LAPACK 3.3.0

dstemr.f

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00001       SUBROUTINE DSTEMR( 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 3.2.2)                                  --
00007 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00008 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00009 *  -- June 2010                                                       --
00010 *
00011 *     .. Scalar Arguments ..
00012       CHARACTER          JOBZ, RANGE
00013       LOGICAL            TRYRAC
00014       INTEGER            IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
00015       DOUBLE PRECISION VL, VU
00016 *     ..
00017 *     .. Array Arguments ..
00018       INTEGER            ISUPPZ( * ), IWORK( * )
00019       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * )
00020       DOUBLE PRECISION   Z( LDZ, * )
00021 *     ..
00022 *
00023 *  Purpose
00024 *  =======
00025 *
00026 *  DSTEMR 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.DSTEMR 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 *  Arguments
00078 *  =========
00079 *
00080 *  JOBZ    (input) CHARACTER*1
00081 *          = 'N':  Compute eigenvalues only;
00082 *          = 'V':  Compute eigenvalues and eigenvectors.
00083 *
00084 *  RANGE   (input) CHARACTER*1
00085 *          = 'A': all eigenvalues will be found.
00086 *          = 'V': all eigenvalues in the half-open interval (VL,VU]
00087 *                 will be found.
00088 *          = 'I': the IL-th through IU-th eigenvalues will be found.
00089 *
00090 *  N       (input) INTEGER
00091 *          The order of the matrix.  N >= 0.
00092 *
00093 *  D       (input/output) DOUBLE PRECISION array, dimension (N)
00094 *          On entry, the N diagonal elements of the tridiagonal matrix
00095 *          T. On exit, D is overwritten.
00096 *
00097 *  E       (input/output) DOUBLE PRECISION array, dimension (N)
00098 *          On entry, the (N-1) subdiagonal elements of the tridiagonal
00099 *          matrix T in elements 1 to N-1 of E. E(N) need not be set on
00100 *          input, but is used internally as workspace.
00101 *          On exit, E is overwritten.
00102 *
00103 *  VL      (input) DOUBLE PRECISION
00104 *  VU      (input) DOUBLE PRECISION
00105 *          If RANGE='V', the lower and upper bounds of the interval to
00106 *          be searched for eigenvalues. VL < VU.
00107 *          Not referenced if RANGE = 'A' or 'I'.
00108 *
00109 *  IL      (input) INTEGER
00110 *  IU      (input) INTEGER
00111 *          If RANGE='I', the indices (in ascending order) of the
00112 *          smallest and largest eigenvalues to be returned.
00113 *          1 <= IL <= IU <= N, if N > 0.
00114 *          Not referenced if RANGE = 'A' or 'V'.
00115 *
00116 *  M       (output) INTEGER
00117 *          The total number of eigenvalues found.  0 <= M <= N.
00118 *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00119 *
00120 *  W       (output) DOUBLE PRECISION array, dimension (N)
00121 *          The first M elements contain the selected eigenvalues in
00122 *          ascending order.
00123 *
00124 *  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
00125 *          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
00126 *          contain the orthonormal eigenvectors of the matrix T
00127 *          corresponding to the selected eigenvalues, with the i-th
00128 *          column of Z holding the eigenvector associated with W(i).
00129 *          If JOBZ = 'N', then Z is not referenced.
00130 *          Note: the user must ensure that at least max(1,M) columns are
00131 *          supplied in the array Z; if RANGE = 'V', the exact value of M
00132 *          is not known in advance and can be computed with a workspace
00133 *          query by setting NZC = -1, see below.
00134 *
00135 *  LDZ     (input) INTEGER
00136 *          The leading dimension of the array Z.  LDZ >= 1, and if
00137 *          JOBZ = 'V', then LDZ >= max(1,N).
00138 *
00139 *  NZC     (input) INTEGER
00140 *          The number of eigenvectors to be held in the array Z.
00141 *          If RANGE = 'A', then NZC >= max(1,N).
00142 *          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
00143 *          If RANGE = 'I', then NZC >= IU-IL+1.
00144 *          If NZC = -1, then a workspace query is assumed; the
00145 *          routine calculates the number of columns of the array Z that
00146 *          are needed to hold the eigenvectors.
00147 *          This value is returned as the first entry of the Z array, and
00148 *          no error message related to NZC is issued by XERBLA.
00149 *
00150 *  ISUPPZ  (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
00151 *          The support of the eigenvectors in Z, i.e., the indices
00152 *          indicating the nonzero elements in Z. The i-th computed eigenvector
00153 *          is nonzero only in elements ISUPPZ( 2*i-1 ) through
00154 *          ISUPPZ( 2*i ). This is relevant in the case when the matrix
00155 *          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
00156 *
00157 *  TRYRAC  (input/output) LOGICAL
00158 *          If TRYRAC.EQ..TRUE., indicates that the code should check whether
00159 *          the tridiagonal matrix defines its eigenvalues to high relative
00160 *          accuracy.  If so, the code uses relative-accuracy preserving
00161 *          algorithms that might be (a bit) slower depending on the matrix.
00162 *          If the matrix does not define its eigenvalues to high relative
00163 *          accuracy, the code can uses possibly faster algorithms.
00164 *          If TRYRAC.EQ..FALSE., the code is not required to guarantee
00165 *          relatively accurate eigenvalues and can use the fastest possible
00166 *          techniques.
00167 *          On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
00168 *          does not define its eigenvalues to high relative accuracy.
00169 *
00170 *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
00171 *          On exit, if INFO = 0, WORK(1) returns the optimal
00172 *          (and minimal) LWORK.
00173 *
00174 *  LWORK   (input) INTEGER
00175 *          The dimension of the array WORK. LWORK >= max(1,18*N)
00176 *          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
00177 *          If LWORK = -1, then a workspace query is assumed; the routine
00178 *          only calculates the optimal size of the WORK array, returns
00179 *          this value as the first entry of the WORK array, and no error
00180 *          message related to LWORK is issued by XERBLA.
00181 *
00182 *  IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
00183 *          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
00184 *
00185 *  LIWORK  (input) INTEGER
00186 *          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
00187 *          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
00188 *          if only the eigenvalues are to be computed.
00189 *          If LIWORK = -1, then a workspace query is assumed; the
00190 *          routine only calculates the optimal size of the IWORK array,
00191 *          returns this value as the first entry of the IWORK array, and
00192 *          no error message related to LIWORK is issued by XERBLA.
00193 *
00194 *  INFO    (output) INTEGER
00195 *          On exit, INFO
00196 *          = 0:  successful exit
00197 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00198 *          > 0:  if INFO = 1X, internal error in DLARRE,
00199 *                if INFO = 2X, internal error in DLARRV.
00200 *                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
00201 *                the nonzero error code returned by DLARRE or
00202 *                DLARRV, respectively.
00203 *
00204 *
00205 *  Further Details
00206 *  ===============
00207 *
00208 *  Based on contributions by
00209 *     Beresford Parlett, University of California, Berkeley, USA
00210 *     Jim Demmel, University of California, Berkeley, USA
00211 *     Inderjit Dhillon, University of Texas, Austin, USA
00212 *     Osni Marques, LBNL/NERSC, USA
00213 *     Christof Voemel, University of California, Berkeley, USA
00214 *
00215 *  =====================================================================
00216 *
00217 *     .. Parameters ..
00218       DOUBLE PRECISION   ZERO, ONE, FOUR, MINRGP
00219       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0,
00220      $                     FOUR = 4.0D0,
00221      $                     MINRGP = 1.0D-3 )
00222 *     ..
00223 *     .. Local Scalars ..
00224       LOGICAL            ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
00225       INTEGER            I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
00226      $                   IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
00227      $                   INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
00228      $                   ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
00229      $                   NZCMIN, OFFSET, WBEGIN, WEND
00230       DOUBLE PRECISION   BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
00231      $                   RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
00232      $                   THRESH, TMP, TNRM, WL, WU
00233 *     ..
00234 *     ..
00235 *     .. External Functions ..
00236       LOGICAL            LSAME
00237       DOUBLE PRECISION   DLAMCH, DLANST
00238       EXTERNAL           LSAME, DLAMCH, DLANST
00239 *     ..
00240 *     .. External Subroutines ..
00241       EXTERNAL           DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ,
00242      $                   DLARRR, DLARRV, DLASRT, DSCAL, DSWAP, XERBLA
00243 *     ..
00244 *     .. Intrinsic Functions ..
00245       INTRINSIC          MAX, MIN, SQRT
00246 
00247 
00248 *     ..
00249 *     .. Executable Statements ..
00250 *
00251 *     Test the input parameters.
00252 *
00253       WANTZ = LSAME( JOBZ, 'V' )
00254       ALLEIG = LSAME( RANGE, 'A' )
00255       VALEIG = LSAME( RANGE, 'V' )
00256       INDEIG = LSAME( RANGE, 'I' )
00257 *
00258       LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
00259       ZQUERY = ( NZC.EQ.-1 )
00260 
00261 *     DSTEMR needs WORK of size 6*N, IWORK of size 3*N.
00262 *     In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N.
00263 *     Furthermore, DLARRV needs WORK of size 12*N, IWORK of size 7*N.
00264       IF( WANTZ ) THEN
00265          LWMIN = 18*N
00266          LIWMIN = 10*N
00267       ELSE
00268 *        need less workspace if only the eigenvalues are wanted
00269          LWMIN = 12*N
00270          LIWMIN = 8*N
00271       ENDIF
00272 
00273       WL = ZERO
00274       WU = ZERO
00275       IIL = 0
00276       IIU = 0
00277 
00278       IF( VALEIG ) THEN
00279 *        We do not reference VL, VU in the cases RANGE = 'I','A'
00280 *        The interval (WL, WU] contains all the wanted eigenvalues.
00281 *        It is either given by the user or computed in DLARRE.
00282          WL = VL
00283          WU = VU
00284       ELSEIF( INDEIG ) THEN
00285 *        We do not reference IL, IU in the cases RANGE = 'V','A'
00286          IIL = IL
00287          IIU = IU
00288       ENDIF
00289 *
00290       INFO = 0
00291       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00292          INFO = -1
00293       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00294          INFO = -2
00295       ELSE IF( N.LT.0 ) THEN
00296          INFO = -3
00297       ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
00298          INFO = -7
00299       ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
00300          INFO = -8
00301       ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
00302          INFO = -9
00303       ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
00304          INFO = -13
00305       ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
00306          INFO = -17
00307       ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
00308          INFO = -19
00309       END IF
00310 *
00311 *     Get machine constants.
00312 *
00313       SAFMIN = DLAMCH( 'Safe minimum' )
00314       EPS = DLAMCH( 'Precision' )
00315       SMLNUM = SAFMIN / EPS
00316       BIGNUM = ONE / SMLNUM
00317       RMIN = SQRT( SMLNUM )
00318       RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
00319 *
00320       IF( INFO.EQ.0 ) THEN
00321          WORK( 1 ) = LWMIN
00322          IWORK( 1 ) = LIWMIN
00323 *
00324          IF( WANTZ .AND. ALLEIG ) THEN
00325             NZCMIN = N
00326          ELSE IF( WANTZ .AND. VALEIG ) THEN
00327             CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN,
00328      $                            NZCMIN, ITMP, ITMP2, INFO )
00329          ELSE IF( WANTZ .AND. INDEIG ) THEN
00330             NZCMIN = IIU-IIL+1
00331          ELSE
00332 *           WANTZ .EQ. FALSE.
00333             NZCMIN = 0
00334          ENDIF
00335          IF( ZQUERY .AND. INFO.EQ.0 ) THEN
00336             Z( 1,1 ) = NZCMIN
00337          ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
00338             INFO = -14
00339          END IF
00340       END IF
00341 
00342       IF( INFO.NE.0 ) THEN
00343 *
00344          CALL XERBLA( 'DSTEMR', -INFO )
00345 *
00346          RETURN
00347       ELSE IF( LQUERY .OR. ZQUERY ) THEN
00348          RETURN
00349       END IF
00350 *
00351 *     Handle N = 0, 1, and 2 cases immediately
00352 *
00353       M = 0
00354       IF( N.EQ.0 )
00355      $   RETURN
00356 *
00357       IF( N.EQ.1 ) THEN
00358          IF( ALLEIG .OR. INDEIG ) THEN
00359             M = 1
00360             W( 1 ) = D( 1 )
00361          ELSE
00362             IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
00363                M = 1
00364                W( 1 ) = D( 1 )
00365             END IF
00366          END IF
00367          IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00368             Z( 1, 1 ) = ONE
00369             ISUPPZ(1) = 1
00370             ISUPPZ(2) = 1
00371          END IF
00372          RETURN
00373       END IF
00374 *
00375       IF( N.EQ.2 ) THEN
00376          IF( .NOT.WANTZ ) THEN
00377             CALL DLAE2( D(1), E(1), D(2), R1, R2 )
00378          ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00379             CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
00380          END IF
00381          IF( ALLEIG.OR.
00382      $      (VALEIG.AND.(R2.GT.WL).AND.
00383      $                  (R2.LE.WU)).OR.
00384      $      (INDEIG.AND.(IIL.EQ.1)) ) THEN
00385             M = M+1
00386             W( M ) = R2
00387             IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00388                Z( 1, M ) = -SN
00389                Z( 2, M ) = CS
00390 *              Note: At most one of SN and CS can be zero.
00391                IF (SN.NE.ZERO) THEN
00392                   IF (CS.NE.ZERO) THEN
00393                      ISUPPZ(2*M-1) = 1
00394                      ISUPPZ(2*M) = 2
00395                   ELSE
00396                      ISUPPZ(2*M-1) = 1
00397                      ISUPPZ(2*M) = 1
00398                   END IF
00399                ELSE
00400                   ISUPPZ(2*M-1) = 2
00401                   ISUPPZ(2*M) = 2
00402                END IF
00403             ENDIF
00404          ENDIF
00405          IF( ALLEIG.OR.
00406      $      (VALEIG.AND.(R1.GT.WL).AND.
00407      $                  (R1.LE.WU)).OR.
00408      $      (INDEIG.AND.(IIU.EQ.2)) ) THEN
00409             M = M+1
00410             W( M ) = R1
00411             IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
00412                Z( 1, M ) = CS
00413                Z( 2, M ) = SN
00414 *              Note: At most one of SN and CS can be zero.
00415                IF (SN.NE.ZERO) THEN
00416                   IF (CS.NE.ZERO) THEN
00417                      ISUPPZ(2*M-1) = 1
00418                      ISUPPZ(2*M) = 2
00419                   ELSE
00420                      ISUPPZ(2*M-1) = 1
00421                      ISUPPZ(2*M) = 1
00422                   END IF
00423                ELSE
00424                   ISUPPZ(2*M-1) = 2
00425                   ISUPPZ(2*M) = 2
00426                END IF
00427             ENDIF
00428          ENDIF
00429          RETURN
00430       END IF
00431 
00432 *     Continue with general N
00433 
00434       INDGRS = 1
00435       INDERR = 2*N + 1
00436       INDGP = 3*N + 1
00437       INDD = 4*N + 1
00438       INDE2 = 5*N + 1
00439       INDWRK = 6*N + 1
00440 *
00441       IINSPL = 1
00442       IINDBL = N + 1
00443       IINDW = 2*N + 1
00444       IINDWK = 3*N + 1
00445 *
00446 *     Scale matrix to allowable range, if necessary.
00447 *     The allowable range is related to the PIVMIN parameter; see the
00448 *     comments in DLARRD.  The preference for scaling small values
00449 *     up is heuristic; we expect users' matrices not to be close to the
00450 *     RMAX threshold.
00451 *
00452       SCALE = ONE
00453       TNRM = DLANST( 'M', N, D, E )
00454       IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
00455          SCALE = RMIN / TNRM
00456       ELSE IF( TNRM.GT.RMAX ) THEN
00457          SCALE = RMAX / TNRM
00458       END IF
00459       IF( SCALE.NE.ONE ) THEN
00460          CALL DSCAL( N, SCALE, D, 1 )
00461          CALL DSCAL( N-1, SCALE, E, 1 )
00462          TNRM = TNRM*SCALE
00463          IF( VALEIG ) THEN
00464 *           If eigenvalues in interval have to be found,
00465 *           scale (WL, WU] accordingly
00466             WL = WL*SCALE
00467             WU = WU*SCALE
00468          ENDIF
00469       END IF
00470 *
00471 *     Compute the desired eigenvalues of the tridiagonal after splitting
00472 *     into smaller subblocks if the corresponding off-diagonal elements
00473 *     are small
00474 *     THRESH is the splitting parameter for DLARRE
00475 *     A negative THRESH forces the old splitting criterion based on the
00476 *     size of the off-diagonal. A positive THRESH switches to splitting
00477 *     which preserves relative accuracy.
00478 *
00479       IF( TRYRAC ) THEN
00480 *        Test whether the matrix warrants the more expensive relative approach.
00481          CALL DLARRR( N, D, E, IINFO )
00482       ELSE
00483 *        The user does not care about relative accurately eigenvalues
00484          IINFO = -1
00485       ENDIF
00486 *     Set the splitting criterion
00487       IF (IINFO.EQ.0) THEN
00488          THRESH = EPS
00489       ELSE
00490          THRESH = -EPS
00491 *        relative accuracy is desired but T does not guarantee it
00492          TRYRAC = .FALSE.
00493       ENDIF
00494 *
00495       IF( TRYRAC ) THEN
00496 *        Copy original diagonal, needed to guarantee relative accuracy
00497          CALL DCOPY(N,D,1,WORK(INDD),1)
00498       ENDIF
00499 *     Store the squares of the offdiagonal values of T
00500       DO 5 J = 1, N-1
00501          WORK( INDE2+J-1 ) = E(J)**2
00502  5    CONTINUE
00503 
00504 *     Set the tolerance parameters for bisection
00505       IF( .NOT.WANTZ ) THEN
00506 *        DLARRE computes the eigenvalues to full precision.
00507          RTOL1 = FOUR * EPS
00508          RTOL2 = FOUR * EPS
00509       ELSE
00510 *        DLARRE computes the eigenvalues to less than full precision.
00511 *        DLARRV will refine the eigenvalue approximations, and we can
00512 *        need less accurate initial bisection in DLARRE.
00513 *        Note: these settings do only affect the subset case and DLARRE
00514          RTOL1 = SQRT(EPS)
00515          RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
00516       ENDIF
00517       CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
00518      $             WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
00519      $             IWORK( IINSPL ), M, W, WORK( INDERR ),
00520      $             WORK( INDGP ), IWORK( IINDBL ),
00521      $             IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
00522      $             WORK( INDWRK ), IWORK( IINDWK ), IINFO )
00523       IF( IINFO.NE.0 ) THEN
00524          INFO = 10 + ABS( IINFO )
00525          RETURN
00526       END IF
00527 *     Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
00528 *     part of the spectrum. All desired eigenvalues are contained in
00529 *     (WL,WU]
00530 
00531 
00532       IF( WANTZ ) THEN
00533 *
00534 *        Compute the desired eigenvectors corresponding to the computed
00535 *        eigenvalues
00536 *
00537          CALL DLARRV( N, WL, WU, D, E,
00538      $                PIVMIN, IWORK( IINSPL ), M,
00539      $                1, M, MINRGP, RTOL1, RTOL2,
00540      $                W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
00541      $                IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
00542      $                ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
00543          IF( IINFO.NE.0 ) THEN
00544             INFO = 20 + ABS( IINFO )
00545             RETURN
00546          END IF
00547       ELSE
00548 *        DLARRE computes eigenvalues of the (shifted) root representation
00549 *        DLARRV returns the eigenvalues of the unshifted matrix.
00550 *        However, if the eigenvectors are not desired by the user, we need
00551 *        to apply the corresponding shifts from DLARRE to obtain the
00552 *        eigenvalues of the original matrix.
00553          DO 20 J = 1, M
00554             ITMP = IWORK( IINDBL+J-1 )
00555             W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
00556  20      CONTINUE
00557       END IF
00558 *
00559 
00560       IF ( TRYRAC ) THEN
00561 *        Refine computed eigenvalues so that they are relatively accurate
00562 *        with respect to the original matrix T.
00563          IBEGIN = 1
00564          WBEGIN = 1
00565          DO 39  JBLK = 1, IWORK( IINDBL+M-1 )
00566             IEND = IWORK( IINSPL+JBLK-1 )
00567             IN = IEND - IBEGIN + 1
00568             WEND = WBEGIN - 1
00569 *           check if any eigenvalues have to be refined in this block
00570  36         CONTINUE
00571             IF( WEND.LT.M ) THEN
00572                IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
00573                   WEND = WEND + 1
00574                   GO TO 36
00575                END IF
00576             END IF
00577             IF( WEND.LT.WBEGIN ) THEN
00578                IBEGIN = IEND + 1
00579                GO TO 39
00580             END IF
00581 
00582             OFFSET = IWORK(IINDW+WBEGIN-1)-1
00583             IFIRST = IWORK(IINDW+WBEGIN-1)
00584             ILAST = IWORK(IINDW+WEND-1)
00585             RTOL2 = FOUR * EPS
00586             CALL DLARRJ( IN,
00587      $                   WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
00588      $                   IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
00589      $                   WORK( INDERR+WBEGIN-1 ),
00590      $                   WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
00591      $                   TNRM, IINFO )
00592             IBEGIN = IEND + 1
00593             WBEGIN = WEND + 1
00594  39      CONTINUE
00595       ENDIF
00596 *
00597 *     If matrix was scaled, then rescale eigenvalues appropriately.
00598 *
00599       IF( SCALE.NE.ONE ) THEN
00600          CALL DSCAL( M, ONE / SCALE, W, 1 )
00601       END IF
00602 *
00603 *     If eigenvalues are not in increasing order, then sort them,
00604 *     possibly along with eigenvectors.
00605 *
00606       IF( NSPLIT.GT.1 ) THEN
00607          IF( .NOT. WANTZ ) THEN
00608             CALL DLASRT( 'I', M, W, IINFO )
00609             IF( IINFO.NE.0 ) THEN
00610                INFO = 3
00611                RETURN
00612             END IF
00613          ELSE
00614             DO 60 J = 1, M - 1
00615                I = 0
00616                TMP = W( J )
00617                DO 50 JJ = J + 1, M
00618                   IF( W( JJ ).LT.TMP ) THEN
00619                      I = JJ
00620                      TMP = W( JJ )
00621                   END IF
00622  50            CONTINUE
00623                IF( I.NE.0 ) THEN
00624                   W( I ) = W( J )
00625                   W( J ) = TMP
00626                   IF( WANTZ ) THEN
00627                      CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
00628                      ITMP = ISUPPZ( 2*I-1 )
00629                      ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
00630                      ISUPPZ( 2*J-1 ) = ITMP
00631                      ITMP = ISUPPZ( 2*I )
00632                      ISUPPZ( 2*I ) = ISUPPZ( 2*J )
00633                      ISUPPZ( 2*J ) = ITMP
00634                   END IF
00635                END IF
00636  60         CONTINUE
00637          END IF
00638       ENDIF
00639 *
00640 *
00641       WORK( 1 ) = LWMIN
00642       IWORK( 1 ) = LIWMIN
00643       RETURN
00644 *
00645 *     End of DSTEMR
00646 *
00647       END
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