1*> \brief <b> DSYEVX 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 DSYEVX + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyevx.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyevx.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyevx.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, 22* ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, 23* IFAIL, INFO ) 24* 25* .. Scalar Arguments .. 26* CHARACTER JOBZ, RANGE, UPLO 27* INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N 28* DOUBLE PRECISION ABSTOL, VL, VU 29* .. 30* .. Array Arguments .. 31* INTEGER IFAIL( * ), IWORK( * ) 32* DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * ) 33* .. 34* 35* 36*> \par Purpose: 37* ============= 38*> 39*> \verbatim 40*> 41*> DSYEVX 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 indices 44*> for the desired eigenvalues. 45*> \endverbatim 46* 47* Arguments: 48* ========== 49* 50*> \param[in] JOBZ 51*> \verbatim 52*> JOBZ is CHARACTER*1 53*> = 'N': Compute eigenvalues only; 54*> = 'V': Compute eigenvalues and eigenvectors. 55*> \endverbatim 56*> 57*> \param[in] RANGE 58*> \verbatim 59*> RANGE is CHARACTER*1 60*> = 'A': all eigenvalues will be found. 61*> = 'V': all eigenvalues in the half-open interval (VL,VU] 62*> will be found. 63*> = 'I': the IL-th through IU-th eigenvalues will be found. 64*> \endverbatim 65*> 66*> \param[in] UPLO 67*> \verbatim 68*> UPLO is CHARACTER*1 69*> = 'U': Upper triangle of A is stored; 70*> = 'L': Lower triangle of A is stored. 71*> \endverbatim 72*> 73*> \param[in] N 74*> \verbatim 75*> N is INTEGER 76*> The order of the matrix A. N >= 0. 77*> \endverbatim 78*> 79*> \param[in,out] A 80*> \verbatim 81*> A is DOUBLE PRECISION array, dimension (LDA, N) 82*> On entry, the symmetric matrix A. If UPLO = 'U', the 83*> leading N-by-N upper triangular part of A contains the 84*> upper triangular part of the matrix A. If UPLO = 'L', 85*> the leading N-by-N lower triangular part of A contains 86*> the lower triangular part of the matrix A. 87*> On exit, the lower triangle (if UPLO='L') or the upper 88*> triangle (if UPLO='U') of A, including the diagonal, is 89*> destroyed. 90*> \endverbatim 91*> 92*> \param[in] LDA 93*> \verbatim 94*> LDA is INTEGER 95*> The leading dimension of the array A. LDA >= max(1,N). 96*> \endverbatim 97*> 98*> \param[in] VL 99*> \verbatim 100*> VL is DOUBLE PRECISION 101*> If RANGE='V', the lower bound of the interval to 102*> be searched for eigenvalues. VL < VU. 103*> Not referenced if RANGE = 'A' or 'I'. 104*> \endverbatim 105*> 106*> \param[in] VU 107*> \verbatim 108*> VU is DOUBLE PRECISION 109*> If RANGE='V', the upper bound of the interval to 110*> be searched for eigenvalues. VL < VU. 111*> Not referenced if RANGE = 'A' or 'I'. 112*> \endverbatim 113*> 114*> \param[in] IL 115*> \verbatim 116*> IL is INTEGER 117*> If RANGE='I', the index of the 118*> smallest eigenvalue to be returned. 119*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 120*> Not referenced if RANGE = 'A' or 'V'. 121*> \endverbatim 122*> 123*> \param[in] IU 124*> \verbatim 125*> IU is INTEGER 126*> If RANGE='I', the index of the 127*> largest eigenvalue to be returned. 128*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 129*> Not referenced if RANGE = 'A' or 'V'. 130*> \endverbatim 131*> 132*> \param[in] ABSTOL 133*> \verbatim 134*> ABSTOL is DOUBLE PRECISION 135*> The absolute error tolerance for the eigenvalues. 136*> An approximate eigenvalue is accepted as converged 137*> when it is determined to lie in an interval [a,b] 138*> of width less than or equal to 139*> 140*> ABSTOL + EPS * max( |a|,|b| ) , 141*> 142*> where EPS is the machine precision. If ABSTOL is less than 143*> or equal to zero, then EPS*|T| will be used in its place, 144*> where |T| is the 1-norm of the tridiagonal matrix obtained 145*> by reducing A to tridiagonal form. 146*> 147*> Eigenvalues will be computed most accurately when ABSTOL is 148*> set to twice the underflow threshold 2*DLAMCH('S'), not zero. 149*> If this routine returns with INFO>0, indicating that some 150*> eigenvectors did not converge, try setting ABSTOL to 151*> 2*DLAMCH('S'). 152*> 153*> See "Computing Small Singular Values of Bidiagonal Matrices 154*> with Guaranteed High Relative Accuracy," by Demmel and 155*> Kahan, LAPACK Working Note #3. 156*> \endverbatim 157*> 158*> \param[out] M 159*> \verbatim 160*> M is INTEGER 161*> The total number of eigenvalues found. 0 <= M <= N. 162*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 163*> \endverbatim 164*> 165*> \param[out] W 166*> \verbatim 167*> W is DOUBLE PRECISION array, dimension (N) 168*> On normal exit, the first M elements contain the selected 169*> eigenvalues in ascending order. 170*> \endverbatim 171*> 172*> \param[out] Z 173*> \verbatim 174*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M)) 175*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z 176*> contain the orthonormal eigenvectors of the matrix A 177*> corresponding to the selected eigenvalues, with the i-th 178*> column of Z holding the eigenvector associated with W(i). 179*> If an eigenvector fails to converge, then that column of Z 180*> contains the latest approximation to the eigenvector, and the 181*> index of the eigenvector is returned in IFAIL. 182*> If JOBZ = 'N', then Z is not referenced. 183*> Note: the user must ensure that at least max(1,M) columns are 184*> supplied in the array Z; if RANGE = 'V', the exact value of M 185*> is not known in advance and an upper bound must be used. 186*> \endverbatim 187*> 188*> \param[in] LDZ 189*> \verbatim 190*> LDZ is INTEGER 191*> The leading dimension of the array Z. LDZ >= 1, and if 192*> JOBZ = 'V', LDZ >= max(1,N). 193*> \endverbatim 194*> 195*> \param[out] WORK 196*> \verbatim 197*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) 198*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 199*> \endverbatim 200*> 201*> \param[in] LWORK 202*> \verbatim 203*> LWORK is INTEGER 204*> The length of the array WORK. LWORK >= 1, when N <= 1; 205*> otherwise 8*N. 206*> For optimal efficiency, LWORK >= (NB+3)*N, 207*> where NB is the max of the blocksize for DSYTRD and DORMTR 208*> returned by ILAENV. 209*> 210*> If LWORK = -1, then a workspace query is assumed; the routine 211*> only calculates the optimal size of the WORK array, returns 212*> this value as the first entry of the WORK array, and no error 213*> message related to LWORK is issued by XERBLA. 214*> \endverbatim 215*> 216*> \param[out] IWORK 217*> \verbatim 218*> IWORK is INTEGER array, dimension (5*N) 219*> \endverbatim 220*> 221*> \param[out] IFAIL 222*> \verbatim 223*> IFAIL is INTEGER array, dimension (N) 224*> If JOBZ = 'V', then if INFO = 0, the first M elements of 225*> IFAIL are zero. If INFO > 0, then IFAIL contains the 226*> indices of the eigenvectors that failed to converge. 227*> If JOBZ = 'N', then IFAIL is not referenced. 228*> \endverbatim 229*> 230*> \param[out] INFO 231*> \verbatim 232*> INFO is INTEGER 233*> = 0: successful exit 234*> < 0: if INFO = -i, the i-th argument had an illegal value 235*> > 0: if INFO = i, then i eigenvectors failed to converge. 236*> Their indices are stored in array IFAIL. 237*> \endverbatim 238* 239* Authors: 240* ======== 241* 242*> \author Univ. of Tennessee 243*> \author Univ. of California Berkeley 244*> \author Univ. of Colorado Denver 245*> \author NAG Ltd. 246* 247*> \ingroup doubleSYeigen 248* 249* ===================================================================== 250 SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, 251 $ ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, 252 $ IFAIL, INFO ) 253* 254* -- LAPACK driver routine -- 255* -- LAPACK is a software package provided by Univ. of Tennessee, -- 256* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 257* 258* .. Scalar Arguments .. 259 CHARACTER JOBZ, RANGE, UPLO 260 INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N 261 DOUBLE PRECISION ABSTOL, VL, VU 262* .. 263* .. Array Arguments .. 264 INTEGER IFAIL( * ), IWORK( * ) 265 DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * ) 266* .. 267* 268* ===================================================================== 269* 270* .. Parameters .. 271 DOUBLE PRECISION ZERO, ONE 272 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 273* .. 274* .. Local Scalars .. 275 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG, 276 $ WANTZ 277 CHARACTER ORDER 278 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL, 279 $ INDISP, INDIWO, INDTAU, INDWKN, INDWRK, ISCALE, 280 $ ITMP1, J, JJ, LLWORK, LLWRKN, LWKMIN, 281 $ LWKOPT, NB, NSPLIT 282 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, 283 $ SIGMA, SMLNUM, TMP1, VLL, VUU 284* .. 285* .. External Functions .. 286 LOGICAL LSAME 287 INTEGER ILAENV 288 DOUBLE PRECISION DLAMCH, DLANSY 289 EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY 290* .. 291* .. External Subroutines .. 292 EXTERNAL DCOPY, DLACPY, DORGTR, DORMTR, DSCAL, DSTEBZ, 293 $ DSTEIN, DSTEQR, DSTERF, DSWAP, DSYTRD, XERBLA 294* .. 295* .. Intrinsic Functions .. 296 INTRINSIC MAX, MIN, SQRT 297* .. 298* .. Executable Statements .. 299* 300* Test the input parameters. 301* 302 LOWER = LSAME( UPLO, 'L' ) 303 WANTZ = LSAME( JOBZ, 'V' ) 304 ALLEIG = LSAME( RANGE, 'A' ) 305 VALEIG = LSAME( RANGE, 'V' ) 306 INDEIG = LSAME( RANGE, 'I' ) 307 LQUERY = ( LWORK.EQ.-1 ) 308* 309 INFO = 0 310 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 311 INFO = -1 312 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 313 INFO = -2 314 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN 315 INFO = -3 316 ELSE IF( N.LT.0 ) THEN 317 INFO = -4 318 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 319 INFO = -6 320 ELSE 321 IF( VALEIG ) THEN 322 IF( N.GT.0 .AND. VU.LE.VL ) 323 $ INFO = -8 324 ELSE IF( INDEIG ) THEN 325 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 326 INFO = -9 327 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 328 INFO = -10 329 END IF 330 END IF 331 END IF 332 IF( INFO.EQ.0 ) THEN 333 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN 334 INFO = -15 335 END IF 336 END IF 337* 338 IF( INFO.EQ.0 ) THEN 339 IF( N.LE.1 ) THEN 340 LWKMIN = 1 341 WORK( 1 ) = LWKMIN 342 ELSE 343 LWKMIN = 8*N 344 NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 ) 345 NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) ) 346 LWKOPT = MAX( LWKMIN, ( NB + 3 )*N ) 347 WORK( 1 ) = LWKOPT 348 END IF 349* 350 IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) 351 $ INFO = -17 352 END IF 353* 354 IF( INFO.NE.0 ) THEN 355 CALL XERBLA( 'DSYEVX', -INFO ) 356 RETURN 357 ELSE IF( LQUERY ) THEN 358 RETURN 359 END IF 360* 361* Quick return if possible 362* 363 M = 0 364 IF( N.EQ.0 ) THEN 365 RETURN 366 END IF 367* 368 IF( N.EQ.1 ) THEN 369 IF( ALLEIG .OR. INDEIG ) THEN 370 M = 1 371 W( 1 ) = A( 1, 1 ) 372 ELSE 373 IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN 374 M = 1 375 W( 1 ) = A( 1, 1 ) 376 END IF 377 END IF 378 IF( WANTZ ) 379 $ Z( 1, 1 ) = ONE 380 RETURN 381 END IF 382* 383* Get machine constants. 384* 385 SAFMIN = DLAMCH( 'Safe minimum' ) 386 EPS = DLAMCH( 'Precision' ) 387 SMLNUM = SAFMIN / EPS 388 BIGNUM = ONE / SMLNUM 389 RMIN = SQRT( SMLNUM ) 390 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 391* 392* Scale matrix to allowable range, if necessary. 393* 394 ISCALE = 0 395 ABSTLL = ABSTOL 396 IF( VALEIG ) THEN 397 VLL = VL 398 VUU = VU 399 END IF 400 ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK ) 401 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN 402 ISCALE = 1 403 SIGMA = RMIN / ANRM 404 ELSE IF( ANRM.GT.RMAX ) THEN 405 ISCALE = 1 406 SIGMA = RMAX / ANRM 407 END IF 408 IF( ISCALE.EQ.1 ) THEN 409 IF( LOWER ) THEN 410 DO 10 J = 1, N 411 CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 ) 412 10 CONTINUE 413 ELSE 414 DO 20 J = 1, N 415 CALL DSCAL( J, SIGMA, A( 1, J ), 1 ) 416 20 CONTINUE 417 END IF 418 IF( ABSTOL.GT.0 ) 419 $ ABSTLL = ABSTOL*SIGMA 420 IF( VALEIG ) THEN 421 VLL = VL*SIGMA 422 VUU = VU*SIGMA 423 END IF 424 END IF 425* 426* Call DSYTRD to reduce symmetric matrix to tridiagonal form. 427* 428 INDTAU = 1 429 INDE = INDTAU + N 430 INDD = INDE + N 431 INDWRK = INDD + N 432 LLWORK = LWORK - INDWRK + 1 433 CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ), 434 $ WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO ) 435* 436* If all eigenvalues are desired and ABSTOL is less than or equal to 437* zero, then call DSTERF or DORGTR and SSTEQR. If this fails for 438* some eigenvalue, then try DSTEBZ. 439* 440 TEST = .FALSE. 441 IF( INDEIG ) THEN 442 IF( IL.EQ.1 .AND. IU.EQ.N ) THEN 443 TEST = .TRUE. 444 END IF 445 END IF 446 IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN 447 CALL DCOPY( N, WORK( INDD ), 1, W, 1 ) 448 INDEE = INDWRK + 2*N 449 IF( .NOT.WANTZ ) THEN 450 CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) 451 CALL DSTERF( N, W, WORK( INDEE ), INFO ) 452 ELSE 453 CALL DLACPY( 'A', N, N, A, LDA, Z, LDZ ) 454 CALL DORGTR( UPLO, N, Z, LDZ, WORK( INDTAU ), 455 $ WORK( INDWRK ), LLWORK, IINFO ) 456 CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) 457 CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ, 458 $ WORK( INDWRK ), INFO ) 459 IF( INFO.EQ.0 ) THEN 460 DO 30 I = 1, N 461 IFAIL( I ) = 0 462 30 CONTINUE 463 END IF 464 END IF 465 IF( INFO.EQ.0 ) THEN 466 M = N 467 GO TO 40 468 END IF 469 INFO = 0 470 END IF 471* 472* Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN. 473* 474 IF( WANTZ ) THEN 475 ORDER = 'B' 476 ELSE 477 ORDER = 'E' 478 END IF 479 INDIBL = 1 480 INDISP = INDIBL + N 481 INDIWO = INDISP + N 482 CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, 483 $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W, 484 $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ), 485 $ IWORK( INDIWO ), INFO ) 486* 487 IF( WANTZ ) THEN 488 CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W, 489 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, 490 $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO ) 491* 492* Apply orthogonal matrix used in reduction to tridiagonal 493* form to eigenvectors returned by DSTEIN. 494* 495 INDWKN = INDE 496 LLWRKN = LWORK - INDWKN + 1 497 CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z, 498 $ LDZ, WORK( INDWKN ), LLWRKN, IINFO ) 499 END IF 500* 501* If matrix was scaled, then rescale eigenvalues appropriately. 502* 503 40 CONTINUE 504 IF( ISCALE.EQ.1 ) THEN 505 IF( INFO.EQ.0 ) THEN 506 IMAX = M 507 ELSE 508 IMAX = INFO - 1 509 END IF 510 CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) 511 END IF 512* 513* If eigenvalues are not in order, then sort them, along with 514* eigenvectors. 515* 516 IF( WANTZ ) THEN 517 DO 60 J = 1, M - 1 518 I = 0 519 TMP1 = W( J ) 520 DO 50 JJ = J + 1, M 521 IF( W( JJ ).LT.TMP1 ) THEN 522 I = JJ 523 TMP1 = W( JJ ) 524 END IF 525 50 CONTINUE 526* 527 IF( I.NE.0 ) THEN 528 ITMP1 = IWORK( INDIBL+I-1 ) 529 W( I ) = W( J ) 530 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) 531 W( J ) = TMP1 532 IWORK( INDIBL+J-1 ) = ITMP1 533 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 534 IF( INFO.NE.0 ) THEN 535 ITMP1 = IFAIL( I ) 536 IFAIL( I ) = IFAIL( J ) 537 IFAIL( J ) = ITMP1 538 END IF 539 END IF 540 60 CONTINUE 541 END IF 542* 543* Set WORK(1) to optimal workspace size. 544* 545 WORK( 1 ) = LWKOPT 546* 547 RETURN 548* 549* End of DSYEVX 550* 551 END 552