LAPACK 3.3.0

cstemr.f

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00001       SUBROUTINE CSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
00002      $                   M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
00003      $                   IWORK, LIWORK, INFO )
00004       IMPLICIT NONE
00005 *
00006 *  -- LAPACK computational routine (version 3.2.1)                                  --
00007 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00008 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00009 *  -- April 2009                                                      --
00010 *
00011 *     .. Scalar Arguments ..
00012       CHARACTER          JOBZ, RANGE
00013       LOGICAL            TRYRAC
00014       INTEGER            IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
00015       REAL             VL, VU
00016 *     ..
00017 *     .. Array Arguments ..
00018       INTEGER            ISUPPZ( * ), IWORK( * )
00019       REAL               D( * ), E( * ), W( * ), WORK( * )
00020       COMPLEX            Z( LDZ, * )
00021 *     ..
00022 *
00023 *  Purpose
00024 *  =======
00025 *
00026 *  CSTEMR computes selected eigenvalues and, optionally, eigenvectors
00027 *  of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
00028 *  a well defined set of pairwise different real eigenvalues, the corresponding
00029 *  real eigenvectors are pairwise orthogonal.
00030 *
00031 *  The spectrum may be computed either completely or partially by specifying
00032 *  either an interval (VL,VU] or a range of indices IL:IU for the desired
00033 *  eigenvalues.
00034 *
00035 *  Depending on the number of desired eigenvalues, these are computed either
00036 *  by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
00037 *  computed by the use of various suitable L D L^T factorizations near clusters
00038 *  of close eigenvalues (referred to as RRRs, Relatively Robust
00039 *  Representations). An informal sketch of the algorithm follows.
00040 *
00041 *  For each unreduced block (submatrix) of T,
00042 *     (a) Compute T - sigma I  = L D L^T, so that L and D
00043 *         define all the wanted eigenvalues to high relative accuracy.
00044 *         This means that small relative changes in the entries of D and L
00045 *         cause only small relative changes in the eigenvalues and
00046 *         eigenvectors. The standard (unfactored) representation of the
00047 *         tridiagonal matrix T does not have this property in general.
00048 *     (b) Compute the eigenvalues to suitable accuracy.
00049 *         If the eigenvectors are desired, the algorithm attains full
00050 *         accuracy of the computed eigenvalues only right before
00051 *         the corresponding vectors have to be computed, see steps c) and d).
00052 *     (c) For each cluster of close eigenvalues, select a new
00053 *         shift close to the cluster, find a new factorization, and refine
00054 *         the shifted eigenvalues to suitable accuracy.
00055 *     (d) For each eigenvalue with a large enough relative separation compute
00056 *         the corresponding eigenvector by forming a rank revealing twisted
00057 *         factorization. Go back to (c) for any clusters that remain.
00058 *
00059 *  For more details, see:
00060 *  - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
00061 *    to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
00062 *    Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
00063 *  - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
00064 *    Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
00065 *    2004.  Also LAPACK Working Note 154.
00066 *  - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
00067 *    tridiagonal eigenvalue/eigenvector problem",
00068 *    Computer Science Division Technical Report No. UCB/CSD-97-971,
00069 *    UC Berkeley, May 1997.
00070 *
00071 *  Further Details
00072 *  1.CSTEMR works only on machines which follow IEEE-754
00073 *  floating-point standard in their handling of infinities and NaNs.
00074 *  This permits the use of efficient inner loops avoiding a check for
00075 *  zero divisors.
00076 *
00077 *  2. LAPACK routines can be used to reduce a complex Hermitean matrix to
00078 *  real symmetric tridiagonal form.
00079 *
00080 *  (Any complex Hermitean tridiagonal matrix has real values on its diagonal
00081 *  and potentially complex numbers on its off-diagonals. By applying a
00082 *  similarity transform with an appropriate diagonal matrix
00083 *  diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean
00084 *  matrix can be transformed into a real symmetric matrix and complex
00085 *  arithmetic can be entirely avoided.)
00086 *
00087 *  While the eigenvectors of the real symmetric tridiagonal matrix are real,
00088 *  the eigenvectors of original complex Hermitean matrix have complex entries
00089 *  in general.
00090 *  Since LAPACK drivers overwrite the matrix data with the eigenvectors,
00091 *  CSTEMR accepts complex workspace to facilitate interoperability
00092 *  with CUNMTR or CUPMTR.
00093 *
00094 *  Arguments
00095 *  =========
00096 *
00097 *  JOBZ    (input) CHARACTER*1
00098 *          = 'N':  Compute eigenvalues only;
00099 *          = 'V':  Compute eigenvalues and eigenvectors.
00100 *
00101 *  RANGE   (input) CHARACTER*1
00102 *          = 'A': all eigenvalues will be found.
00103 *          = 'V': all eigenvalues in the half-open interval (VL,VU]
00104 *                 will be found.
00105 *          = 'I': the IL-th through IU-th eigenvalues will be found.
00106 *
00107 *  N       (input) INTEGER
00108 *          The order of the matrix.  N >= 0.
00109 *
00110 *  D       (input/output) REAL array, dimension (N)
00111 *          On entry, the N diagonal elements of the tridiagonal matrix
00112 *          T. On exit, D is overwritten.
00113 *
00114 *  E       (input/output) REAL array, dimension (N)
00115 *          On entry, the (N-1) subdiagonal elements of the tridiagonal
00116 *          matrix T in elements 1 to N-1 of E. E(N) need not be set on
00117 *          input, but is used internally as workspace.
00118 *          On exit, E is overwritten.
00119 *
00120 *  VL      (input) REAL
00121 *  VU      (input) REAL
00122 *          If RANGE='V', the lower and upper bounds of the interval to
00123 *          be searched for eigenvalues. VL < VU.
00124 *          Not referenced if RANGE = 'A' or 'I'.
00125 *
00126 *  IL      (input) INTEGER
00127 *  IU      (input) INTEGER
00128 *          If RANGE='I', the indices (in ascending order) of the
00129 *          smallest and largest eigenvalues to be returned.
00130 *          1 <= IL <= IU <= N, if N > 0.
00131 *          Not referenced if RANGE = 'A' or 'V'.
00132 *
00133 *  M       (output) INTEGER
00134 *          The total number of eigenvalues found.  0 <= M <= N.
00135 *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00136 *
00137 *  W       (output) REAL array, dimension (N)
00138 *          The first M elements contain the selected eigenvalues in
00139 *          ascending order.
00140 *
00141 *  Z       (output) COMPLEX array, dimension (LDZ, max(1,M) )
00142 *          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
00143 *          contain the orthonormal eigenvectors of the matrix T
00144 *          corresponding to the selected eigenvalues, with the i-th
00145 *          column of Z holding the eigenvector associated with W(i).
00146 *          If JOBZ = 'N', then Z is not referenced.
00147 *          Note: the user must ensure that at least max(1,M) columns are
00148 *          supplied in the array Z; if RANGE = 'V', the exact value of M
00149 *          is not known in advance and can be computed with a workspace
00150 *          query by setting NZC = -1, see below.
00151 *
00152 *  LDZ     (input) INTEGER
00153 *          The leading dimension of the array Z.  LDZ >= 1, and if
00154 *          JOBZ = 'V', then LDZ >= max(1,N).
00155 *
00156 *  NZC     (input) INTEGER
00157 *          The number of eigenvectors to be held in the array Z.
00158 *          If RANGE = 'A', then NZC >= max(1,N).
00159 *          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
00160 *          If RANGE = 'I', then NZC >= IU-IL+1.
00161 *          If NZC = -1, then a workspace query is assumed; the
00162 *          routine calculates the number of columns of the array Z that
00163 *          are needed to hold the eigenvectors.
00164 *          This value is returned as the first entry of the Z array, and
00165 *          no error message related to NZC is issued by XERBLA.
00166 *
00167 *  ISUPPZ  (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
00168 *          The support of the eigenvectors in Z, i.e., the indices
00169 *          indicating the nonzero elements in Z. The i-th computed eigenvector
00170 *          is nonzero only in elements ISUPPZ( 2*i-1 ) through
00171 *          ISUPPZ( 2*i ). This is relevant in the case when the matrix
00172 *          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
00173 *
00174 *  TRYRAC  (input/output) LOGICAL
00175 *          If TRYRAC.EQ..TRUE., indicates that the code should check whether
00176 *          the tridiagonal matrix defines its eigenvalues to high relative
00177 *          accuracy.  If so, the code uses relative-accuracy preserving
00178 *          algorithms that might be (a bit) slower depending on the matrix.
00179 *          If the matrix does not define its eigenvalues to high relative
00180 *          accuracy, the code can uses possibly faster algorithms.
00181 *          If TRYRAC.EQ..FALSE., the code is not required to guarantee
00182 *          relatively accurate eigenvalues and can use the fastest possible
00183 *          techniques.
00184 *          On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
00185 *          does not define its eigenvalues to high relative accuracy.
00186 *
00187 *  WORK    (workspace/output) REAL array, dimension (LWORK)
00188 *          On exit, if INFO = 0, WORK(1) returns the optimal
00189 *          (and minimal) LWORK.
00190 *
00191 *  LWORK   (input) INTEGER
00192 *          The dimension of the array WORK. LWORK >= max(1,18*N)
00193 *          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
00194 *          If LWORK = -1, then a workspace query is assumed; the routine
00195 *          only calculates the optimal size of the WORK array, returns
00196 *          this value as the first entry of the WORK array, and no error
00197 *          message related to LWORK is issued by XERBLA.
00198 *
00199 *  IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
00200 *          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
00201 *
00202 *  LIWORK  (input) INTEGER
00203 *          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
00204 *          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
00205 *          if only the eigenvalues are to be computed.
00206 *          If LIWORK = -1, then a workspace query is assumed; the
00207 *          routine only calculates the optimal size of the IWORK array,
00208 *          returns this value as the first entry of the IWORK array, and
00209 *          no error message related to LIWORK is issued by XERBLA.
00210 *
00211 *  INFO    (output) INTEGER
00212 *          On exit, INFO
00213 *          = 0:  successful exit
00214 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00215 *          > 0:  if INFO = 1X, internal error in SLARRE,
00216 *                if INFO = 2X, internal error in CLARRV.
00217 *                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
00218 *                the nonzero error code returned by SLARRE or
00219 *                CLARRV, respectively.
00220 *
00221 *
00222 *  Further Details
00223 *  ===============
00224 *
00225 *  Based on contributions by
00226 *     Beresford Parlett, University of California, Berkeley, USA
00227 *     Jim Demmel, University of California, Berkeley, USA
00228 *     Inderjit Dhillon, University of Texas, Austin, USA
00229 *     Osni Marques, LBNL/NERSC, USA
00230 *     Christof Voemel, University of California, Berkeley, USA
00231 *
00232 *  =====================================================================
00233 *
00234 *     .. Parameters ..
00235       REAL               ZERO, ONE, FOUR, MINRGP
00236       PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0,
00237      $                     FOUR = 4.0E0,
00238      $                     MINRGP = 3.0E-3 )
00239 *     ..
00240 *     .. Local Scalars ..
00241       LOGICAL            ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
00242       INTEGER            I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
00243      $                   IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
00244      $                   INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
00245      $                   ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
00246      $                   NZCMIN, OFFSET, WBEGIN, WEND
00247       REAL               BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
00248      $                   RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
00249      $                   THRESH, TMP, TNRM, WL, WU
00250 *     ..
00251 *     ..
00252 *     .. External Functions ..
00253       LOGICAL            LSAME
00254       REAL               SLAMCH, SLANST
00255       EXTERNAL           LSAME, SLAMCH, SLANST
00256 *     ..
00257 *     .. External Subroutines ..
00258       EXTERNAL           CLARRV, CSWAP, SCOPY, SLAE2, SLAEV2, SLARRC,
00259      $                   SLARRE, SLARRJ, SLARRR, SLASRT, SSCAL, XERBLA
00260 *     ..
00261 *     .. Intrinsic Functions ..
00262       INTRINSIC          MAX, MIN, SQRT
00263 
00264 
00265 *     ..
00266 *     .. Executable Statements ..
00267 *
00268 *     Test the input parameters.
00269 *
00270       WANTZ = LSAME( JOBZ, 'V' )
00271       ALLEIG = LSAME( RANGE, 'A' )
00272       VALEIG = LSAME( RANGE, 'V' )
00273       INDEIG = LSAME( RANGE, 'I' )
00274 *
00275       LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
00276       ZQUERY = ( NZC.EQ.-1 )
00277 
00278 *     SSTEMR needs WORK of size 6*N, IWORK of size 3*N.
00279 *     In addition, SLARRE needs WORK of size 6*N, IWORK of size 5*N.
00280 *     Furthermore, CLARRV needs WORK of size 12*N, IWORK of size 7*N.
00281       IF( WANTZ ) THEN
00282          LWMIN = 18*N
00283          LIWMIN = 10*N
00284       ELSE
00285 *        need less workspace if only the eigenvalues are wanted
00286          LWMIN = 12*N
00287          LIWMIN = 8*N
00288       ENDIF
00289 
00290       WL = ZERO
00291       WU = ZERO
00292       IIL = 0
00293       IIU = 0
00294 
00295       IF( VALEIG ) THEN
00296 *        We do not reference VL, VU in the cases RANGE = 'I','A'
00297 *        The interval (WL, WU] contains all the wanted eigenvalues.
00298 *        It is either given by the user or computed in SLARRE.
00299          WL = VL
00300          WU = VU
00301       ELSEIF( INDEIG ) THEN
00302 *        We do not reference IL, IU in the cases RANGE = 'V','A'
00303          IIL = IL
00304          IIU = IU
00305       ENDIF
00306 *
00307       INFO = 0
00308       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00309          INFO = -1
00310       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00311          INFO = -2
00312       ELSE IF( N.LT.0 ) THEN
00313          INFO = -3
00314       ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
00315          INFO = -7
00316       ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
00317          INFO = -8
00318       ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
00319          INFO = -9
00320       ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
00321          INFO = -13
00322       ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
00323          INFO = -17
00324       ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
00325          INFO = -19
00326       END IF
00327 *
00328 *     Get machine constants.
00329 *
00330       SAFMIN = SLAMCH( 'Safe minimum' )
00331       EPS = SLAMCH( 'Precision' )
00332       SMLNUM = SAFMIN / EPS
00333       BIGNUM = ONE / SMLNUM
00334       RMIN = SQRT( SMLNUM )
00335       RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
00336 *
00337       IF( INFO.EQ.0 ) THEN
00338          WORK( 1 ) = LWMIN
00339          IWORK( 1 ) = LIWMIN
00340 *
00341          IF( WANTZ .AND. ALLEIG ) THEN
00342             NZCMIN = N
00343          ELSE IF( WANTZ .AND. VALEIG ) THEN
00344             CALL SLARRC( 'T', N, VL, VU, D, E, SAFMIN,
00345      $                            NZCMIN, ITMP, ITMP2, INFO )
00346          ELSE IF( WANTZ .AND. INDEIG ) THEN
00347             NZCMIN = IIU-IIL+1
00348          ELSE
00349 *           WANTZ .EQ. FALSE.
00350             NZCMIN = 0
00351          ENDIF
00352          IF( ZQUERY .AND. INFO.EQ.0 ) THEN
00353             Z( 1,1 ) = NZCMIN
00354          ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
00355             INFO = -14
00356          END IF
00357       END IF
00358 
00359       IF( INFO.NE.0 ) THEN
00360 *
00361          CALL XERBLA( 'CSTEMR', -INFO )
00362 *
00363          RETURN
00364       ELSE IF( LQUERY .OR. ZQUERY ) THEN
00365          RETURN
00366       END IF
00367 *
00368 *     Handle N = 0, 1, and 2 cases immediately
00369 *
00370       M = 0
00371       IF( N.EQ.0 )
00372      $   RETURN
00373 *
00374       IF( N.EQ.1 ) THEN
00375          IF( ALLEIG .OR. INDEIG ) THEN
00376             M = 1
00377             W( 1 ) = D( 1 )
00378          ELSE
00379             IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
00380                M = 1
00381                W( 1 ) = D( 1 )
00382             END IF
00383          END IF
00384          IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00385             Z( 1, 1 ) = ONE
00386             ISUPPZ(1) = 1
00387             ISUPPZ(2) = 1
00388          END IF
00389          RETURN
00390       END IF
00391 *
00392       IF( N.EQ.2 ) THEN
00393          IF( .NOT.WANTZ ) THEN
00394             CALL SLAE2( D(1), E(1), D(2), R1, R2 )
00395          ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00396             CALL SLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
00397          END IF
00398          IF( ALLEIG.OR.
00399      $      (VALEIG.AND.(R2.GT.WL).AND.
00400      $                  (R2.LE.WU)).OR.
00401      $      (INDEIG.AND.(IIL.EQ.1)) ) THEN
00402             M = M+1
00403             W( M ) = R2
00404             IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00405                Z( 1, M ) = -SN
00406                Z( 2, M ) = CS
00407 *              Note: At most one of SN and CS can be zero.
00408                IF (SN.NE.ZERO) THEN
00409                   IF (CS.NE.ZERO) THEN
00410                      ISUPPZ(2*M-1) = 1
00411                      ISUPPZ(2*M-1) = 2
00412                   ELSE
00413                      ISUPPZ(2*M-1) = 1
00414                      ISUPPZ(2*M-1) = 1
00415                   END IF
00416                ELSE
00417                   ISUPPZ(2*M-1) = 2
00418                   ISUPPZ(2*M) = 2
00419                END IF
00420             ENDIF
00421          ENDIF
00422          IF( ALLEIG.OR.
00423      $      (VALEIG.AND.(R1.GT.WL).AND.
00424      $                  (R1.LE.WU)).OR.
00425      $      (INDEIG.AND.(IIU.EQ.2)) ) THEN
00426             M = M+1
00427             W( M ) = R1
00428             IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00429                Z( 1, M ) = CS
00430                Z( 2, M ) = SN
00431 *              Note: At most one of SN and CS can be zero.
00432                IF (SN.NE.ZERO) THEN
00433                   IF (CS.NE.ZERO) THEN
00434                      ISUPPZ(2*M-1) = 1
00435                      ISUPPZ(2*M-1) = 2
00436                   ELSE
00437                      ISUPPZ(2*M-1) = 1
00438                      ISUPPZ(2*M-1) = 1
00439                   END IF
00440                ELSE
00441                   ISUPPZ(2*M-1) = 2
00442                   ISUPPZ(2*M) = 2
00443                END IF
00444             ENDIF
00445          ENDIF
00446          RETURN
00447       END IF
00448 
00449 *     Continue with general N
00450 
00451       INDGRS = 1
00452       INDERR = 2*N + 1
00453       INDGP = 3*N + 1
00454       INDD = 4*N + 1
00455       INDE2 = 5*N + 1
00456       INDWRK = 6*N + 1
00457 *
00458       IINSPL = 1
00459       IINDBL = N + 1
00460       IINDW = 2*N + 1
00461       IINDWK = 3*N + 1
00462 *
00463 *     Scale matrix to allowable range, if necessary.
00464 *     The allowable range is related to the PIVMIN parameter; see the
00465 *     comments in SLARRD.  The preference for scaling small values
00466 *     up is heuristic; we expect users' matrices not to be close to the
00467 *     RMAX threshold.
00468 *
00469       SCALE = ONE
00470       TNRM = SLANST( 'M', N, D, E )
00471       IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
00472          SCALE = RMIN / TNRM
00473       ELSE IF( TNRM.GT.RMAX ) THEN
00474          SCALE = RMAX / TNRM
00475       END IF
00476       IF( SCALE.NE.ONE ) THEN
00477          CALL SSCAL( N, SCALE, D, 1 )
00478          CALL SSCAL( N-1, SCALE, E, 1 )
00479          TNRM = TNRM*SCALE
00480          IF( VALEIG ) THEN
00481 *           If eigenvalues in interval have to be found,
00482 *           scale (WL, WU] accordingly
00483             WL = WL*SCALE
00484             WU = WU*SCALE
00485          ENDIF
00486       END IF
00487 *
00488 *     Compute the desired eigenvalues of the tridiagonal after splitting
00489 *     into smaller subblocks if the corresponding off-diagonal elements
00490 *     are small
00491 *     THRESH is the splitting parameter for SLARRE
00492 *     A negative THRESH forces the old splitting criterion based on the
00493 *     size of the off-diagonal. A positive THRESH switches to splitting
00494 *     which preserves relative accuracy.
00495 *
00496       IF( TRYRAC ) THEN
00497 *        Test whether the matrix warrants the more expensive relative approach.
00498          CALL SLARRR( N, D, E, IINFO )
00499       ELSE
00500 *        The user does not care about relative accurately eigenvalues
00501          IINFO = -1
00502       ENDIF
00503 *     Set the splitting criterion
00504       IF (IINFO.EQ.0) THEN
00505          THRESH = EPS
00506       ELSE
00507          THRESH = -EPS
00508 *        relative accuracy is desired but T does not guarantee it
00509          TRYRAC = .FALSE.
00510       ENDIF
00511 *
00512       IF( TRYRAC ) THEN
00513 *        Copy original diagonal, needed to guarantee relative accuracy
00514          CALL SCOPY(N,D,1,WORK(INDD),1)
00515       ENDIF
00516 *     Store the squares of the offdiagonal values of T
00517       DO 5 J = 1, N-1
00518          WORK( INDE2+J-1 ) = E(J)**2
00519  5    CONTINUE
00520 
00521 *     Set the tolerance parameters for bisection
00522       IF( .NOT.WANTZ ) THEN
00523 *        SLARRE computes the eigenvalues to full precision.
00524          RTOL1 = FOUR * EPS
00525          RTOL2 = FOUR * EPS
00526       ELSE
00527 *        SLARRE computes the eigenvalues to less than full precision.
00528 *        CLARRV will refine the eigenvalue approximations, and we only
00529 *        need less accurate initial bisection in SLARRE.
00530 *        Note: these settings do only affect the subset case and SLARRE
00531          RTOL1 = MAX( SQRT(EPS)*5.0E-2, FOUR * EPS )
00532          RTOL2 = MAX( SQRT(EPS)*5.0E-3, FOUR * EPS )
00533       ENDIF
00534       CALL SLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
00535      $             WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
00536      $             IWORK( IINSPL ), M, W, WORK( INDERR ),
00537      $             WORK( INDGP ), IWORK( IINDBL ),
00538      $             IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
00539      $             WORK( INDWRK ), IWORK( IINDWK ), IINFO )
00540       IF( IINFO.NE.0 ) THEN
00541          INFO = 10 + ABS( IINFO )
00542          RETURN
00543       END IF
00544 *     Note that if RANGE .NE. 'V', SLARRE computes bounds on the desired
00545 *     part of the spectrum. All desired eigenvalues are contained in
00546 *     (WL,WU]
00547 
00548 
00549       IF( WANTZ ) THEN
00550 *
00551 *        Compute the desired eigenvectors corresponding to the computed
00552 *        eigenvalues
00553 *
00554          CALL CLARRV( N, WL, WU, D, E,
00555      $                PIVMIN, IWORK( IINSPL ), M,
00556      $                1, M, MINRGP, RTOL1, RTOL2,
00557      $                W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
00558      $                IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
00559      $                ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
00560          IF( IINFO.NE.0 ) THEN
00561             INFO = 20 + ABS( IINFO )
00562             RETURN
00563          END IF
00564       ELSE
00565 *        SLARRE computes eigenvalues of the (shifted) root representation
00566 *        CLARRV returns the eigenvalues of the unshifted matrix.
00567 *        However, if the eigenvectors are not desired by the user, we need
00568 *        to apply the corresponding shifts from SLARRE to obtain the
00569 *        eigenvalues of the original matrix.
00570          DO 20 J = 1, M
00571             ITMP = IWORK( IINDBL+J-1 )
00572             W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
00573  20      CONTINUE
00574       END IF
00575 *
00576 
00577       IF ( TRYRAC ) THEN
00578 *        Refine computed eigenvalues so that they are relatively accurate
00579 *        with respect to the original matrix T.
00580          IBEGIN = 1
00581          WBEGIN = 1
00582          DO 39  JBLK = 1, IWORK( IINDBL+M-1 )
00583             IEND = IWORK( IINSPL+JBLK-1 )
00584             IN = IEND - IBEGIN + 1
00585             WEND = WBEGIN - 1
00586 *           check if any eigenvalues have to be refined in this block
00587  36         CONTINUE
00588             IF( WEND.LT.M ) THEN
00589                IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
00590                   WEND = WEND + 1
00591                   GO TO 36
00592                END IF
00593             END IF
00594             IF( WEND.LT.WBEGIN ) THEN
00595                IBEGIN = IEND + 1
00596                GO TO 39
00597             END IF
00598 
00599             OFFSET = IWORK(IINDW+WBEGIN-1)-1
00600             IFIRST = IWORK(IINDW+WBEGIN-1)
00601             ILAST = IWORK(IINDW+WEND-1)
00602             RTOL2 = FOUR * EPS
00603             CALL SLARRJ( IN,
00604      $                   WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
00605      $                   IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
00606      $                   WORK( INDERR+WBEGIN-1 ),
00607      $                   WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
00608      $                   TNRM, IINFO )
00609             IBEGIN = IEND + 1
00610             WBEGIN = WEND + 1
00611  39      CONTINUE
00612       ENDIF
00613 *
00614 *     If matrix was scaled, then rescale eigenvalues appropriately.
00615 *
00616       IF( SCALE.NE.ONE ) THEN
00617          CALL SSCAL( M, ONE / SCALE, W, 1 )
00618       END IF
00619 *
00620 *     If eigenvalues are not in increasing order, then sort them,
00621 *     possibly along with eigenvectors.
00622 *
00623       IF( NSPLIT.GT.1 ) THEN
00624          IF( .NOT. WANTZ ) THEN
00625             CALL SLASRT( 'I', M, W, IINFO )
00626             IF( IINFO.NE.0 ) THEN
00627                INFO = 3
00628                RETURN
00629             END IF
00630          ELSE
00631             DO 60 J = 1, M - 1
00632                I = 0
00633                TMP = W( J )
00634                DO 50 JJ = J + 1, M
00635                   IF( W( JJ ).LT.TMP ) THEN
00636                      I = JJ
00637                      TMP = W( JJ )
00638                   END IF
00639  50            CONTINUE
00640                IF( I.NE.0 ) THEN
00641                   W( I ) = W( J )
00642                   W( J ) = TMP
00643                   IF( WANTZ ) THEN
00644                      CALL CSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
00645                      ITMP = ISUPPZ( 2*I-1 )
00646                      ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
00647                      ISUPPZ( 2*J-1 ) = ITMP
00648                      ITMP = ISUPPZ( 2*I )
00649                      ISUPPZ( 2*I ) = ISUPPZ( 2*J )
00650                      ISUPPZ( 2*J ) = ITMP
00651                   END IF
00652                END IF
00653  60         CONTINUE
00654          END IF
00655       ENDIF
00656 *
00657 *
00658       WORK( 1 ) = LWMIN
00659       IWORK( 1 ) = LIWMIN
00660       RETURN
00661 *
00662 *     End of CSTEMR
00663 *
00664       END
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