endverbatim 299*> 300*> \param[out] IWORK 301*> \verbatim 302*> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) 303*> On exit, if INFO = 0, IWORK(1) returns the optimal LWORK. 304*> \endverbatim 305*> 306*> \param[in] LIWORK 307*> \verbatim 308*> LIWORK is INTEGER 309*> The dimension of the array IWORK. LIWORK >= max(1,10*N). 310*> 311*> If LIWORK = -1, then a workspace query is assumed; the 312*> routine only calculates the optimal size of the IWORK array, 313*> returns this value as the first entry of the IWORK array, and 314*> no error message related to LIWORK is issued by XERBLA. 315*> \endverbatim 316*> 317*> \param[out] INFO 318*> \verbatim 319*> INFO is INTEGER 320*> = 0: successful exit 321*> < 0: if INFO = -i, the i-th argument had an illegal value 322*> > 0: Internal error 323*> \endverbatim 324* 325* Authors: 326* ======== 327* 328*> \author Univ. of Tennessee 329*> \author Univ. of California Berkeley 330*> \author Univ. of Colorado Denver 331*> \author NAG Ltd. 332* 333*> \ingroup realSYeigen 334* 335*> \par Contributors: 336* ================== 337*> 338*> Inderjit Dhillon, IBM Almaden, USA \n 339*> Osni Marques, LBNL/NERSC, USA \n 340*> Ken Stanley, Computer Science Division, University of 341*> California at Berkeley, USA \n 342*> Jason Riedy, Computer Science Division, University of 343*> California at Berkeley, USA \n 344*> 345*> \par Further Details: 346* ===================== 347*> 348*> \verbatim 349*> 350*> All details about the 2stage techniques are available in: 351*> 352*> Azzam Haidar, Hatem Ltaief, and Jack Dongarra. 353*> Parallel reduction to condensed forms for symmetric eigenvalue problems 354*> using aggregated fine-grained and memory-aware kernels. In Proceedings 355*> of 2011 International Conference for High Performance Computing, 356*> Networking, Storage and Analysis (SC '11), New York, NY, USA, 357*> Article 8 , 11 pages. 358*> http://doi.acm.org/10.1145/2063384.2063394 359*> 360*> A. Haidar, J. Kurzak, P. Luszczek, 2013. 361*> An improved parallel singular value algorithm and its implementation 362*> for multicore hardware, In Proceedings of 2013 International Conference 363*> for High Performance Computing, Networking, Storage and Analysis (SC '13). 364*> Denver, Colorado, USA, 2013. 365*> Article 90, 12 pages. 366*> http://doi.acm.org/10.1145/2503210.2503292 367*> 368*> A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. 369*> A novel hybrid CPU-GPU generalized eigensolver for electronic structure 370*> calculations based on fine-grained memory aware tasks. 371*> International Journal of High Performance Computing Applications. 372*> Volume 28 Issue 2, Pages 196-209, May 2014. 373*> http://hpc.sagepub.com/content/28/2/196 374*> 375*> \endverbatim 376* 377* ===================================================================== 378 SUBROUTINE SSYEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, 379 $ IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, 380 $ LWORK, IWORK, LIWORK, INFO ) 381* 382 IMPLICIT NONE 383* 384* -- LAPACK driver routine -- 385* -- LAPACK is a software package provided by Univ. of Tennessee, -- 386* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 387* 388* .. Scalar Arguments .. 389 CHARACTER JOBZ, RANGE, UPLO 390 INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N 391 REAL ABSTOL, VL, VU 392* .. 393* .. Array Arguments .. 394 INTEGER ISUPPZ( * ), IWORK( * ) 395 REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * ) 396* .. 397* 398* ===================================================================== 399* 400* .. Parameters .. 401 REAL ZERO, ONE, TWO 402 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 ) 403* .. 404* .. Local Scalars .. 405 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ, 406 $ TRYRAC, TEST 407 CHARACTER ORDER 408 INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE, 409 $ INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU, 410 $ INDWK, INDWKN, ISCALE, J, JJ, LIWMIN, 411 $ LLWORK, LLWRKN, LWMIN, NSPLIT, 412 $ LHTRD, LWTRD, KD, IB, INDHOUS 413 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, 414 $ SIGMA, SMLNUM, TMP1, VLL, VUU 415* .. 416* .. External Functions .. 417 LOGICAL LSAME 418 INTEGER ILAENV, ILAENV2STAGE 419 REAL SLAMCH, SLANSY 420 EXTERNAL LSAME, SLAMCH, SLANSY, ILAENV, ILAENV2STAGE 421* .. 422* .. External Subroutines .. 423 EXTERNAL SCOPY, SORMTR, SSCAL, SSTEBZ, SSTEMR, SSTEIN, 424 $ SSTERF, SSWAP, SSYTRD_2STAGE, XERBLA 425* .. 426* .. Intrinsic Functions .. 427 INTRINSIC MAX, MIN, SQRT 428* .. 429* .. Executable Statements .. 430* 431* Test the input parameters. 432* 433 IEEEOK = ILAENV( 10, 'SSYEVR', 'N', 1, 2, 3, 4 ) 434* 435 LOWER = LSAME( UPLO, 'L' ) 436 WANTZ = LSAME( JOBZ, 'V' ) 437 ALLEIG = LSAME( RANGE, 'A' ) 438 VALEIG = LSAME( RANGE, 'V' ) 439 INDEIG = LSAME( RANGE, 'I' ) 440* 441 LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) ) 442* 443 KD = ILAENV2STAGE( 1, 'SSYTRD_2STAGE', JOBZ, N, -1, -1, -1 ) 444 IB = ILAENV2STAGE( 2, 'SSYTRD_2STAGE', JOBZ, N, KD, -1, -1 ) 445 LHTRD = ILAENV2STAGE( 3, 'SSYTRD_2STAGE', JOBZ, N, KD, IB, -1 ) 446 LWTRD = ILAENV2STAGE( 4, 'SSYTRD_2STAGE', JOBZ, N, KD, IB, -1 ) 447 LWMIN = MAX( 26*N, 5*N + LHTRD + LWTRD ) 448 LIWMIN = MAX( 1, 10*N ) 449* 450 INFO = 0 451 IF( .NOT.( LSAME( JOBZ, 'N' ) ) ) THEN 452 INFO = -1 453 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 454 INFO = -2 455 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN 456 INFO = -3 457 ELSE IF( N.LT.0 ) THEN 458 INFO = -4 459 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 460 INFO = -6 461 ELSE 462 IF( VALEIG ) THEN 463 IF( N.GT.0 .AND. VU.LE.VL ) 464 $ INFO = -8 465 ELSE IF( INDEIG ) THEN 466 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 467 INFO = -9 468 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 469 INFO = -10 470 END IF 471 END IF 472 END IF 473 IF( INFO.EQ.0 ) THEN 474 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 475 INFO = -15 476 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 477 INFO = -18 478 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 479 INFO = -20 480 END IF 481 END IF 482* 483 IF( INFO.EQ.0 ) THEN 484* NB = ILAENV( 1, 'SSYTRD', UPLO, N, -1, -1, -1 ) 485* NB = MAX( NB, ILAENV( 1, 'SORMTR', UPLO, N, -1, -1, -1 ) ) 486* LWKOPT = MAX( ( NB+1 )*N, LWMIN ) 487 WORK( 1 ) = LWMIN 488 IWORK( 1 ) = LIWMIN 489 END IF 490* 491 IF( INFO.NE.0 ) THEN 492 CALL XERBLA( 'SSYEVR_2STAGE', -INFO ) 493 RETURN 494 ELSE IF( LQUERY ) THEN 495 RETURN 496 END IF 497* 498* Quick return if possible 499* 500 M = 0 501 IF( N.EQ.0 ) THEN 502 WORK( 1 ) = 1 503 RETURN 504 END IF 505* 506 IF( N.EQ.1 ) THEN 507 WORK( 1 ) = 26 508 IF( ALLEIG .OR. INDEIG ) THEN 509 M = 1 510 W( 1 ) = A( 1, 1 ) 511 ELSE 512 IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN 513 M = 1 514 W( 1 ) = A( 1, 1 ) 515 END IF 516 END IF 517 IF( WANTZ ) THEN 518 Z( 1, 1 ) = ONE 519 ISUPPZ( 1 ) = 1 520 ISUPPZ( 2 ) = 1 521 END IF 522 RETURN 523 END IF 524* 525* Get machine constants. 526* 527 SAFMIN = SLAMCH( 'Safe minimum' ) 528 EPS = SLAMCH( 'Precision' ) 529 SMLNUM = SAFMIN / EPS 530 BIGNUM = ONE / SMLNUM 531 RMIN = SQRT( SMLNUM ) 532 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 533* 534* Scale matrix to allowable range, if necessary. 535* 536 ISCALE = 0 537 ABSTLL = ABSTOL 538 IF (VALEIG) THEN 539 VLL = VL 540 VUU = VU 541 END IF 542 ANRM = SLANSY( 'M', UPLO, N, A, LDA, WORK ) 543 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN 544 ISCALE = 1 545 SIGMA = RMIN / ANRM 546 ELSE IF( ANRM.GT.RMAX ) THEN 547 ISCALE = 1 548 SIGMA = RMAX / ANRM 549 END IF 550 IF( ISCALE.EQ.1 ) THEN 551 IF( LOWER ) THEN 552 DO 10 J = 1, N 553 CALL SSCAL( N-J+1, SIGMA, A( J, J ), 1 ) 554 10 CONTINUE 555 ELSE 556 DO 20 J = 1, N 557 CALL SSCAL( J, SIGMA, A( 1, J ), 1 ) 558 20 CONTINUE 559 END IF 560 IF( ABSTOL.GT.0 ) 561 $ ABSTLL = ABSTOL*SIGMA 562 IF( VALEIG ) THEN 563 VLL = VL*SIGMA 564 VUU = VU*SIGMA 565 END IF 566 END IF 567 568* Initialize indices into workspaces. Note: The IWORK indices are 569* used only if SSTERF or SSTEMR fail. 570 571* WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the 572* elementary reflectors used in SSYTRD. 573 INDTAU = 1 574* WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries. 575 INDD = INDTAU + N 576* WORK(INDE:INDE+N-1) stores the off-diagonal entries of the 577* tridiagonal matrix from SSYTRD. 578 INDE = INDD + N 579* WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over 580* -written by SSTEMR (the SSTERF path copies the diagonal to W). 581 INDDD = INDE + N 582* WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over 583* -written while computing the eigenvalues in SSTERF and SSTEMR. 584 INDEE = INDDD + N 585* INDHOUS is the starting offset Householder storage of stage 2 586 INDHOUS = INDEE + N 587* INDWK is the starting offset of the left-over workspace, and 588* LLWORK is the remaining workspace size. 589 INDWK = INDHOUS + LHTRD 590 LLWORK = LWORK - INDWK + 1 591 592 593* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and 594* stores the block indices of each of the M<=N eigenvalues. 595 INDIBL = 1 596* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and 597* stores the starting and finishing indices of each block. 598 INDISP = INDIBL + N 599* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors 600* that corresponding to eigenvectors that fail to converge in 601* SSTEIN. This information is discarded; if any fail, the driver 602* returns INFO > 0. 603 INDIFL = INDISP + N 604* INDIWO is the offset of the remaining integer workspace. 605 INDIWO = INDIFL + N 606 607* 608* Call SSYTRD_2STAGE to reduce symmetric matrix to tridiagonal form. 609* 610* 611 CALL SSYTRD_2STAGE( JOBZ, UPLO, N, A, LDA, WORK( INDD ), 612 $ WORK( INDE ), WORK( INDTAU ), WORK( INDHOUS ), 613 $ LHTRD, WORK( INDWK ), LLWORK, IINFO ) 614* 615* If all eigenvalues are desired 616* then call SSTERF or SSTEMR and SORMTR. 617* 618 TEST = .FALSE. 619 IF( INDEIG ) THEN 620 IF( IL.EQ.1 .AND. IU.EQ.N ) THEN 621 TEST = .TRUE. 622 END IF 623 END IF 624 IF( ( ALLEIG.OR.TEST ) .AND. ( IEEEOK.EQ.1 ) ) THEN 625 IF( .NOT.WANTZ ) THEN 626 CALL SCOPY( N, WORK( INDD ), 1, W, 1 ) 627 CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) 628 CALL SSTERF( N, W, WORK( INDEE ), INFO ) 629 ELSE 630 CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) 631 CALL SCOPY( N, WORK( INDD ), 1, WORK( INDDD ), 1 ) 632* 633 IF (ABSTOL .LE. TWO*N*EPS) THEN 634 TRYRAC = .TRUE. 635 ELSE 636 TRYRAC = .FALSE. 637 END IF 638 CALL SSTEMR( JOBZ, 'A', N, WORK( INDDD ), WORK( INDEE ), 639 $ VL, VU, IL, IU, M, W, Z, LDZ, N, ISUPPZ, 640 $ TRYRAC, WORK( INDWK ), LWORK, IWORK, LIWORK, 641 $ INFO ) 642* 643* 644* 645* Apply orthogonal matrix used in reduction to tridiagonal 646* form to eigenvectors returned by SSTEMR. 647* 648 IF( WANTZ .AND. INFO.EQ.0 ) THEN 649 INDWKN = INDE 650 LLWRKN = LWORK - INDWKN + 1 651 CALL SORMTR( 'L', UPLO, 'N', N, M, A, LDA, 652 $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ), 653 $ LLWRKN, IINFO ) 654 END IF 655 END IF 656* 657* 658 IF( INFO.EQ.0 ) THEN 659* Everything worked. Skip SSTEBZ/SSTEIN. IWORK(:) are 660* undefined. 661 M = N 662 GO TO 30 663 END IF 664 INFO = 0 665 END IF 666* 667* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. 668* Also call SSTEBZ and SSTEIN if SSTEMR fails. 669* 670 IF( WANTZ ) THEN 671 ORDER = 'B' 672 ELSE 673 ORDER = 'E' 674 END IF 675 676 CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, 677 $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W, 678 $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWK ), 679 $ IWORK( INDIWO ), INFO ) 680* 681 IF( WANTZ ) THEN 682 CALL SSTEIN( N, WORK( INDD ), WORK( INDE ), M, W, 683 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, 684 $ WORK( INDWK ), IWORK( INDIWO ), IWORK( INDIFL ), 685 $ INFO ) 686* 687* Apply orthogonal matrix used in reduction to tridiagonal 688* form to eigenvectors returned by SSTEIN. 689* 690 INDWKN = INDE 691 LLWRKN = LWORK - INDWKN + 1 692 CALL SORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z, 693 $ LDZ, WORK( INDWKN ), LLWRKN, IINFO ) 694 END IF 695* 696* If matrix was scaled, then rescale eigenvalues appropriately. 697* 698* Jump here if SSTEMR/SSTEIN succeeded. 699 30 CONTINUE 700 IF( ISCALE.EQ.1 ) THEN 701 IF( INFO.EQ.0 ) THEN 702 IMAX = M 703 ELSE 704 IMAX = INFO - 1 705 END IF 706 CALL SSCAL( IMAX, ONE / SIGMA, W, 1 ) 707 END IF 708* 709* If eigenvalues are not in order, then sort them, along with 710* eigenvectors. Note: We do not sort the IFAIL portion of IWORK. 711* It may not be initialized (if SSTEMR/SSTEIN succeeded), and we do 712* not return this detailed information to the user. 713* 714 IF( WANTZ ) THEN 715 DO 50 J = 1, M - 1 716 I = 0 717 TMP1 = W( J ) 718 DO 40 JJ = J + 1, M 719 IF( W( JJ ).LT.TMP1 ) THEN 720 I = JJ 721 TMP1 = W( JJ ) 722 END IF 723 40 CONTINUE 724* 725 IF( I.NE.0 ) THEN 726 W( I ) = W( J ) 727 W( J ) = TMP1 728 CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 729 END IF 730 50 CONTINUE 731 END IF 732* 733* Set WORK(1) to optimal workspace size. 734* 735 WORK( 1 ) = LWMIN 736 IWORK( 1 ) = LIWMIN 737* 738 RETURN 739* 740* End of SSYEVR_2STAGE 741* 742 END 743