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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief <b> DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b> 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download DSBEVX + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbevx.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbevx.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbevx.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, 00022 * VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, 00023 * IFAIL, INFO ) 00024 * 00025 * .. Scalar Arguments .. 00026 * CHARACTER JOBZ, RANGE, UPLO 00027 * INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N 00028 * DOUBLE PRECISION ABSTOL, VL, VU 00029 * .. 00030 * .. Array Arguments .. 00031 * INTEGER IFAIL( * ), IWORK( * ) 00032 * DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ), 00033 * $ Z( LDZ, * ) 00034 * .. 00035 * 00036 * 00037 *> \par Purpose: 00038 * ============= 00039 *> 00040 *> \verbatim 00041 *> 00042 *> DSBEVX computes selected eigenvalues and, optionally, eigenvectors 00043 *> of a real symmetric band matrix A. Eigenvalues and eigenvectors can 00044 *> be selected by specifying either a range of values or a range of 00045 *> indices for the desired eigenvalues. 00046 *> \endverbatim 00047 * 00048 * Arguments: 00049 * ========== 00050 * 00051 *> \param[in] JOBZ 00052 *> \verbatim 00053 *> JOBZ is CHARACTER*1 00054 *> = 'N': Compute eigenvalues only; 00055 *> = 'V': Compute eigenvalues and eigenvectors. 00056 *> \endverbatim 00057 *> 00058 *> \param[in] RANGE 00059 *> \verbatim 00060 *> RANGE is CHARACTER*1 00061 *> = 'A': all eigenvalues will be found; 00062 *> = 'V': all eigenvalues in the half-open interval (VL,VU] 00063 *> will be found; 00064 *> = 'I': the IL-th through IU-th eigenvalues will be found. 00065 *> \endverbatim 00066 *> 00067 *> \param[in] UPLO 00068 *> \verbatim 00069 *> UPLO is CHARACTER*1 00070 *> = 'U': Upper triangle of A is stored; 00071 *> = 'L': Lower triangle of A is stored. 00072 *> \endverbatim 00073 *> 00074 *> \param[in] N 00075 *> \verbatim 00076 *> N is INTEGER 00077 *> The order of the matrix A. N >= 0. 00078 *> \endverbatim 00079 *> 00080 *> \param[in] KD 00081 *> \verbatim 00082 *> KD is INTEGER 00083 *> The number of superdiagonals of the matrix A if UPLO = 'U', 00084 *> or the number of subdiagonals if UPLO = 'L'. KD >= 0. 00085 *> \endverbatim 00086 *> 00087 *> \param[in,out] AB 00088 *> \verbatim 00089 *> AB is DOUBLE PRECISION array, dimension (LDAB, N) 00090 *> On entry, the upper or lower triangle of the symmetric band 00091 *> matrix A, stored in the first KD+1 rows of the array. The 00092 *> j-th column of A is stored in the j-th column of the array AB 00093 *> as follows: 00094 *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 00095 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 00096 *> 00097 *> On exit, AB is overwritten by values generated during the 00098 *> reduction to tridiagonal form. If UPLO = 'U', the first 00099 *> superdiagonal and the diagonal of the tridiagonal matrix T 00100 *> are returned in rows KD and KD+1 of AB, and if UPLO = 'L', 00101 *> the diagonal and first subdiagonal of T are returned in the 00102 *> first two rows of AB. 00103 *> \endverbatim 00104 *> 00105 *> \param[in] LDAB 00106 *> \verbatim 00107 *> LDAB is INTEGER 00108 *> The leading dimension of the array AB. LDAB >= KD + 1. 00109 *> \endverbatim 00110 *> 00111 *> \param[out] Q 00112 *> \verbatim 00113 *> Q is DOUBLE PRECISION array, dimension (LDQ, N) 00114 *> If JOBZ = 'V', the N-by-N orthogonal matrix used in the 00115 *> reduction to tridiagonal form. 00116 *> If JOBZ = 'N', the array Q is not referenced. 00117 *> \endverbatim 00118 *> 00119 *> \param[in] LDQ 00120 *> \verbatim 00121 *> LDQ is INTEGER 00122 *> The leading dimension of the array Q. If JOBZ = 'V', then 00123 *> LDQ >= max(1,N). 00124 *> \endverbatim 00125 *> 00126 *> \param[in] VL 00127 *> \verbatim 00128 *> VL is DOUBLE PRECISION 00129 *> \endverbatim 00130 *> 00131 *> \param[in] VU 00132 *> \verbatim 00133 *> VU is DOUBLE PRECISION 00134 *> If RANGE='V', the lower and upper bounds of the interval to 00135 *> be searched for eigenvalues. VL < VU. 00136 *> Not referenced if RANGE = 'A' or 'I'. 00137 *> \endverbatim 00138 *> 00139 *> \param[in] IL 00140 *> \verbatim 00141 *> IL is INTEGER 00142 *> \endverbatim 00143 *> 00144 *> \param[in] IU 00145 *> \verbatim 00146 *> IU is INTEGER 00147 *> If RANGE='I', the indices (in ascending order) of the 00148 *> smallest and largest eigenvalues to be returned. 00149 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. 00150 *> Not referenced if RANGE = 'A' or 'V'. 00151 *> \endverbatim 00152 *> 00153 *> \param[in] ABSTOL 00154 *> \verbatim 00155 *> ABSTOL is DOUBLE PRECISION 00156 *> The absolute error tolerance for the eigenvalues. 00157 *> An approximate eigenvalue is accepted as converged 00158 *> when it is determined to lie in an interval [a,b] 00159 *> of width less than or equal to 00160 *> 00161 *> ABSTOL + EPS * max( |a|,|b| ) , 00162 *> 00163 *> where EPS is the machine precision. If ABSTOL is less than 00164 *> or equal to zero, then EPS*|T| will be used in its place, 00165 *> where |T| is the 1-norm of the tridiagonal matrix obtained 00166 *> by reducing AB to tridiagonal form. 00167 *> 00168 *> Eigenvalues will be computed most accurately when ABSTOL is 00169 *> set to twice the underflow threshold 2*DLAMCH('S'), not zero. 00170 *> If this routine returns with INFO>0, indicating that some 00171 *> eigenvectors did not converge, try setting ABSTOL to 00172 *> 2*DLAMCH('S'). 00173 *> 00174 *> See "Computing Small Singular Values of Bidiagonal Matrices 00175 *> with Guaranteed High Relative Accuracy," by Demmel and 00176 *> Kahan, LAPACK Working Note #3. 00177 *> \endverbatim 00178 *> 00179 *> \param[out] M 00180 *> \verbatim 00181 *> M is INTEGER 00182 *> The total number of eigenvalues found. 0 <= M <= N. 00183 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 00184 *> \endverbatim 00185 *> 00186 *> \param[out] W 00187 *> \verbatim 00188 *> W is DOUBLE PRECISION array, dimension (N) 00189 *> The first M elements contain the selected eigenvalues in 00190 *> ascending order. 00191 *> \endverbatim 00192 *> 00193 *> \param[out] Z 00194 *> \verbatim 00195 *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M)) 00196 *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z 00197 *> contain the orthonormal eigenvectors of the matrix A 00198 *> corresponding to the selected eigenvalues, with the i-th 00199 *> column of Z holding the eigenvector associated with W(i). 00200 *> If an eigenvector fails to converge, then that column of Z 00201 *> contains the latest approximation to the eigenvector, and the 00202 *> index of the eigenvector is returned in IFAIL. 00203 *> If JOBZ = 'N', then Z is not referenced. 00204 *> Note: the user must ensure that at least max(1,M) columns are 00205 *> supplied in the array Z; if RANGE = 'V', the exact value of M 00206 *> is not known in advance and an upper bound must be used. 00207 *> \endverbatim 00208 *> 00209 *> \param[in] LDZ 00210 *> \verbatim 00211 *> LDZ is INTEGER 00212 *> The leading dimension of the array Z. LDZ >= 1, and if 00213 *> JOBZ = 'V', LDZ >= max(1,N). 00214 *> \endverbatim 00215 *> 00216 *> \param[out] WORK 00217 *> \verbatim 00218 *> WORK is DOUBLE PRECISION array, dimension (7*N) 00219 *> \endverbatim 00220 *> 00221 *> \param[out] IWORK 00222 *> \verbatim 00223 *> IWORK is INTEGER array, dimension (5*N) 00224 *> \endverbatim 00225 *> 00226 *> \param[out] IFAIL 00227 *> \verbatim 00228 *> IFAIL is INTEGER array, dimension (N) 00229 *> If JOBZ = 'V', then if INFO = 0, the first M elements of 00230 *> IFAIL are zero. If INFO > 0, then IFAIL contains the 00231 *> indices of the eigenvectors that failed to converge. 00232 *> If JOBZ = 'N', then IFAIL is not referenced. 00233 *> \endverbatim 00234 *> 00235 *> \param[out] INFO 00236 *> \verbatim 00237 *> INFO is INTEGER 00238 *> = 0: successful exit. 00239 *> < 0: if INFO = -i, the i-th argument had an illegal value. 00240 *> > 0: if INFO = i, then i eigenvectors failed to converge. 00241 *> Their indices are stored in array IFAIL. 00242 *> \endverbatim 00243 * 00244 * Authors: 00245 * ======== 00246 * 00247 *> \author Univ. of Tennessee 00248 *> \author Univ. of California Berkeley 00249 *> \author Univ. of Colorado Denver 00250 *> \author NAG Ltd. 00251 * 00252 *> \date November 2011 00253 * 00254 *> \ingroup doubleOTHEReigen 00255 * 00256 * ===================================================================== 00257 SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, 00258 $ VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, 00259 $ IFAIL, INFO ) 00260 * 00261 * -- LAPACK driver routine (version 3.4.0) -- 00262 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00263 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00264 * November 2011 00265 * 00266 * .. Scalar Arguments .. 00267 CHARACTER JOBZ, RANGE, UPLO 00268 INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N 00269 DOUBLE PRECISION ABSTOL, VL, VU 00270 * .. 00271 * .. Array Arguments .. 00272 INTEGER IFAIL( * ), IWORK( * ) 00273 DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ), 00274 $ Z( LDZ, * ) 00275 * .. 00276 * 00277 * ===================================================================== 00278 * 00279 * .. Parameters .. 00280 DOUBLE PRECISION ZERO, ONE 00281 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) 00282 * .. 00283 * .. Local Scalars .. 00284 LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ 00285 CHARACTER ORDER 00286 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL, 00287 $ INDISP, INDIWO, INDWRK, ISCALE, ITMP1, J, JJ, 00288 $ NSPLIT 00289 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, 00290 $ SIGMA, SMLNUM, TMP1, VLL, VUU 00291 * .. 00292 * .. External Functions .. 00293 LOGICAL LSAME 00294 DOUBLE PRECISION DLAMCH, DLANSB 00295 EXTERNAL LSAME, DLAMCH, DLANSB 00296 * .. 00297 * .. External Subroutines .. 00298 EXTERNAL DCOPY, DGEMV, DLACPY, DLASCL, DSBTRD, DSCAL, 00299 $ DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA 00300 * .. 00301 * .. Intrinsic Functions .. 00302 INTRINSIC MAX, MIN, SQRT 00303 * .. 00304 * .. Executable Statements .. 00305 * 00306 * Test the input parameters. 00307 * 00308 WANTZ = LSAME( JOBZ, 'V' ) 00309 ALLEIG = LSAME( RANGE, 'A' ) 00310 VALEIG = LSAME( RANGE, 'V' ) 00311 INDEIG = LSAME( RANGE, 'I' ) 00312 LOWER = LSAME( UPLO, 'L' ) 00313 * 00314 INFO = 0 00315 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN 00316 INFO = -1 00317 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN 00318 INFO = -2 00319 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN 00320 INFO = -3 00321 ELSE IF( N.LT.0 ) THEN 00322 INFO = -4 00323 ELSE IF( KD.LT.0 ) THEN 00324 INFO = -5 00325 ELSE IF( LDAB.LT.KD+1 ) THEN 00326 INFO = -7 00327 ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN 00328 INFO = -9 00329 ELSE 00330 IF( VALEIG ) THEN 00331 IF( N.GT.0 .AND. VU.LE.VL ) 00332 $ INFO = -11 00333 ELSE IF( INDEIG ) THEN 00334 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN 00335 INFO = -12 00336 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN 00337 INFO = -13 00338 END IF 00339 END IF 00340 END IF 00341 IF( INFO.EQ.0 ) THEN 00342 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) 00343 $ INFO = -18 00344 END IF 00345 * 00346 IF( INFO.NE.0 ) THEN 00347 CALL XERBLA( 'DSBEVX', -INFO ) 00348 RETURN 00349 END IF 00350 * 00351 * Quick return if possible 00352 * 00353 M = 0 00354 IF( N.EQ.0 ) 00355 $ RETURN 00356 * 00357 IF( N.EQ.1 ) THEN 00358 M = 1 00359 IF( LOWER ) THEN 00360 TMP1 = AB( 1, 1 ) 00361 ELSE 00362 TMP1 = AB( KD+1, 1 ) 00363 END IF 00364 IF( VALEIG ) THEN 00365 IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) ) 00366 $ M = 0 00367 END IF 00368 IF( M.EQ.1 ) THEN 00369 W( 1 ) = TMP1 00370 IF( WANTZ ) 00371 $ Z( 1, 1 ) = ONE 00372 END IF 00373 RETURN 00374 END IF 00375 * 00376 * Get machine constants. 00377 * 00378 SAFMIN = DLAMCH( 'Safe minimum' ) 00379 EPS = DLAMCH( 'Precision' ) 00380 SMLNUM = SAFMIN / EPS 00381 BIGNUM = ONE / SMLNUM 00382 RMIN = SQRT( SMLNUM ) 00383 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) 00384 * 00385 * Scale matrix to allowable range, if necessary. 00386 * 00387 ISCALE = 0 00388 ABSTLL = ABSTOL 00389 IF( VALEIG ) THEN 00390 VLL = VL 00391 VUU = VU 00392 ELSE 00393 VLL = ZERO 00394 VUU = ZERO 00395 END IF 00396 ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK ) 00397 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN 00398 ISCALE = 1 00399 SIGMA = RMIN / ANRM 00400 ELSE IF( ANRM.GT.RMAX ) THEN 00401 ISCALE = 1 00402 SIGMA = RMAX / ANRM 00403 END IF 00404 IF( ISCALE.EQ.1 ) THEN 00405 IF( LOWER ) THEN 00406 CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) 00407 ELSE 00408 CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) 00409 END IF 00410 IF( ABSTOL.GT.0 ) 00411 $ ABSTLL = ABSTOL*SIGMA 00412 IF( VALEIG ) THEN 00413 VLL = VL*SIGMA 00414 VUU = VU*SIGMA 00415 END IF 00416 END IF 00417 * 00418 * Call DSBTRD to reduce symmetric band matrix to tridiagonal form. 00419 * 00420 INDD = 1 00421 INDE = INDD + N 00422 INDWRK = INDE + N 00423 CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, WORK( INDD ), 00424 $ WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO ) 00425 * 00426 * If all eigenvalues are desired and ABSTOL is less than or equal 00427 * to zero, then call DSTERF or SSTEQR. If this fails for some 00428 * eigenvalue, then try DSTEBZ. 00429 * 00430 TEST = .FALSE. 00431 IF (INDEIG) THEN 00432 IF (IL.EQ.1 .AND. IU.EQ.N) THEN 00433 TEST = .TRUE. 00434 END IF 00435 END IF 00436 IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN 00437 CALL DCOPY( N, WORK( INDD ), 1, W, 1 ) 00438 INDEE = INDWRK + 2*N 00439 IF( .NOT.WANTZ ) THEN 00440 CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) 00441 CALL DSTERF( N, W, WORK( INDEE ), INFO ) 00442 ELSE 00443 CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ ) 00444 CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) 00445 CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ, 00446 $ WORK( INDWRK ), INFO ) 00447 IF( INFO.EQ.0 ) THEN 00448 DO 10 I = 1, N 00449 IFAIL( I ) = 0 00450 10 CONTINUE 00451 END IF 00452 END IF 00453 IF( INFO.EQ.0 ) THEN 00454 M = N 00455 GO TO 30 00456 END IF 00457 INFO = 0 00458 END IF 00459 * 00460 * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN. 00461 * 00462 IF( WANTZ ) THEN 00463 ORDER = 'B' 00464 ELSE 00465 ORDER = 'E' 00466 END IF 00467 INDIBL = 1 00468 INDISP = INDIBL + N 00469 INDIWO = INDISP + N 00470 CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, 00471 $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W, 00472 $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ), 00473 $ IWORK( INDIWO ), INFO ) 00474 * 00475 IF( WANTZ ) THEN 00476 CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W, 00477 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, 00478 $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO ) 00479 * 00480 * Apply orthogonal matrix used in reduction to tridiagonal 00481 * form to eigenvectors returned by DSTEIN. 00482 * 00483 DO 20 J = 1, M 00484 CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 ) 00485 CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO, 00486 $ Z( 1, J ), 1 ) 00487 20 CONTINUE 00488 END IF 00489 * 00490 * If matrix was scaled, then rescale eigenvalues appropriately. 00491 * 00492 30 CONTINUE 00493 IF( ISCALE.EQ.1 ) THEN 00494 IF( INFO.EQ.0 ) THEN 00495 IMAX = M 00496 ELSE 00497 IMAX = INFO - 1 00498 END IF 00499 CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) 00500 END IF 00501 * 00502 * If eigenvalues are not in order, then sort them, along with 00503 * eigenvectors. 00504 * 00505 IF( WANTZ ) THEN 00506 DO 50 J = 1, M - 1 00507 I = 0 00508 TMP1 = W( J ) 00509 DO 40 JJ = J + 1, M 00510 IF( W( JJ ).LT.TMP1 ) THEN 00511 I = JJ 00512 TMP1 = W( JJ ) 00513 END IF 00514 40 CONTINUE 00515 * 00516 IF( I.NE.0 ) THEN 00517 ITMP1 = IWORK( INDIBL+I-1 ) 00518 W( I ) = W( J ) 00519 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) 00520 W( J ) = TMP1 00521 IWORK( INDIBL+J-1 ) = ITMP1 00522 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) 00523 IF( INFO.NE.0 ) THEN 00524 ITMP1 = IFAIL( I ) 00525 IFAIL( I ) = IFAIL( J ) 00526 IFAIL( J ) = ITMP1 00527 END IF 00528 END IF 00529 50 CONTINUE 00530 END IF 00531 * 00532 RETURN 00533 * 00534 * End of DSBEVX 00535 * 00536 END