1*> \brief <b> SSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER 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 SSPEVX + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sspevx.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sspevx.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sspevx.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE SSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, 22* ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, 23* INFO ) 24* 25* .. Scalar Arguments .. 26* CHARACTER JOBZ, RANGE, UPLO 27* INTEGER IL, INFO, IU, LDZ, M, N 28* REAL ABSTOL, VL, VU 29* .. 30* .. Array Arguments .. 31* INTEGER IFAIL( * ), IWORK( * ) 32* REAL AP( * ), W( * ), WORK( * ), Z( LDZ, * ) 33* .. 34* 35* 36*> \par Purpose: 37* ============= 38*> 39*> \verbatim 40*> 41*> SSPEVX computes selected eigenvalues and, optionally, eigenvectors 42*> of a real symmetric matrix A in packed storage. Eigenvalues/vectors 43*> can be selected by specifying either a range of values or a range of 44*> indices 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] AP 80*> \verbatim 81*> AP is REAL array, dimension (N*(N+1)/2) 82*> On entry, the upper or lower triangle of the symmetric matrix 83*> A, packed columnwise in a linear array. The j-th column of A 84*> is stored in the array AP as follows: 85*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 86*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. 87*> 88*> On exit, AP is overwritten by values generated during the 89*> reduction to tridiagonal form. If UPLO = 'U', the diagonal 90*> and first superdiagonal of the tridiagonal matrix T overwrite 91*> the corresponding elements of A, and if UPLO = 'L', the 92*> diagonal and first subdiagonal of T overwrite the 93*> corresponding elements of A. 94*> \endverbatim 95*> 96*> \param[in] VL 97*> \verbatim 98*> VL is REAL 99*> If RANGE='V', the lower bound of the interval to 100*> be searched for eigenvalues. VL < VU. 101*> Not referenced if RANGE = 'A' or 'I'. 102*> \endverbatim 103*> 104*> \param[in] VU 105*> \verbatim 106*> VU is REAL 107*> If RANGE='V', the upper bound of the interval to 108*> be searched for eigenvalues. VL < VU. 109*> Not referenced if RANGE = 'A' or 'I'. 110*> \endverbatim 111*> 112*> \param[in] IL 113*> \verbatim 114*> IL is INTEGER 115*> If RANGE='I', the index of the 116*> smallest eigenvalue to be returned. 117*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 118*> Not referenced if RANGE = 'A' or 'V'. 119*> \endverbatim 120*> 121*> \param[in] IU 122*> \verbatim 123*> IU is INTEGER 124*> If RANGE='I', the index of the 125*> largest eigenvalue to be returned. 126*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 127*> Not referenced if RANGE = 'A' or 'V'. 128*> \endverbatim 129*> 130*> \param[in] ABSTOL 131*> \verbatim 132*> ABSTOL is REAL 133*> The absolute error tolerance for the eigenvalues. 134*> An approximate eigenvalue is accepted as converged 135*> when it is determined to lie in an interval [a,b] 136*> of width less than or equal to 137*> 138*> ABSTOL + EPS * max( |a|,|b| ) , 139*> 140*> where EPS is the machine precision. If ABSTOL is less than 141*> or equal to zero, then EPS*|T| will be used in its place, 142*> where |T| is the 1-norm of the tridiagonal matrix obtained 143*> by reducing AP to tridiagonal form. 144*> 145*> Eigenvalues will be computed most accurately when ABSTOL is 146*> set to twice the underflow threshold 2*SLAMCH('S'), not zero. 147*> If this routine returns with INFO>0, indicating that some 148*> eigenvectors did not converge, try setting ABSTOL to 149*> 2*SLAMCH('S'). 150*> 151*> See "Computing Small Singular Values of Bidiagonal Matrices 152*> with Guaranteed High Relative Accuracy," by Demmel and 153*> Kahan, LAPACK Working Note #3. 154*> \endverbatim 155*> 156*> \param[out] M 157*> \verbatim 158*> M is INTEGER 159*> The total number of eigenvalues found. 0 <= M <= N. 160*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 161*> \endverbatim 162*> 163*> \param[out] W 164*> \verbatim 165*> W is REAL array, dimension (N) 166*> If INFO = 0, the selected eigenvalues in ascending order. 167*> \endverbatim 168*> 169*> \param[out] Z 170*> \verbatim 171*> Z is REAL array, dimension (LDZ, max(1,M)) 172*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z 173*> contain the orthonormal eigenvectors of the matrix A 174*> corresponding to the selected eigenvalues, with the i-th 175*> column of Z holding the eigenvector associated with W(i). 176*> If an eigenvector fails to converge, then that column of Z 177*> contains the latest approximation to the eigenvector, and the 178*> index of the eigenvector is returned in IFAIL. 179*> If JOBZ = 'N', then Z is not referenced. 180*> Note: the user must ensure that at least max(1,M) columns are 181*> supplied in the array Z; if RANGE = 'V', the exact value of M 182*> is not known in advance and an upper bound must be used. 183*> \endverbatim 184*> 185*> \param[in] LDZ 186*> \verbatim 187*> LDZ is INTEGER 188*> The leading dimension of the array Z. LDZ >= 1, and if 189*> JOBZ = 'V', LDZ >= max(1,N). 190*> \endverbatim 191*> 192*> \param[out] WORK 193*> \verbatim 194*> WORK is REAL array, dimension (8*N) 195*> \endverbatim 196*> 197*> \param[out] IWORK 198*> \verbatim 199*> IWORK is INTEGER array, dimension (5*N) 200*> \endverbatim 201*> 202*> \param[out] IFAIL 203*> \verbatim 204*> IFAIL is INTEGER array, dimension (N) 205*> If JOBZ = 'V', then if INFO = 0, the first M elements of 206*> IFAIL are zero. If INFO > 0, then IFAIL contains the 207*> indices of the eigenvectors that failed to converge. 208*> If JOBZ = 'N', then IFAIL is not referenced. 209*> \endverbatim 210*> 211*> \param[out] INFO 212*> \verbatim 213*> INFO is INTEGER 214*> = 0: successful exit 215*> < 0: if INFO = -i, the i-th argument had an illegal value 216*> > 0: if INFO = i, then i eigenvectors failed to converge. 217*> Their indices are stored in array IFAIL. 218*> \endverbatim 219* 220* Authors: 221* ======== 222* 223*> \author Univ. of Tennessee 224*> \author Univ. of California Berkeley 225*> \author Univ. of Colorado Denver 226*> \author NAG Ltd. 227* 228*> \ingroup realOTHEReigen 229* 230* ===================================================================== 231 SUBROUTINE SSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, 232 $ ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, 233 $ INFO ) 234* 235* -- LAPACK driver routine -- 236* -- LAPACK is a software package provided by Univ. of Tennessee, -- 237* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 238* 239* .. Scalar Arguments .. 240 CHARACTER JOBZ, RANGE, UPLO 241 INTEGER IL, INFO, IU, LDZ, M, N 242 REAL ABSTOL, VL, VU 243* .. 244* .. Array Arguments .. 245 INTEGER IFAIL( * ), IWORK( * ) 246 REAL AP( * ), W( * ), WORK( * ), Z( LDZ, * ) 247* .. 248* 249* ===================================================================== 250* 251* .. Parameters .. 252 REAL ZERO, ONE 253 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) 254* .. 255* .. Local Scalars .. 256 LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ 257 CHARACTER ORDER 258 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL, 259 $ INDISP, INDIWO, INDTAU, INDWRK, ISCALE, ITMP1, 260 $ J, JJ, NSPLIT 261 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, 262 $ SIGMA, SMLNUM, TMP1, VLL, VUU 263* .. 264* .. External Functions .. 265 LOGICAL LSAME 266 REAL SLAMCH, SLANSP 267 EXTERNAL LSAME, SLAMCH, SLANSP 268* .. 269* .. External Subroutines .. 270 EXTERNAL SCOPY, SOPGTR, SOPMTR, SSCAL, SSPTRD, SSTEBZ, 271 $ SSTEIN, SSTEQR, SSTERF, SSWAP, XERBLA 272* .. 273* .. Intrinsic Functions .. 274 INTRINSIC MAX, MIN, SQRT 275* .. 276* .. Executable Statements .. 277* 278* Test the input parameters. 279* 280 WANTZ = LSAME( JOBZ, 'V' ) 281 ALLEIG = LSAME( RANGE, 'A' ) 282 VALEIG = LSAME( RANGE, 'V' ) 283 INDEIG = LSAME( RANGE, 'I' ) 284* 285 INFO = 0 286 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 287 INFO = -1 288 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 289 INFO = -2 290 ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) ) 291 $ THEN 292 INFO = -3 293 ELSE IF( N.LT.0 ) THEN 294 INFO = -4 295 ELSE 296 IF( VALEIG ) THEN 297 IF( N.GT.0 .AND. VU.LE.VL ) 298 $ INFO = -7 299 ELSE IF( INDEIG ) THEN 300 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 301 INFO = -8 302 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 303 INFO = -9 304 END IF 305 END IF 306 END IF 307 IF( INFO.EQ.0 ) THEN 308 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) 309 $ INFO = -14 310 END IF 311* 312 IF( INFO.NE.0 ) THEN 313 CALL XERBLA( 'SSPEVX', -INFO ) 314 RETURN 315 END IF 316* 317* Quick return if possible 318* 319 M = 0 320 IF( N.EQ.0 ) 321 $ RETURN 322* 323 IF( N.EQ.1 ) THEN 324 IF( ALLEIG .OR. INDEIG ) THEN 325 M = 1 326 W( 1 ) = AP( 1 ) 327 ELSE 328 IF( VL.LT.AP( 1 ) .AND. VU.GE.AP( 1 ) ) THEN 329 M = 1 330 W( 1 ) = AP( 1 ) 331 END IF 332 END IF 333 IF( WANTZ ) 334 $ Z( 1, 1 ) = ONE 335 RETURN 336 END IF 337* 338* Get machine constants. 339* 340 SAFMIN = SLAMCH( 'Safe minimum' ) 341 EPS = SLAMCH( 'Precision' ) 342 SMLNUM = SAFMIN / EPS 343 BIGNUM = ONE / SMLNUM 344 RMIN = SQRT( SMLNUM ) 345 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 346* 347* Scale matrix to allowable range, if necessary. 348* 349 ISCALE = 0 350 ABSTLL = ABSTOL 351 IF ( VALEIG ) THEN 352 VLL = VL 353 VUU = VU 354 ELSE 355 VLL = ZERO 356 VUU = ZERO 357 ENDIF 358 ANRM = SLANSP( 'M', UPLO, N, AP, WORK ) 359 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN 360 ISCALE = 1 361 SIGMA = RMIN / ANRM 362 ELSE IF( ANRM.GT.RMAX ) THEN 363 ISCALE = 1 364 SIGMA = RMAX / ANRM 365 END IF 366 IF( ISCALE.EQ.1 ) THEN 367 CALL SSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 ) 368 IF( ABSTOL.GT.0 ) 369 $ ABSTLL = ABSTOL*SIGMA 370 IF( VALEIG ) THEN 371 VLL = VL*SIGMA 372 VUU = VU*SIGMA 373 END IF 374 END IF 375* 376* Call SSPTRD to reduce symmetric packed matrix to tridiagonal form. 377* 378 INDTAU = 1 379 INDE = INDTAU + N 380 INDD = INDE + N 381 INDWRK = INDD + N 382 CALL SSPTRD( UPLO, N, AP, WORK( INDD ), WORK( INDE ), 383 $ WORK( INDTAU ), IINFO ) 384* 385* If all eigenvalues are desired and ABSTOL is less than or equal 386* to zero, then call SSTERF or SOPGTR and SSTEQR. If this fails 387* for some eigenvalue, then try SSTEBZ. 388* 389 TEST = .FALSE. 390 IF (INDEIG) THEN 391 IF (IL.EQ.1 .AND. IU.EQ.N) THEN 392 TEST = .TRUE. 393 END IF 394 END IF 395 IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN 396 CALL SCOPY( N, WORK( INDD ), 1, W, 1 ) 397 INDEE = INDWRK + 2*N 398 IF( .NOT.WANTZ ) THEN 399 CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) 400 CALL SSTERF( N, W, WORK( INDEE ), INFO ) 401 ELSE 402 CALL SOPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ, 403 $ WORK( INDWRK ), IINFO ) 404 CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) 405 CALL SSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ, 406 $ WORK( INDWRK ), INFO ) 407 IF( INFO.EQ.0 ) THEN 408 DO 10 I = 1, N 409 IFAIL( I ) = 0 410 10 CONTINUE 411 END IF 412 END IF 413 IF( INFO.EQ.0 ) THEN 414 M = N 415 GO TO 20 416 END IF 417 INFO = 0 418 END IF 419* 420* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. 421* 422 IF( WANTZ ) THEN 423 ORDER = 'B' 424 ELSE 425 ORDER = 'E' 426 END IF 427 INDIBL = 1 428 INDISP = INDIBL + N 429 INDIWO = INDISP + N 430 CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, 431 $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W, 432 $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ), 433 $ IWORK( INDIWO ), INFO ) 434* 435 IF( WANTZ ) THEN 436 CALL SSTEIN( N, WORK( INDD ), WORK( INDE ), M, W, 437 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, 438 $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO ) 439* 440* Apply orthogonal matrix used in reduction to tridiagonal 441* form to eigenvectors returned by SSTEIN. 442* 443 CALL SOPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z, LDZ, 444 $ WORK( INDWRK ), IINFO ) 445 END IF 446* 447* If matrix was scaled, then rescale eigenvalues appropriately. 448* 449 20 CONTINUE 450 IF( ISCALE.EQ.1 ) THEN 451 IF( INFO.EQ.0 ) THEN 452 IMAX = M 453 ELSE 454 IMAX = INFO - 1 455 END IF 456 CALL SSCAL( IMAX, ONE / SIGMA, W, 1 ) 457 END IF 458* 459* If eigenvalues are not in order, then sort them, along with 460* eigenvectors. 461* 462 IF( WANTZ ) THEN 463 DO 40 J = 1, M - 1 464 I = 0 465 TMP1 = W( J ) 466 DO 30 JJ = J + 1, M 467 IF( W( JJ ).LT.TMP1 ) THEN 468 I = JJ 469 TMP1 = W( JJ ) 470 END IF 471 30 CONTINUE 472* 473 IF( I.NE.0 ) THEN 474 ITMP1 = IWORK( INDIBL+I-1 ) 475 W( I ) = W( J ) 476 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) 477 W( J ) = TMP1 478 IWORK( INDIBL+J-1 ) = ITMP1 479 CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 480 IF( INFO.NE.0 ) THEN 481 ITMP1 = IFAIL( I ) 482 IFAIL( I ) = IFAIL( J ) 483 IFAIL( J ) = ITMP1 484 END IF 485 END IF 486 40 CONTINUE 487 END IF 488* 489 RETURN 490* 491* End of SSPEVX 492* 493 END 494