LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zstemr.f
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00001 *> \brief \b ZSTEMR
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download ZSTEMR + dependencies 
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00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zstemr.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zstemr.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
00022 *                          M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
00023 *                          IWORK, LIWORK, INFO )
00024 * 
00025 *       .. Scalar Arguments ..
00026 *       CHARACTER          JOBZ, RANGE
00027 *       LOGICAL            TRYRAC
00028 *       INTEGER            IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
00029 *       DOUBLE PRECISION VL, VU
00030 *       ..
00031 *       .. Array Arguments ..
00032 *       INTEGER            ISUPPZ( * ), IWORK( * )
00033 *       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * )
00034 *       COMPLEX*16         Z( LDZ, * )
00035 *       ..
00036 *  
00037 *
00038 *> \par Purpose:
00039 *  =============
00040 *>
00041 *> \verbatim
00042 *>
00043 *> ZSTEMR computes selected eigenvalues and, optionally, eigenvectors
00044 *> of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
00045 *> a well defined set of pairwise different real eigenvalues, the corresponding
00046 *> real eigenvectors are pairwise orthogonal.
00047 *>
00048 *> The spectrum may be computed either completely or partially by specifying
00049 *> either an interval (VL,VU] or a range of indices IL:IU for the desired
00050 *> eigenvalues.
00051 *>
00052 *> Depending on the number of desired eigenvalues, these are computed either
00053 *> by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
00054 *> computed by the use of various suitable L D L^T factorizations near clusters
00055 *> of close eigenvalues (referred to as RRRs, Relatively Robust
00056 *> Representations). An informal sketch of the algorithm follows.
00057 *>
00058 *> For each unreduced block (submatrix) of T,
00059 *>    (a) Compute T - sigma I  = L D L^T, so that L and D
00060 *>        define all the wanted eigenvalues to high relative accuracy.
00061 *>        This means that small relative changes in the entries of D and L
00062 *>        cause only small relative changes in the eigenvalues and
00063 *>        eigenvectors. The standard (unfactored) representation of the
00064 *>        tridiagonal matrix T does not have this property in general.
00065 *>    (b) Compute the eigenvalues to suitable accuracy.
00066 *>        If the eigenvectors are desired, the algorithm attains full
00067 *>        accuracy of the computed eigenvalues only right before
00068 *>        the corresponding vectors have to be computed, see steps c) and d).
00069 *>    (c) For each cluster of close eigenvalues, select a new
00070 *>        shift close to the cluster, find a new factorization, and refine
00071 *>        the shifted eigenvalues to suitable accuracy.
00072 *>    (d) For each eigenvalue with a large enough relative separation compute
00073 *>        the corresponding eigenvector by forming a rank revealing twisted
00074 *>        factorization. Go back to (c) for any clusters that remain.
00075 *>
00076 *> For more details, see:
00077 *> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
00078 *>   to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
00079 *>   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
00080 *> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
00081 *>   Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
00082 *>   2004.  Also LAPACK Working Note 154.
00083 *> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
00084 *>   tridiagonal eigenvalue/eigenvector problem",
00085 *>   Computer Science Division Technical Report No. UCB/CSD-97-971,
00086 *>   UC Berkeley, May 1997.
00087 *>
00088 *> Further Details
00089 *> 1.ZSTEMR works only on machines which follow IEEE-754
00090 *> floating-point standard in their handling of infinities and NaNs.
00091 *> This permits the use of efficient inner loops avoiding a check for
00092 *> zero divisors.
00093 *>
00094 *> 2. LAPACK routines can be used to reduce a complex Hermitean matrix to
00095 *> real symmetric tridiagonal form.
00096 *>
00097 *> (Any complex Hermitean tridiagonal matrix has real values on its diagonal
00098 *> and potentially complex numbers on its off-diagonals. By applying a
00099 *> similarity transform with an appropriate diagonal matrix
00100 *> diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean
00101 *> matrix can be transformed into a real symmetric matrix and complex
00102 *> arithmetic can be entirely avoided.)
00103 *>
00104 *> While the eigenvectors of the real symmetric tridiagonal matrix are real,
00105 *> the eigenvectors of original complex Hermitean matrix have complex entries
00106 *> in general.
00107 *> Since LAPACK drivers overwrite the matrix data with the eigenvectors,
00108 *> ZSTEMR accepts complex workspace to facilitate interoperability
00109 *> with ZUNMTR or ZUPMTR.
00110 *> \endverbatim
00111 *
00112 *  Arguments:
00113 *  ==========
00114 *
00115 *> \param[in] JOBZ
00116 *> \verbatim
00117 *>          JOBZ is CHARACTER*1
00118 *>          = 'N':  Compute eigenvalues only;
00119 *>          = 'V':  Compute eigenvalues and eigenvectors.
00120 *> \endverbatim
00121 *>
00122 *> \param[in] RANGE
00123 *> \verbatim
00124 *>          RANGE is CHARACTER*1
00125 *>          = 'A': all eigenvalues will be found.
00126 *>          = 'V': all eigenvalues in the half-open interval (VL,VU]
00127 *>                 will be found.
00128 *>          = 'I': the IL-th through IU-th eigenvalues will be found.
00129 *> \endverbatim
00130 *>
00131 *> \param[in] N
00132 *> \verbatim
00133 *>          N is INTEGER
00134 *>          The order of the matrix.  N >= 0.
00135 *> \endverbatim
00136 *>
00137 *> \param[in,out] D
00138 *> \verbatim
00139 *>          D is DOUBLE PRECISION array, dimension (N)
00140 *>          On entry, the N diagonal elements of the tridiagonal matrix
00141 *>          T. On exit, D is overwritten.
00142 *> \endverbatim
00143 *>
00144 *> \param[in,out] E
00145 *> \verbatim
00146 *>          E is DOUBLE PRECISION array, dimension (N)
00147 *>          On entry, the (N-1) subdiagonal elements of the tridiagonal
00148 *>          matrix T in elements 1 to N-1 of E. E(N) need not be set on
00149 *>          input, but is used internally as workspace.
00150 *>          On exit, E is overwritten.
00151 *> \endverbatim
00152 *>
00153 *> \param[in] VL
00154 *> \verbatim
00155 *>          VL is DOUBLE PRECISION
00156 *> \endverbatim
00157 *>
00158 *> \param[in] VU
00159 *> \verbatim
00160 *>          VU is DOUBLE PRECISION
00161 *>
00162 *>          If RANGE='V', the lower and upper bounds of the interval to
00163 *>          be searched for eigenvalues. VL < VU.
00164 *>          Not referenced if RANGE = 'A' or 'I'.
00165 *> \endverbatim
00166 *>
00167 *> \param[in] IL
00168 *> \verbatim
00169 *>          IL is INTEGER
00170 *> \endverbatim
00171 *>
00172 *> \param[in] IU
00173 *> \verbatim
00174 *>          IU is INTEGER
00175 *>
00176 *>          If RANGE='I', the indices (in ascending order) of the
00177 *>          smallest and largest eigenvalues to be returned.
00178 *>          1 <= IL <= IU <= N, if N > 0.
00179 *>          Not referenced if RANGE = 'A' or 'V'.
00180 *> \endverbatim
00181 *>
00182 *> \param[out] M
00183 *> \verbatim
00184 *>          M is INTEGER
00185 *>          The total number of eigenvalues found.  0 <= M <= N.
00186 *>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00187 *> \endverbatim
00188 *>
00189 *> \param[out] W
00190 *> \verbatim
00191 *>          W is DOUBLE PRECISION array, dimension (N)
00192 *>          The first M elements contain the selected eigenvalues in
00193 *>          ascending order.
00194 *> \endverbatim
00195 *>
00196 *> \param[out] Z
00197 *> \verbatim
00198 *>          Z is COMPLEX*16 array, dimension (LDZ, max(1,M) )
00199 *>          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
00200 *>          contain the orthonormal eigenvectors of the matrix T
00201 *>          corresponding to the selected eigenvalues, with the i-th
00202 *>          column of Z holding the eigenvector associated with W(i).
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 can be computed with a workspace
00207 *>          query by setting NZC = -1, see below.
00208 *> \endverbatim
00209 *>
00210 *> \param[in] LDZ
00211 *> \verbatim
00212 *>          LDZ is INTEGER
00213 *>          The leading dimension of the array Z.  LDZ >= 1, and if
00214 *>          JOBZ = 'V', then LDZ >= max(1,N).
00215 *> \endverbatim
00216 *>
00217 *> \param[in] NZC
00218 *> \verbatim
00219 *>          NZC is INTEGER
00220 *>          The number of eigenvectors to be held in the array Z.
00221 *>          If RANGE = 'A', then NZC >= max(1,N).
00222 *>          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
00223 *>          If RANGE = 'I', then NZC >= IU-IL+1.
00224 *>          If NZC = -1, then a workspace query is assumed; the
00225 *>          routine calculates the number of columns of the array Z that
00226 *>          are needed to hold the eigenvectors.
00227 *>          This value is returned as the first entry of the Z array, and
00228 *>          no error message related to NZC is issued by XERBLA.
00229 *> \endverbatim
00230 *>
00231 *> \param[out] ISUPPZ
00232 *> \verbatim
00233 *>          ISUPPZ is INTEGER ARRAY, dimension ( 2*max(1,M) )
00234 *>          The support of the eigenvectors in Z, i.e., the indices
00235 *>          indicating the nonzero elements in Z. The i-th computed eigenvector
00236 *>          is nonzero only in elements ISUPPZ( 2*i-1 ) through
00237 *>          ISUPPZ( 2*i ). This is relevant in the case when the matrix
00238 *>          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
00239 *> \endverbatim
00240 *>
00241 *> \param[in,out] TRYRAC
00242 *> \verbatim
00243 *>          TRYRAC is LOGICAL
00244 *>          If TRYRAC.EQ..TRUE., indicates that the code should check whether
00245 *>          the tridiagonal matrix defines its eigenvalues to high relative
00246 *>          accuracy.  If so, the code uses relative-accuracy preserving
00247 *>          algorithms that might be (a bit) slower depending on the matrix.
00248 *>          If the matrix does not define its eigenvalues to high relative
00249 *>          accuracy, the code can uses possibly faster algorithms.
00250 *>          If TRYRAC.EQ..FALSE., the code is not required to guarantee
00251 *>          relatively accurate eigenvalues and can use the fastest possible
00252 *>          techniques.
00253 *>          On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
00254 *>          does not define its eigenvalues to high relative accuracy.
00255 *> \endverbatim
00256 *>
00257 *> \param[out] WORK
00258 *> \verbatim
00259 *>          WORK is DOUBLE PRECISION array, dimension (LWORK)
00260 *>          On exit, if INFO = 0, WORK(1) returns the optimal
00261 *>          (and minimal) LWORK.
00262 *> \endverbatim
00263 *>
00264 *> \param[in] LWORK
00265 *> \verbatim
00266 *>          LWORK is INTEGER
00267 *>          The dimension of the array WORK. LWORK >= max(1,18*N)
00268 *>          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
00269 *>          If LWORK = -1, then a workspace query is assumed; the routine
00270 *>          only calculates the optimal size of the WORK array, returns
00271 *>          this value as the first entry of the WORK array, and no error
00272 *>          message related to LWORK is issued by XERBLA.
00273 *> \endverbatim
00274 *>
00275 *> \param[out] IWORK
00276 *> \verbatim
00277 *>          IWORK is INTEGER array, dimension (LIWORK)
00278 *>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
00279 *> \endverbatim
00280 *>
00281 *> \param[in] LIWORK
00282 *> \verbatim
00283 *>          LIWORK is INTEGER
00284 *>          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
00285 *>          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
00286 *>          if only the eigenvalues are to be computed.
00287 *>          If LIWORK = -1, then a workspace query is assumed; the
00288 *>          routine only calculates the optimal size of the IWORK array,
00289 *>          returns this value as the first entry of the IWORK array, and
00290 *>          no error message related to LIWORK is issued by XERBLA.
00291 *> \endverbatim
00292 *>
00293 *> \param[out] INFO
00294 *> \verbatim
00295 *>          INFO is INTEGER
00296 *>          On exit, INFO
00297 *>          = 0:  successful exit
00298 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00299 *>          > 0:  if INFO = 1X, internal error in DLARRE,
00300 *>                if INFO = 2X, internal error in ZLARRV.
00301 *>                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
00302 *>                the nonzero error code returned by DLARRE or
00303 *>                ZLARRV, respectively.
00304 *> \endverbatim
00305 *
00306 *  Authors:
00307 *  ========
00308 *
00309 *> \author Univ. of Tennessee 
00310 *> \author Univ. of California Berkeley 
00311 *> \author Univ. of Colorado Denver 
00312 *> \author NAG Ltd. 
00313 *
00314 *> \date November 2011
00315 *
00316 *> \ingroup complex16OTHERcomputational
00317 *
00318 *> \par Contributors:
00319 *  ==================
00320 *>
00321 *> Beresford Parlett, University of California, Berkeley, USA \n
00322 *> Jim Demmel, University of California, Berkeley, USA \n
00323 *> Inderjit Dhillon, University of Texas, Austin, USA \n
00324 *> Osni Marques, LBNL/NERSC, USA \n
00325 *> Christof Voemel, University of California, Berkeley, USA \n
00326 *
00327 *  =====================================================================
00328       SUBROUTINE ZSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
00329      $                   M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
00330      $                   IWORK, LIWORK, INFO )
00331 *
00332 *  -- LAPACK computational routine (version 3.4.0) --
00333 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00334 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00335 *     November 2011
00336 *
00337 *     .. Scalar Arguments ..
00338       CHARACTER          JOBZ, RANGE
00339       LOGICAL            TRYRAC
00340       INTEGER            IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
00341       DOUBLE PRECISION VL, VU
00342 *     ..
00343 *     .. Array Arguments ..
00344       INTEGER            ISUPPZ( * ), IWORK( * )
00345       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * )
00346       COMPLEX*16         Z( LDZ, * )
00347 *     ..
00348 *
00349 *  =====================================================================
00350 *
00351 *     .. Parameters ..
00352       DOUBLE PRECISION   ZERO, ONE, FOUR, MINRGP
00353       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0,
00354      $                     FOUR = 4.0D0,
00355      $                     MINRGP = 1.0D-3 )
00356 *     ..
00357 *     .. Local Scalars ..
00358       LOGICAL            ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
00359       INTEGER            I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
00360      $                   IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
00361      $                   INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
00362      $                   ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
00363      $                   NZCMIN, OFFSET, WBEGIN, WEND
00364       DOUBLE PRECISION   BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
00365      $                   RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
00366      $                   THRESH, TMP, TNRM, WL, WU
00367 *     ..
00368 *     ..
00369 *     .. External Functions ..
00370       LOGICAL            LSAME
00371       DOUBLE PRECISION   DLAMCH, DLANST
00372       EXTERNAL           LSAME, DLAMCH, DLANST
00373 *     ..
00374 *     .. External Subroutines ..
00375       EXTERNAL           DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ,
00376      $                   DLARRR, DLASRT, DSCAL, XERBLA, ZLARRV, ZSWAP
00377 *     ..
00378 *     .. Intrinsic Functions ..
00379       INTRINSIC          MAX, MIN, SQRT
00380 
00381 
00382 *     ..
00383 *     .. Executable Statements ..
00384 *
00385 *     Test the input parameters.
00386 *
00387       WANTZ = LSAME( JOBZ, 'V' )
00388       ALLEIG = LSAME( RANGE, 'A' )
00389       VALEIG = LSAME( RANGE, 'V' )
00390       INDEIG = LSAME( RANGE, 'I' )
00391 *
00392       LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
00393       ZQUERY = ( NZC.EQ.-1 )
00394 
00395 *     DSTEMR needs WORK of size 6*N, IWORK of size 3*N.
00396 *     In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N.
00397 *     Furthermore, ZLARRV needs WORK of size 12*N, IWORK of size 7*N.
00398       IF( WANTZ ) THEN
00399          LWMIN = 18*N
00400          LIWMIN = 10*N
00401       ELSE
00402 *        need less workspace if only the eigenvalues are wanted
00403          LWMIN = 12*N
00404          LIWMIN = 8*N
00405       ENDIF
00406 
00407       WL = ZERO
00408       WU = ZERO
00409       IIL = 0
00410       IIU = 0
00411 
00412       IF( VALEIG ) THEN
00413 *        We do not reference VL, VU in the cases RANGE = 'I','A'
00414 *        The interval (WL, WU] contains all the wanted eigenvalues.
00415 *        It is either given by the user or computed in DLARRE.
00416          WL = VL
00417          WU = VU
00418       ELSEIF( INDEIG ) THEN
00419 *        We do not reference IL, IU in the cases RANGE = 'V','A'
00420          IIL = IL
00421          IIU = IU
00422       ENDIF
00423 *
00424       INFO = 0
00425       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00426          INFO = -1
00427       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00428          INFO = -2
00429       ELSE IF( N.LT.0 ) THEN
00430          INFO = -3
00431       ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
00432          INFO = -7
00433       ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
00434          INFO = -8
00435       ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
00436          INFO = -9
00437       ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
00438          INFO = -13
00439       ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
00440          INFO = -17
00441       ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
00442          INFO = -19
00443       END IF
00444 *
00445 *     Get machine constants.
00446 *
00447       SAFMIN = DLAMCH( 'Safe minimum' )
00448       EPS = DLAMCH( 'Precision' )
00449       SMLNUM = SAFMIN / EPS
00450       BIGNUM = ONE / SMLNUM
00451       RMIN = SQRT( SMLNUM )
00452       RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
00453 *
00454       IF( INFO.EQ.0 ) THEN
00455          WORK( 1 ) = LWMIN
00456          IWORK( 1 ) = LIWMIN
00457 *
00458          IF( WANTZ .AND. ALLEIG ) THEN
00459             NZCMIN = N
00460          ELSE IF( WANTZ .AND. VALEIG ) THEN
00461             CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN,
00462      $                            NZCMIN, ITMP, ITMP2, INFO )
00463          ELSE IF( WANTZ .AND. INDEIG ) THEN
00464             NZCMIN = IIU-IIL+1
00465          ELSE
00466 *           WANTZ .EQ. FALSE.
00467             NZCMIN = 0
00468          ENDIF
00469          IF( ZQUERY .AND. INFO.EQ.0 ) THEN
00470             Z( 1,1 ) = NZCMIN
00471          ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
00472             INFO = -14
00473          END IF
00474       END IF
00475 
00476       IF( INFO.NE.0 ) THEN
00477 *
00478          CALL XERBLA( 'ZSTEMR', -INFO )
00479 *
00480          RETURN
00481       ELSE IF( LQUERY .OR. ZQUERY ) THEN
00482          RETURN
00483       END IF
00484 *
00485 *     Handle N = 0, 1, and 2 cases immediately
00486 *
00487       M = 0
00488       IF( N.EQ.0 )
00489      $   RETURN
00490 *
00491       IF( N.EQ.1 ) THEN
00492          IF( ALLEIG .OR. INDEIG ) THEN
00493             M = 1
00494             W( 1 ) = D( 1 )
00495          ELSE
00496             IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
00497                M = 1
00498                W( 1 ) = D( 1 )
00499             END IF
00500          END IF
00501          IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00502             Z( 1, 1 ) = ONE
00503             ISUPPZ(1) = 1
00504             ISUPPZ(2) = 1
00505          END IF
00506          RETURN
00507       END IF
00508 *
00509       IF( N.EQ.2 ) THEN
00510          IF( .NOT.WANTZ ) THEN
00511             CALL DLAE2( D(1), E(1), D(2), R1, R2 )
00512          ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00513             CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
00514          END IF
00515          IF( ALLEIG.OR.
00516      $      (VALEIG.AND.(R2.GT.WL).AND.
00517      $                  (R2.LE.WU)).OR.
00518      $      (INDEIG.AND.(IIL.EQ.1)) ) THEN
00519             M = M+1
00520             W( M ) = R2
00521             IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00522                Z( 1, M ) = -SN
00523                Z( 2, M ) = CS
00524 *              Note: At most one of SN and CS can be zero.
00525                IF (SN.NE.ZERO) THEN
00526                   IF (CS.NE.ZERO) THEN
00527                      ISUPPZ(2*M-1) = 1
00528                      ISUPPZ(2*M-1) = 2
00529                   ELSE
00530                      ISUPPZ(2*M-1) = 1
00531                      ISUPPZ(2*M-1) = 1
00532                   END IF
00533                ELSE
00534                   ISUPPZ(2*M-1) = 2
00535                   ISUPPZ(2*M) = 2
00536                END IF
00537             ENDIF
00538          ENDIF
00539          IF( ALLEIG.OR.
00540      $      (VALEIG.AND.(R1.GT.WL).AND.
00541      $                  (R1.LE.WU)).OR.
00542      $      (INDEIG.AND.(IIU.EQ.2)) ) THEN
00543             M = M+1
00544             W( M ) = R1
00545             IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00546                Z( 1, M ) = CS
00547                Z( 2, M ) = SN
00548 *              Note: At most one of SN and CS can be zero.
00549                IF (SN.NE.ZERO) THEN
00550                   IF (CS.NE.ZERO) THEN
00551                      ISUPPZ(2*M-1) = 1
00552                      ISUPPZ(2*M-1) = 2
00553                   ELSE
00554                      ISUPPZ(2*M-1) = 1
00555                      ISUPPZ(2*M-1) = 1
00556                   END IF
00557                ELSE
00558                   ISUPPZ(2*M-1) = 2
00559                   ISUPPZ(2*M) = 2
00560                END IF
00561             ENDIF
00562          ENDIF
00563          RETURN
00564       END IF
00565 
00566 *     Continue with general N
00567 
00568       INDGRS = 1
00569       INDERR = 2*N + 1
00570       INDGP = 3*N + 1
00571       INDD = 4*N + 1
00572       INDE2 = 5*N + 1
00573       INDWRK = 6*N + 1
00574 *
00575       IINSPL = 1
00576       IINDBL = N + 1
00577       IINDW = 2*N + 1
00578       IINDWK = 3*N + 1
00579 *
00580 *     Scale matrix to allowable range, if necessary.
00581 *     The allowable range is related to the PIVMIN parameter; see the
00582 *     comments in DLARRD.  The preference for scaling small values
00583 *     up is heuristic; we expect users' matrices not to be close to the
00584 *     RMAX threshold.
00585 *
00586       SCALE = ONE
00587       TNRM = DLANST( 'M', N, D, E )
00588       IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
00589          SCALE = RMIN / TNRM
00590       ELSE IF( TNRM.GT.RMAX ) THEN
00591          SCALE = RMAX / TNRM
00592       END IF
00593       IF( SCALE.NE.ONE ) THEN
00594          CALL DSCAL( N, SCALE, D, 1 )
00595          CALL DSCAL( N-1, SCALE, E, 1 )
00596          TNRM = TNRM*SCALE
00597          IF( VALEIG ) THEN
00598 *           If eigenvalues in interval have to be found,
00599 *           scale (WL, WU] accordingly
00600             WL = WL*SCALE
00601             WU = WU*SCALE
00602          ENDIF
00603       END IF
00604 *
00605 *     Compute the desired eigenvalues of the tridiagonal after splitting
00606 *     into smaller subblocks if the corresponding off-diagonal elements
00607 *     are small
00608 *     THRESH is the splitting parameter for DLARRE
00609 *     A negative THRESH forces the old splitting criterion based on the
00610 *     size of the off-diagonal. A positive THRESH switches to splitting
00611 *     which preserves relative accuracy.
00612 *
00613       IF( TRYRAC ) THEN
00614 *        Test whether the matrix warrants the more expensive relative approach.
00615          CALL DLARRR( N, D, E, IINFO )
00616       ELSE
00617 *        The user does not care about relative accurately eigenvalues
00618          IINFO = -1
00619       ENDIF
00620 *     Set the splitting criterion
00621       IF (IINFO.EQ.0) THEN
00622          THRESH = EPS
00623       ELSE
00624          THRESH = -EPS
00625 *        relative accuracy is desired but T does not guarantee it
00626          TRYRAC = .FALSE.
00627       ENDIF
00628 *
00629       IF( TRYRAC ) THEN
00630 *        Copy original diagonal, needed to guarantee relative accuracy
00631          CALL DCOPY(N,D,1,WORK(INDD),1)
00632       ENDIF
00633 *     Store the squares of the offdiagonal values of T
00634       DO 5 J = 1, N-1
00635          WORK( INDE2+J-1 ) = E(J)**2
00636  5    CONTINUE
00637 
00638 *     Set the tolerance parameters for bisection
00639       IF( .NOT.WANTZ ) THEN
00640 *        DLARRE computes the eigenvalues to full precision.
00641          RTOL1 = FOUR * EPS
00642          RTOL2 = FOUR * EPS
00643       ELSE
00644 *        DLARRE computes the eigenvalues to less than full precision.
00645 *        ZLARRV will refine the eigenvalue approximations, and we only
00646 *        need less accurate initial bisection in DLARRE.
00647 *        Note: these settings do only affect the subset case and DLARRE
00648          RTOL1 = SQRT(EPS)
00649          RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
00650       ENDIF
00651       CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
00652      $             WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
00653      $             IWORK( IINSPL ), M, W, WORK( INDERR ),
00654      $             WORK( INDGP ), IWORK( IINDBL ),
00655      $             IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
00656      $             WORK( INDWRK ), IWORK( IINDWK ), IINFO )
00657       IF( IINFO.NE.0 ) THEN
00658          INFO = 10 + ABS( IINFO )
00659          RETURN
00660       END IF
00661 *     Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
00662 *     part of the spectrum. All desired eigenvalues are contained in
00663 *     (WL,WU]
00664 
00665 
00666       IF( WANTZ ) THEN
00667 *
00668 *        Compute the desired eigenvectors corresponding to the computed
00669 *        eigenvalues
00670 *
00671          CALL ZLARRV( N, WL, WU, D, E,
00672      $                PIVMIN, IWORK( IINSPL ), M,
00673      $                1, M, MINRGP, RTOL1, RTOL2,
00674      $                W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
00675      $                IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
00676      $                ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
00677          IF( IINFO.NE.0 ) THEN
00678             INFO = 20 + ABS( IINFO )
00679             RETURN
00680          END IF
00681       ELSE
00682 *        DLARRE computes eigenvalues of the (shifted) root representation
00683 *        ZLARRV returns the eigenvalues of the unshifted matrix.
00684 *        However, if the eigenvectors are not desired by the user, we need
00685 *        to apply the corresponding shifts from DLARRE to obtain the
00686 *        eigenvalues of the original matrix.
00687          DO 20 J = 1, M
00688             ITMP = IWORK( IINDBL+J-1 )
00689             W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
00690  20      CONTINUE
00691       END IF
00692 *
00693 
00694       IF ( TRYRAC ) THEN
00695 *        Refine computed eigenvalues so that they are relatively accurate
00696 *        with respect to the original matrix T.
00697          IBEGIN = 1
00698          WBEGIN = 1
00699          DO 39  JBLK = 1, IWORK( IINDBL+M-1 )
00700             IEND = IWORK( IINSPL+JBLK-1 )
00701             IN = IEND - IBEGIN + 1
00702             WEND = WBEGIN - 1
00703 *           check if any eigenvalues have to be refined in this block
00704  36         CONTINUE
00705             IF( WEND.LT.M ) THEN
00706                IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
00707                   WEND = WEND + 1
00708                   GO TO 36
00709                END IF
00710             END IF
00711             IF( WEND.LT.WBEGIN ) THEN
00712                IBEGIN = IEND + 1
00713                GO TO 39
00714             END IF
00715 
00716             OFFSET = IWORK(IINDW+WBEGIN-1)-1
00717             IFIRST = IWORK(IINDW+WBEGIN-1)
00718             ILAST = IWORK(IINDW+WEND-1)
00719             RTOL2 = FOUR * EPS
00720             CALL DLARRJ( IN,
00721      $                   WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
00722      $                   IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
00723      $                   WORK( INDERR+WBEGIN-1 ),
00724      $                   WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
00725      $                   TNRM, IINFO )
00726             IBEGIN = IEND + 1
00727             WBEGIN = WEND + 1
00728  39      CONTINUE
00729       ENDIF
00730 *
00731 *     If matrix was scaled, then rescale eigenvalues appropriately.
00732 *
00733       IF( SCALE.NE.ONE ) THEN
00734          CALL DSCAL( M, ONE / SCALE, W, 1 )
00735       END IF
00736 *
00737 *     If eigenvalues are not in increasing order, then sort them,
00738 *     possibly along with eigenvectors.
00739 *
00740       IF( NSPLIT.GT.1 ) THEN
00741          IF( .NOT. WANTZ ) THEN
00742             CALL DLASRT( 'I', M, W, IINFO )
00743             IF( IINFO.NE.0 ) THEN
00744                INFO = 3
00745                RETURN
00746             END IF
00747          ELSE
00748             DO 60 J = 1, M - 1
00749                I = 0
00750                TMP = W( J )
00751                DO 50 JJ = J + 1, M
00752                   IF( W( JJ ).LT.TMP ) THEN
00753                      I = JJ
00754                      TMP = W( JJ )
00755                   END IF
00756  50            CONTINUE
00757                IF( I.NE.0 ) THEN
00758                   W( I ) = W( J )
00759                   W( J ) = TMP
00760                   IF( WANTZ ) THEN
00761                      CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
00762                      ITMP = ISUPPZ( 2*I-1 )
00763                      ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
00764                      ISUPPZ( 2*J-1 ) = ITMP
00765                      ITMP = ISUPPZ( 2*I )
00766                      ISUPPZ( 2*I ) = ISUPPZ( 2*J )
00767                      ISUPPZ( 2*J ) = ITMP
00768                   END IF
00769                END IF
00770  60         CONTINUE
00771          END IF
00772       ENDIF
00773 *
00774 *
00775       WORK( 1 ) = LWMIN
00776       IWORK( 1 ) = LIWMIN
00777       RETURN
00778 *
00779 *     End of ZSTEMR
00780 *
00781       END
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