LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
sstemr.f
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00001 *> \brief \b SSTEMR
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download SSTEMR + dependencies 
00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sstemr.f"> 
00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sstemr.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sstemr.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SSTEMR( 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 *       REAL               VL, VU
00030 *       ..
00031 *       .. Array Arguments ..
00032 *       INTEGER            ISUPPZ( * ), IWORK( * )
00033 *       REAL               D( * ), E( * ), W( * ), WORK( * )
00034 *       REAL               Z( LDZ, * )
00035 *       ..
00036 *  
00037 *
00038 *> \par Purpose:
00039 *  =============
00040 *>
00041 *> \verbatim
00042 *>
00043 *> SSTEMR 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.SSTEMR 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 *> \endverbatim
00094 *
00095 *  Arguments:
00096 *  ==========
00097 *
00098 *> \param[in] JOBZ
00099 *> \verbatim
00100 *>          JOBZ is CHARACTER*1
00101 *>          = 'N':  Compute eigenvalues only;
00102 *>          = 'V':  Compute eigenvalues and eigenvectors.
00103 *> \endverbatim
00104 *>
00105 *> \param[in] RANGE
00106 *> \verbatim
00107 *>          RANGE is CHARACTER*1
00108 *>          = 'A': all eigenvalues will be found.
00109 *>          = 'V': all eigenvalues in the half-open interval (VL,VU]
00110 *>                 will be found.
00111 *>          = 'I': the IL-th through IU-th eigenvalues will be found.
00112 *> \endverbatim
00113 *>
00114 *> \param[in] N
00115 *> \verbatim
00116 *>          N is INTEGER
00117 *>          The order of the matrix.  N >= 0.
00118 *> \endverbatim
00119 *>
00120 *> \param[in,out] D
00121 *> \verbatim
00122 *>          D is REAL array, dimension (N)
00123 *>          On entry, the N diagonal elements of the tridiagonal matrix
00124 *>          T. On exit, D is overwritten.
00125 *> \endverbatim
00126 *>
00127 *> \param[in,out] E
00128 *> \verbatim
00129 *>          E is REAL array, dimension (N)
00130 *>          On entry, the (N-1) subdiagonal elements of the tridiagonal
00131 *>          matrix T in elements 1 to N-1 of E. E(N) need not be set on
00132 *>          input, but is used internally as workspace.
00133 *>          On exit, E is overwritten.
00134 *> \endverbatim
00135 *>
00136 *> \param[in] VL
00137 *> \verbatim
00138 *>          VL is REAL
00139 *> \endverbatim
00140 *>
00141 *> \param[in] VU
00142 *> \verbatim
00143 *>          VU is REAL
00144 *>
00145 *>          If RANGE='V', the lower and upper bounds of the interval to
00146 *>          be searched for eigenvalues. VL < VU.
00147 *>          Not referenced if RANGE = 'A' or 'I'.
00148 *> \endverbatim
00149 *>
00150 *> \param[in] IL
00151 *> \verbatim
00152 *>          IL is INTEGER
00153 *> \endverbatim
00154 *>
00155 *> \param[in] IU
00156 *> \verbatim
00157 *>          IU is INTEGER
00158 *>
00159 *>          If RANGE='I', the indices (in ascending order) of the
00160 *>          smallest and largest eigenvalues to be returned.
00161 *>          1 <= IL <= IU <= N, if N > 0.
00162 *>          Not referenced if RANGE = 'A' or 'V'.
00163 *> \endverbatim
00164 *>
00165 *> \param[out] M
00166 *> \verbatim
00167 *>          M is INTEGER
00168 *>          The total number of eigenvalues found.  0 <= M <= N.
00169 *>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00170 *> \endverbatim
00171 *>
00172 *> \param[out] W
00173 *> \verbatim
00174 *>          W is REAL array, dimension (N)
00175 *>          The first M elements contain the selected eigenvalues in
00176 *>          ascending order.
00177 *> \endverbatim
00178 *>
00179 *> \param[out] Z
00180 *> \verbatim
00181 *>          Z is REAL array, dimension (LDZ, max(1,M) )
00182 *>          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
00183 *>          contain the orthonormal eigenvectors of the matrix T
00184 *>          corresponding to the selected eigenvalues, with the i-th
00185 *>          column of Z holding the eigenvector associated with W(i).
00186 *>          If JOBZ = 'N', then Z is not referenced.
00187 *>          Note: the user must ensure that at least max(1,M) columns are
00188 *>          supplied in the array Z; if RANGE = 'V', the exact value of M
00189 *>          is not known in advance and can be computed with a workspace
00190 *>          query by setting NZC = -1, see below.
00191 *> \endverbatim
00192 *>
00193 *> \param[in] LDZ
00194 *> \verbatim
00195 *>          LDZ is INTEGER
00196 *>          The leading dimension of the array Z.  LDZ >= 1, and if
00197 *>          JOBZ = 'V', then LDZ >= max(1,N).
00198 *> \endverbatim
00199 *>
00200 *> \param[in] NZC
00201 *> \verbatim
00202 *>          NZC is INTEGER
00203 *>          The number of eigenvectors to be held in the array Z.
00204 *>          If RANGE = 'A', then NZC >= max(1,N).
00205 *>          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
00206 *>          If RANGE = 'I', then NZC >= IU-IL+1.
00207 *>          If NZC = -1, then a workspace query is assumed; the
00208 *>          routine calculates the number of columns of the array Z that
00209 *>          are needed to hold the eigenvectors.
00210 *>          This value is returned as the first entry of the Z array, and
00211 *>          no error message related to NZC is issued by XERBLA.
00212 *> \endverbatim
00213 *>
00214 *> \param[out] ISUPPZ
00215 *> \verbatim
00216 *>          ISUPPZ is INTEGER ARRAY, dimension ( 2*max(1,M) )
00217 *>          The support of the eigenvectors in Z, i.e., the indices
00218 *>          indicating the nonzero elements in Z. The i-th computed eigenvector
00219 *>          is nonzero only in elements ISUPPZ( 2*i-1 ) through
00220 *>          ISUPPZ( 2*i ). This is relevant in the case when the matrix
00221 *>          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
00222 *> \endverbatim
00223 *>
00224 *> \param[in,out] TRYRAC
00225 *> \verbatim
00226 *>          TRYRAC is LOGICAL
00227 *>          If TRYRAC.EQ..TRUE., indicates that the code should check whether
00228 *>          the tridiagonal matrix defines its eigenvalues to high relative
00229 *>          accuracy.  If so, the code uses relative-accuracy preserving
00230 *>          algorithms that might be (a bit) slower depending on the matrix.
00231 *>          If the matrix does not define its eigenvalues to high relative
00232 *>          accuracy, the code can uses possibly faster algorithms.
00233 *>          If TRYRAC.EQ..FALSE., the code is not required to guarantee
00234 *>          relatively accurate eigenvalues and can use the fastest possible
00235 *>          techniques.
00236 *>          On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
00237 *>          does not define its eigenvalues to high relative accuracy.
00238 *> \endverbatim
00239 *>
00240 *> \param[out] WORK
00241 *> \verbatim
00242 *>          WORK is REAL array, dimension (LWORK)
00243 *>          On exit, if INFO = 0, WORK(1) returns the optimal
00244 *>          (and minimal) LWORK.
00245 *> \endverbatim
00246 *>
00247 *> \param[in] LWORK
00248 *> \verbatim
00249 *>          LWORK is INTEGER
00250 *>          The dimension of the array WORK. LWORK >= max(1,18*N)
00251 *>          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
00252 *>          If LWORK = -1, then a workspace query is assumed; the routine
00253 *>          only calculates the optimal size of the WORK array, returns
00254 *>          this value as the first entry of the WORK array, and no error
00255 *>          message related to LWORK is issued by XERBLA.
00256 *> \endverbatim
00257 *>
00258 *> \param[out] IWORK
00259 *> \verbatim
00260 *>          IWORK is INTEGER array, dimension (LIWORK)
00261 *>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
00262 *> \endverbatim
00263 *>
00264 *> \param[in] LIWORK
00265 *> \verbatim
00266 *>          LIWORK is INTEGER
00267 *>          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
00268 *>          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
00269 *>          if only the eigenvalues are to be computed.
00270 *>          If LIWORK = -1, then a workspace query is assumed; the
00271 *>          routine only calculates the optimal size of the IWORK array,
00272 *>          returns this value as the first entry of the IWORK array, and
00273 *>          no error message related to LIWORK is issued by XERBLA.
00274 *> \endverbatim
00275 *>
00276 *> \param[out] INFO
00277 *> \verbatim
00278 *>          INFO is INTEGER
00279 *>          On exit, INFO
00280 *>          = 0:  successful exit
00281 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00282 *>          > 0:  if INFO = 1X, internal error in SLARRE,
00283 *>                if INFO = 2X, internal error in SLARRV.
00284 *>                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
00285 *>                the nonzero error code returned by SLARRE or
00286 *>                SLARRV, respectively.
00287 *> \endverbatim
00288 *
00289 *  Authors:
00290 *  ========
00291 *
00292 *> \author Univ. of Tennessee 
00293 *> \author Univ. of California Berkeley 
00294 *> \author Univ. of Colorado Denver 
00295 *> \author NAG Ltd. 
00296 *
00297 *> \date November 2011
00298 *
00299 *> \ingroup realOTHERcomputational
00300 *
00301 *> \par Contributors:
00302 *  ==================
00303 *>
00304 *> Beresford Parlett, University of California, Berkeley, USA \n
00305 *> Jim Demmel, University of California, Berkeley, USA \n
00306 *> Inderjit Dhillon, University of Texas, Austin, USA \n
00307 *> Osni Marques, LBNL/NERSC, USA \n
00308 *> Christof Voemel, University of California, Berkeley, USA
00309 *
00310 *  =====================================================================
00311       SUBROUTINE SSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
00312      $                   M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
00313      $                   IWORK, LIWORK, INFO )
00314 *
00315 *  -- LAPACK computational routine (version 3.4.0) --
00316 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00317 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00318 *     November 2011
00319 *
00320 *     .. Scalar Arguments ..
00321       CHARACTER          JOBZ, RANGE
00322       LOGICAL            TRYRAC
00323       INTEGER            IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
00324       REAL               VL, VU
00325 *     ..
00326 *     .. Array Arguments ..
00327       INTEGER            ISUPPZ( * ), IWORK( * )
00328       REAL               D( * ), E( * ), W( * ), WORK( * )
00329       REAL               Z( LDZ, * )
00330 *     ..
00331 *
00332 *  =====================================================================
00333 *
00334 *     .. Parameters ..
00335       REAL               ZERO, ONE, FOUR, MINRGP
00336       PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0,
00337      $                     FOUR = 4.0E0,
00338      $                     MINRGP = 3.0E-3 )
00339 *     ..
00340 *     .. Local Scalars ..
00341       LOGICAL            ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
00342       INTEGER            I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
00343      $                   IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
00344      $                   INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
00345      $                   ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
00346      $                   NZCMIN, OFFSET, WBEGIN, WEND
00347       REAL               BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
00348      $                   RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
00349      $                   THRESH, TMP, TNRM, WL, WU
00350 *     ..
00351 *     ..
00352 *     .. External Functions ..
00353       LOGICAL            LSAME
00354       REAL               SLAMCH, SLANST
00355       EXTERNAL           LSAME, SLAMCH, SLANST
00356 *     ..
00357 *     .. External Subroutines ..
00358       EXTERNAL           SCOPY, SLAE2, SLAEV2, SLARRC, SLARRE, SLARRJ,
00359      $                   SLARRR, SLARRV, SLASRT, SSCAL, SSWAP, XERBLA
00360 *     ..
00361 *     .. Intrinsic Functions ..
00362       INTRINSIC          MAX, MIN, SQRT
00363 *     ..
00364 *     .. Executable Statements ..
00365 *
00366 *     Test the input parameters.
00367 *
00368       WANTZ = LSAME( JOBZ, 'V' )
00369       ALLEIG = LSAME( RANGE, 'A' )
00370       VALEIG = LSAME( RANGE, 'V' )
00371       INDEIG = LSAME( RANGE, 'I' )
00372 *
00373       LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
00374       ZQUERY = ( NZC.EQ.-1 )
00375 
00376 *     SSTEMR needs WORK of size 6*N, IWORK of size 3*N.
00377 *     In addition, SLARRE needs WORK of size 6*N, IWORK of size 5*N.
00378 *     Furthermore, SLARRV needs WORK of size 12*N, IWORK of size 7*N.
00379       IF( WANTZ ) THEN
00380          LWMIN = 18*N
00381          LIWMIN = 10*N
00382       ELSE
00383 *        need less workspace if only the eigenvalues are wanted
00384          LWMIN = 12*N
00385          LIWMIN = 8*N
00386       ENDIF
00387 
00388       WL = ZERO
00389       WU = ZERO
00390       IIL = 0
00391       IIU = 0
00392 
00393       IF( VALEIG ) THEN
00394 *        We do not reference VL, VU in the cases RANGE = 'I','A'
00395 *        The interval (WL, WU] contains all the wanted eigenvalues.
00396 *        It is either given by the user or computed in SLARRE.
00397          WL = VL
00398          WU = VU
00399       ELSEIF( INDEIG ) THEN
00400 *        We do not reference IL, IU in the cases RANGE = 'V','A'
00401          IIL = IL
00402          IIU = IU
00403       ENDIF
00404 *
00405       INFO = 0
00406       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00407          INFO = -1
00408       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00409          INFO = -2
00410       ELSE IF( N.LT.0 ) THEN
00411          INFO = -3
00412       ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
00413          INFO = -7
00414       ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
00415          INFO = -8
00416       ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
00417          INFO = -9
00418       ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
00419          INFO = -13
00420       ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
00421          INFO = -17
00422       ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
00423          INFO = -19
00424       END IF
00425 *
00426 *     Get machine constants.
00427 *
00428       SAFMIN = SLAMCH( 'Safe minimum' )
00429       EPS = SLAMCH( 'Precision' )
00430       SMLNUM = SAFMIN / EPS
00431       BIGNUM = ONE / SMLNUM
00432       RMIN = SQRT( SMLNUM )
00433       RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
00434 *
00435       IF( INFO.EQ.0 ) THEN
00436          WORK( 1 ) = LWMIN
00437          IWORK( 1 ) = LIWMIN
00438 *
00439          IF( WANTZ .AND. ALLEIG ) THEN
00440             NZCMIN = N
00441          ELSE IF( WANTZ .AND. VALEIG ) THEN
00442             CALL SLARRC( 'T', N, VL, VU, D, E, SAFMIN,
00443      $                            NZCMIN, ITMP, ITMP2, INFO )
00444          ELSE IF( WANTZ .AND. INDEIG ) THEN
00445             NZCMIN = IIU-IIL+1
00446          ELSE
00447 *           WANTZ .EQ. FALSE.
00448             NZCMIN = 0
00449          ENDIF
00450          IF( ZQUERY .AND. INFO.EQ.0 ) THEN
00451             Z( 1,1 ) = NZCMIN
00452          ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
00453             INFO = -14
00454          END IF
00455       END IF
00456 
00457       IF( INFO.NE.0 ) THEN
00458 *
00459          CALL XERBLA( 'SSTEMR', -INFO )
00460 *
00461          RETURN
00462       ELSE IF( LQUERY .OR. ZQUERY ) THEN
00463          RETURN
00464       END IF
00465 *
00466 *     Handle N = 0, 1, and 2 cases immediately
00467 *
00468       M = 0
00469       IF( N.EQ.0 )
00470      $   RETURN
00471 *
00472       IF( N.EQ.1 ) THEN
00473          IF( ALLEIG .OR. INDEIG ) THEN
00474             M = 1
00475             W( 1 ) = D( 1 )
00476          ELSE
00477             IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
00478                M = 1
00479                W( 1 ) = D( 1 )
00480             END IF
00481          END IF
00482          IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00483             Z( 1, 1 ) = ONE
00484             ISUPPZ(1) = 1
00485             ISUPPZ(2) = 1
00486          END IF
00487          RETURN
00488       END IF
00489 *
00490       IF( N.EQ.2 ) THEN
00491          IF( .NOT.WANTZ ) THEN
00492             CALL SLAE2( D(1), E(1), D(2), R1, R2 )
00493          ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00494             CALL SLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
00495          END IF
00496          IF( ALLEIG.OR.
00497      $      (VALEIG.AND.(R2.GT.WL).AND.
00498      $                  (R2.LE.WU)).OR.
00499      $      (INDEIG.AND.(IIL.EQ.1)) ) THEN
00500             M = M+1
00501             W( M ) = R2
00502             IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00503                Z( 1, M ) = -SN
00504                Z( 2, M ) = CS
00505 *              Note: At most one of SN and CS can be zero.
00506                IF (SN.NE.ZERO) THEN
00507                   IF (CS.NE.ZERO) THEN
00508                      ISUPPZ(2*M-1) = 1
00509                      ISUPPZ(2*M) = 2
00510                   ELSE
00511                      ISUPPZ(2*M-1) = 1
00512                      ISUPPZ(2*M) = 1
00513                   END IF
00514                ELSE
00515                   ISUPPZ(2*M-1) = 2
00516                   ISUPPZ(2*M) = 2
00517                END IF
00518             ENDIF
00519          ENDIF
00520          IF( ALLEIG.OR.
00521      $      (VALEIG.AND.(R1.GT.WL).AND.
00522      $                  (R1.LE.WU)).OR.
00523      $      (INDEIG.AND.(IIU.EQ.2)) ) THEN
00524             M = M+1
00525             W( M ) = R1
00526             IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00527                Z( 1, M ) = CS
00528                Z( 2, M ) = SN
00529 *              Note: At most one of SN and CS can be zero.
00530                IF (SN.NE.ZERO) THEN
00531                   IF (CS.NE.ZERO) THEN
00532                      ISUPPZ(2*M-1) = 1
00533                      ISUPPZ(2*M) = 2
00534                   ELSE
00535                      ISUPPZ(2*M-1) = 1
00536                      ISUPPZ(2*M) = 1
00537                   END IF
00538                ELSE
00539                   ISUPPZ(2*M-1) = 2
00540                   ISUPPZ(2*M) = 2
00541                END IF
00542             ENDIF
00543          ENDIF
00544          RETURN
00545       END IF
00546 
00547 *     Continue with general N
00548 
00549       INDGRS = 1
00550       INDERR = 2*N + 1
00551       INDGP = 3*N + 1
00552       INDD = 4*N + 1
00553       INDE2 = 5*N + 1
00554       INDWRK = 6*N + 1
00555 *
00556       IINSPL = 1
00557       IINDBL = N + 1
00558       IINDW = 2*N + 1
00559       IINDWK = 3*N + 1
00560 *
00561 *     Scale matrix to allowable range, if necessary.
00562 *     The allowable range is related to the PIVMIN parameter; see the
00563 *     comments in SLARRD.  The preference for scaling small values
00564 *     up is heuristic; we expect users' matrices not to be close to the
00565 *     RMAX threshold.
00566 *
00567       SCALE = ONE
00568       TNRM = SLANST( 'M', N, D, E )
00569       IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
00570          SCALE = RMIN / TNRM
00571       ELSE IF( TNRM.GT.RMAX ) THEN
00572          SCALE = RMAX / TNRM
00573       END IF
00574       IF( SCALE.NE.ONE ) THEN
00575          CALL SSCAL( N, SCALE, D, 1 )
00576          CALL SSCAL( N-1, SCALE, E, 1 )
00577          TNRM = TNRM*SCALE
00578          IF( VALEIG ) THEN
00579 *           If eigenvalues in interval have to be found,
00580 *           scale (WL, WU] accordingly
00581             WL = WL*SCALE
00582             WU = WU*SCALE
00583          ENDIF
00584       END IF
00585 *
00586 *     Compute the desired eigenvalues of the tridiagonal after splitting
00587 *     into smaller subblocks if the corresponding off-diagonal elements
00588 *     are small
00589 *     THRESH is the splitting parameter for SLARRE
00590 *     A negative THRESH forces the old splitting criterion based on the
00591 *     size of the off-diagonal. A positive THRESH switches to splitting
00592 *     which preserves relative accuracy.
00593 *
00594       IF( TRYRAC ) THEN
00595 *        Test whether the matrix warrants the more expensive relative approach.
00596          CALL SLARRR( N, D, E, IINFO )
00597       ELSE
00598 *        The user does not care about relative accurately eigenvalues
00599          IINFO = -1
00600       ENDIF
00601 *     Set the splitting criterion
00602       IF (IINFO.EQ.0) THEN
00603          THRESH = EPS
00604       ELSE
00605          THRESH = -EPS
00606 *        relative accuracy is desired but T does not guarantee it
00607          TRYRAC = .FALSE.
00608       ENDIF
00609 *
00610       IF( TRYRAC ) THEN
00611 *        Copy original diagonal, needed to guarantee relative accuracy
00612          CALL SCOPY(N,D,1,WORK(INDD),1)
00613       ENDIF
00614 *     Store the squares of the offdiagonal values of T
00615       DO 5 J = 1, N-1
00616          WORK( INDE2+J-1 ) = E(J)**2
00617  5    CONTINUE
00618 
00619 *     Set the tolerance parameters for bisection
00620       IF( .NOT.WANTZ ) THEN
00621 *        SLARRE computes the eigenvalues to full precision.
00622          RTOL1 = FOUR * EPS
00623          RTOL2 = FOUR * EPS
00624       ELSE
00625 *        SLARRE computes the eigenvalues to less than full precision.
00626 *        SLARRV will refine the eigenvalue approximations, and we can
00627 *        need less accurate initial bisection in SLARRE.
00628 *        Note: these settings do only affect the subset case and SLARRE
00629          RTOL1 = MAX( SQRT(EPS)*5.0E-2, FOUR * EPS )
00630          RTOL2 = MAX( SQRT(EPS)*5.0E-3, FOUR * EPS )
00631       ENDIF
00632       CALL SLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
00633      $             WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
00634      $             IWORK( IINSPL ), M, W, WORK( INDERR ),
00635      $             WORK( INDGP ), IWORK( IINDBL ),
00636      $             IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
00637      $             WORK( INDWRK ), IWORK( IINDWK ), IINFO )
00638       IF( IINFO.NE.0 ) THEN
00639          INFO = 10 + ABS( IINFO )
00640          RETURN
00641       END IF
00642 *     Note that if RANGE .NE. 'V', SLARRE computes bounds on the desired
00643 *     part of the spectrum. All desired eigenvalues are contained in
00644 *     (WL,WU]
00645 
00646 
00647       IF( WANTZ ) THEN
00648 *
00649 *        Compute the desired eigenvectors corresponding to the computed
00650 *        eigenvalues
00651 *
00652          CALL SLARRV( N, WL, WU, D, E,
00653      $                PIVMIN, IWORK( IINSPL ), M,
00654      $                1, M, MINRGP, RTOL1, RTOL2,
00655      $                W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
00656      $                IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
00657      $                ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
00658          IF( IINFO.NE.0 ) THEN
00659             INFO = 20 + ABS( IINFO )
00660             RETURN
00661          END IF
00662       ELSE
00663 *        SLARRE computes eigenvalues of the (shifted) root representation
00664 *        SLARRV returns the eigenvalues of the unshifted matrix.
00665 *        However, if the eigenvectors are not desired by the user, we need
00666 *        to apply the corresponding shifts from SLARRE to obtain the
00667 *        eigenvalues of the original matrix.
00668          DO 20 J = 1, M
00669             ITMP = IWORK( IINDBL+J-1 )
00670             W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
00671  20      CONTINUE
00672       END IF
00673 *
00674 
00675       IF ( TRYRAC ) THEN
00676 *        Refine computed eigenvalues so that they are relatively accurate
00677 *        with respect to the original matrix T.
00678          IBEGIN = 1
00679          WBEGIN = 1
00680          DO 39  JBLK = 1, IWORK( IINDBL+M-1 )
00681             IEND = IWORK( IINSPL+JBLK-1 )
00682             IN = IEND - IBEGIN + 1
00683             WEND = WBEGIN - 1
00684 *           check if any eigenvalues have to be refined in this block
00685  36         CONTINUE
00686             IF( WEND.LT.M ) THEN
00687                IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
00688                   WEND = WEND + 1
00689                   GO TO 36
00690                END IF
00691             END IF
00692             IF( WEND.LT.WBEGIN ) THEN
00693                IBEGIN = IEND + 1
00694                GO TO 39
00695             END IF
00696 
00697             OFFSET = IWORK(IINDW+WBEGIN-1)-1
00698             IFIRST = IWORK(IINDW+WBEGIN-1)
00699             ILAST = IWORK(IINDW+WEND-1)
00700             RTOL2 = FOUR * EPS
00701             CALL SLARRJ( IN,
00702      $                   WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
00703      $                   IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
00704      $                   WORK( INDERR+WBEGIN-1 ),
00705      $                   WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
00706      $                   TNRM, IINFO )
00707             IBEGIN = IEND + 1
00708             WBEGIN = WEND + 1
00709  39      CONTINUE
00710       ENDIF
00711 *
00712 *     If matrix was scaled, then rescale eigenvalues appropriately.
00713 *
00714       IF( SCALE.NE.ONE ) THEN
00715          CALL SSCAL( M, ONE / SCALE, W, 1 )
00716       END IF
00717 *
00718 *     If eigenvalues are not in increasing order, then sort them,
00719 *     possibly along with eigenvectors.
00720 *
00721       IF( NSPLIT.GT.1 ) THEN
00722          IF( .NOT. WANTZ ) THEN
00723             CALL SLASRT( 'I', M, W, IINFO )
00724             IF( IINFO.NE.0 ) THEN
00725                INFO = 3
00726                RETURN
00727             END IF
00728          ELSE
00729             DO 60 J = 1, M - 1
00730                I = 0
00731                TMP = W( J )
00732                DO 50 JJ = J + 1, M
00733                   IF( W( JJ ).LT.TMP ) THEN
00734                      I = JJ
00735                      TMP = W( JJ )
00736                   END IF
00737  50            CONTINUE
00738                IF( I.NE.0 ) THEN
00739                   W( I ) = W( J )
00740                   W( J ) = TMP
00741                   IF( WANTZ ) THEN
00742                      CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
00743                      ITMP = ISUPPZ( 2*I-1 )
00744                      ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
00745                      ISUPPZ( 2*J-1 ) = ITMP
00746                      ITMP = ISUPPZ( 2*I )
00747                      ISUPPZ( 2*I ) = ISUPPZ( 2*J )
00748                      ISUPPZ( 2*J ) = ITMP
00749                   END IF
00750                END IF
00751  60         CONTINUE
00752          END IF
00753       ENDIF
00754 *
00755 *
00756       WORK( 1 ) = LWMIN
00757       IWORK( 1 ) = LIWMIN
00758       RETURN
00759 *
00760 *     End of SSTEMR
00761 *
00762       END
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