LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zheevr.f
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00001 *> \brief <b> ZHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices</b>
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download ZHEEVR + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zheevr.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZHEEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
00022 *                          ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
00023 *                          RWORK, LRWORK, IWORK, LIWORK, INFO )
00024 * 
00025 *       .. Scalar Arguments ..
00026 *       CHARACTER          JOBZ, RANGE, UPLO
00027 *       INTEGER            IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
00028 *      $                   M, N
00029 *       DOUBLE PRECISION   ABSTOL, VL, VU
00030 *       ..
00031 *       .. Array Arguments ..
00032 *       INTEGER            ISUPPZ( * ), IWORK( * )
00033 *       DOUBLE PRECISION   RWORK( * ), W( * )
00034 *       COMPLEX*16         A( LDA, * ), WORK( * ), Z( LDZ, * )
00035 *       ..
00036 *  
00037 *
00038 *> \par Purpose:
00039 *  =============
00040 *>
00041 *> \verbatim
00042 *>
00043 *> ZHEEVR computes selected eigenvalues and, optionally, eigenvectors
00044 *> of a complex Hermitian matrix A.  Eigenvalues and eigenvectors can
00045 *> be selected by specifying either a range of values or a range of
00046 *> indices for the desired eigenvalues.
00047 *>
00048 *> ZHEEVR first reduces the matrix A to tridiagonal form T with a call
00049 *> to ZHETRD.  Then, whenever possible, ZHEEVR calls ZSTEMR to compute
00050 *> eigenspectrum using Relatively Robust Representations.  ZSTEMR
00051 *> computes eigenvalues by the dqds algorithm, while orthogonal
00052 *> eigenvectors are computed from various "good" L D L^T representations
00053 *> (also known as Relatively Robust Representations). Gram-Schmidt
00054 *> orthogonalization is avoided as far as possible. More specifically,
00055 *> the various steps of the algorithm are as follows.
00056 *>
00057 *> For each unreduced block (submatrix) of T,
00058 *>    (a) Compute T - sigma I  = L D L^T, so that L and D
00059 *>        define all the wanted eigenvalues to high relative accuracy.
00060 *>        This means that small relative changes in the entries of D and L
00061 *>        cause only small relative changes in the eigenvalues and
00062 *>        eigenvectors. The standard (unfactored) representation of the
00063 *>        tridiagonal matrix T does not have this property in general.
00064 *>    (b) Compute the eigenvalues to suitable accuracy.
00065 *>        If the eigenvectors are desired, the algorithm attains full
00066 *>        accuracy of the computed eigenvalues only right before
00067 *>        the corresponding vectors have to be computed, see steps c) and d).
00068 *>    (c) For each cluster of close eigenvalues, select a new
00069 *>        shift close to the cluster, find a new factorization, and refine
00070 *>        the shifted eigenvalues to suitable accuracy.
00071 *>    (d) For each eigenvalue with a large enough relative separation compute
00072 *>        the corresponding eigenvector by forming a rank revealing twisted
00073 *>        factorization. Go back to (c) for any clusters that remain.
00074 *>
00075 *> The desired accuracy of the output can be specified by the input
00076 *> parameter ABSTOL.
00077 *>
00078 *> For more details, see DSTEMR's documentation and:
00079 *> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
00080 *>   to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
00081 *>   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
00082 *> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
00083 *>   Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
00084 *>   2004.  Also LAPACK Working Note 154.
00085 *> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
00086 *>   tridiagonal eigenvalue/eigenvector problem",
00087 *>   Computer Science Division Technical Report No. UCB/CSD-97-971,
00088 *>   UC Berkeley, May 1997.
00089 *>
00090 *>
00091 *> Note 1 : ZHEEVR calls ZSTEMR when the full spectrum is requested
00092 *> on machines which conform to the ieee-754 floating point standard.
00093 *> ZHEEVR calls DSTEBZ and ZSTEIN on non-ieee machines and
00094 *> when partial spectrum requests are made.
00095 *>
00096 *> Normal execution of ZSTEMR may create NaNs and infinities and
00097 *> hence may abort due to a floating point exception in environments
00098 *> which do not handle NaNs and infinities in the ieee standard default
00099 *> manner.
00100 *> \endverbatim
00101 *
00102 *  Arguments:
00103 *  ==========
00104 *
00105 *> \param[in] JOBZ
00106 *> \verbatim
00107 *>          JOBZ is CHARACTER*1
00108 *>          = 'N':  Compute eigenvalues only;
00109 *>          = 'V':  Compute eigenvalues and eigenvectors.
00110 *> \endverbatim
00111 *>
00112 *> \param[in] RANGE
00113 *> \verbatim
00114 *>          RANGE is CHARACTER*1
00115 *>          = 'A': all eigenvalues will be found.
00116 *>          = 'V': all eigenvalues in the half-open interval (VL,VU]
00117 *>                 will be found.
00118 *>          = 'I': the IL-th through IU-th eigenvalues will be found.
00119 *>          For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
00120 *>          ZSTEIN are called
00121 *> \endverbatim
00122 *>
00123 *> \param[in] UPLO
00124 *> \verbatim
00125 *>          UPLO is CHARACTER*1
00126 *>          = 'U':  Upper triangle of A is stored;
00127 *>          = 'L':  Lower triangle of A is stored.
00128 *> \endverbatim
00129 *>
00130 *> \param[in] N
00131 *> \verbatim
00132 *>          N is INTEGER
00133 *>          The order of the matrix A.  N >= 0.
00134 *> \endverbatim
00135 *>
00136 *> \param[in,out] A
00137 *> \verbatim
00138 *>          A is COMPLEX*16 array, dimension (LDA, N)
00139 *>          On entry, the Hermitian matrix A.  If UPLO = 'U', the
00140 *>          leading N-by-N upper triangular part of A contains the
00141 *>          upper triangular part of the matrix A.  If UPLO = 'L',
00142 *>          the leading N-by-N lower triangular part of A contains
00143 *>          the lower triangular part of the matrix A.
00144 *>          On exit, the lower triangle (if UPLO='L') or the upper
00145 *>          triangle (if UPLO='U') of A, including the diagonal, is
00146 *>          destroyed.
00147 *> \endverbatim
00148 *>
00149 *> \param[in] LDA
00150 *> \verbatim
00151 *>          LDA is INTEGER
00152 *>          The leading dimension of the array A.  LDA >= max(1,N).
00153 *> \endverbatim
00154 *>
00155 *> \param[in] VL
00156 *> \verbatim
00157 *>          VL is DOUBLE PRECISION
00158 *> \endverbatim
00159 *>
00160 *> \param[in] VU
00161 *> \verbatim
00162 *>          VU is DOUBLE PRECISION
00163 *>          If RANGE='V', the lower and upper bounds of the interval to
00164 *>          be searched for eigenvalues. VL < VU.
00165 *>          Not referenced if RANGE = 'A' or 'I'.
00166 *> \endverbatim
00167 *>
00168 *> \param[in] IL
00169 *> \verbatim
00170 *>          IL is INTEGER
00171 *> \endverbatim
00172 *>
00173 *> \param[in] IU
00174 *> \verbatim
00175 *>          IU is INTEGER
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; IL = 1 and IU = 0 if N = 0.
00179 *>          Not referenced if RANGE = 'A' or 'V'.
00180 *> \endverbatim
00181 *>
00182 *> \param[in] ABSTOL
00183 *> \verbatim
00184 *>          ABSTOL is DOUBLE PRECISION
00185 *>          The absolute error tolerance for the eigenvalues.
00186 *>          An approximate eigenvalue is accepted as converged
00187 *>          when it is determined to lie in an interval [a,b]
00188 *>          of width less than or equal to
00189 *>
00190 *>                  ABSTOL + EPS *   max( |a|,|b| ) ,
00191 *>
00192 *>          where EPS is the machine precision.  If ABSTOL is less than
00193 *>          or equal to zero, then  EPS*|T|  will be used in its place,
00194 *>          where |T| is the 1-norm of the tridiagonal matrix obtained
00195 *>          by reducing A to tridiagonal form.
00196 *>
00197 *>          See "Computing Small Singular Values of Bidiagonal Matrices
00198 *>          with Guaranteed High Relative Accuracy," by Demmel and
00199 *>          Kahan, LAPACK Working Note #3.
00200 *>
00201 *>          If high relative accuracy is important, set ABSTOL to
00202 *>          DLAMCH( 'Safe minimum' ).  Doing so will guarantee that
00203 *>          eigenvalues are computed to high relative accuracy when
00204 *>          possible in future releases.  The current code does not
00205 *>          make any guarantees about high relative accuracy, but
00206 *>          furutre releases will. See J. Barlow and J. Demmel,
00207 *>          "Computing Accurate Eigensystems of Scaled Diagonally
00208 *>          Dominant Matrices", LAPACK Working Note #7, for a discussion
00209 *>          of which matrices define their eigenvalues to high relative
00210 *>          accuracy.
00211 *> \endverbatim
00212 *>
00213 *> \param[out] M
00214 *> \verbatim
00215 *>          M is INTEGER
00216 *>          The total number of eigenvalues found.  0 <= M <= N.
00217 *>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00218 *> \endverbatim
00219 *>
00220 *> \param[out] W
00221 *> \verbatim
00222 *>          W is DOUBLE PRECISION array, dimension (N)
00223 *>          The first M elements contain the selected eigenvalues in
00224 *>          ascending order.
00225 *> \endverbatim
00226 *>
00227 *> \param[out] Z
00228 *> \verbatim
00229 *>          Z is COMPLEX*16 array, dimension (LDZ, max(1,M))
00230 *>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
00231 *>          contain the orthonormal eigenvectors of the matrix A
00232 *>          corresponding to the selected eigenvalues, with the i-th
00233 *>          column of Z holding the eigenvector associated with W(i).
00234 *>          If JOBZ = 'N', then Z is not referenced.
00235 *>          Note: the user must ensure that at least max(1,M) columns are
00236 *>          supplied in the array Z; if RANGE = 'V', the exact value of M
00237 *>          is not known in advance and an upper bound must be used.
00238 *> \endverbatim
00239 *>
00240 *> \param[in] LDZ
00241 *> \verbatim
00242 *>          LDZ is INTEGER
00243 *>          The leading dimension of the array Z.  LDZ >= 1, and if
00244 *>          JOBZ = 'V', LDZ >= max(1,N).
00245 *> \endverbatim
00246 *>
00247 *> \param[out] ISUPPZ
00248 *> \verbatim
00249 *>          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
00250 *>          The support of the eigenvectors in Z, i.e., the indices
00251 *>          indicating the nonzero elements in Z. The i-th eigenvector
00252 *>          is nonzero only in elements ISUPPZ( 2*i-1 ) through
00253 *>          ISUPPZ( 2*i ).
00254 *>          Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
00255 *> \endverbatim
00256 *>
00257 *> \param[out] WORK
00258 *> \verbatim
00259 *>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
00260 *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00261 *> \endverbatim
00262 *>
00263 *> \param[in] LWORK
00264 *> \verbatim
00265 *>          LWORK is INTEGER
00266 *>          The length of the array WORK.  LWORK >= max(1,2*N).
00267 *>          For optimal efficiency, LWORK >= (NB+1)*N,
00268 *>          where NB is the max of the blocksize for ZHETRD and for
00269 *>          ZUNMTR as returned by ILAENV.
00270 *>
00271 *>          If LWORK = -1, then a workspace query is assumed; the routine
00272 *>          only calculates the optimal sizes of the WORK, RWORK and
00273 *>          IWORK arrays, returns these values as the first entries of
00274 *>          the WORK, RWORK and IWORK arrays, and no error message
00275 *>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
00276 *> \endverbatim
00277 *>
00278 *> \param[out] RWORK
00279 *> \verbatim
00280 *>          RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
00281 *>          On exit, if INFO = 0, RWORK(1) returns the optimal
00282 *>          (and minimal) LRWORK.
00283 *> \endverbatim
00284 *>
00285 *> \param[in] LRWORK
00286 *> \verbatim
00287 *>          LRWORK is INTEGER
00288 *>          The length of the array RWORK.  LRWORK >= max(1,24*N).
00289 *>
00290 *>          If LRWORK = -1, then a workspace query is assumed; the
00291 *>          routine only calculates the optimal sizes of the WORK, RWORK
00292 *>          and IWORK arrays, returns these values as the first entries
00293 *>          of the WORK, RWORK and IWORK arrays, and no error message
00294 *>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
00295 *> \endverbatim
00296 *>
00297 *> \param[out] IWORK
00298 *> \verbatim
00299 *>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
00300 *>          On exit, if INFO = 0, IWORK(1) returns the optimal
00301 *>          (and minimal) LIWORK.
00302 *> \endverbatim
00303 *>
00304 *> \param[in] LIWORK
00305 *> \verbatim
00306 *>          LIWORK is INTEGER
00307 *>          The dimension of the array IWORK.  LIWORK >= max(1,10*N).
00308 *>
00309 *>          If LIWORK = -1, then a workspace query is assumed; the
00310 *>          routine only calculates the optimal sizes of the WORK, RWORK
00311 *>          and IWORK arrays, returns these values as the first entries
00312 *>          of the WORK, RWORK and IWORK arrays, and no error message
00313 *>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
00314 *> \endverbatim
00315 *>
00316 *> \param[out] INFO
00317 *> \verbatim
00318 *>          INFO is INTEGER
00319 *>          = 0:  successful exit
00320 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00321 *>          > 0:  Internal error
00322 *> \endverbatim
00323 *
00324 *  Authors:
00325 *  ========
00326 *
00327 *> \author Univ. of Tennessee 
00328 *> \author Univ. of California Berkeley 
00329 *> \author Univ. of Colorado Denver 
00330 *> \author NAG Ltd. 
00331 *
00332 *> \date November 2011
00333 *
00334 *> \ingroup complex16HEeigen
00335 *
00336 *> \par Contributors:
00337 *  ==================
00338 *>
00339 *>     Inderjit Dhillon, IBM Almaden, USA \n
00340 *>     Osni Marques, LBNL/NERSC, USA \n
00341 *>     Ken Stanley, Computer Science Division, University of
00342 *>       California at Berkeley, USA \n
00343 *>     Jason Riedy, Computer Science Division, University of
00344 *>       California at Berkeley, USA \n
00345 *>
00346 *  =====================================================================
00347       SUBROUTINE ZHEEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
00348      $                   ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
00349      $                   RWORK, LRWORK, IWORK, LIWORK, INFO )
00350 *
00351 *  -- LAPACK driver routine (version 3.4.0) --
00352 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00353 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00354 *     November 2011
00355 *
00356 *     .. Scalar Arguments ..
00357       CHARACTER          JOBZ, RANGE, UPLO
00358       INTEGER            IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
00359      $                   M, N
00360       DOUBLE PRECISION   ABSTOL, VL, VU
00361 *     ..
00362 *     .. Array Arguments ..
00363       INTEGER            ISUPPZ( * ), IWORK( * )
00364       DOUBLE PRECISION   RWORK( * ), W( * )
00365       COMPLEX*16         A( LDA, * ), WORK( * ), Z( LDZ, * )
00366 *     ..
00367 *
00368 *  =====================================================================
00369 *
00370 *     .. Parameters ..
00371       DOUBLE PRECISION   ZERO, ONE, TWO
00372       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
00373 *     ..
00374 *     .. Local Scalars ..
00375       LOGICAL            ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
00376      $                   WANTZ, TRYRAC
00377       CHARACTER          ORDER
00378       INTEGER            I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP,
00379      $                   INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK,
00380      $                   INDTAU, INDWK, INDWKN, ISCALE, ITMP1, J, JJ,
00381      $                   LIWMIN, LLWORK, LLRWORK, LLWRKN, LRWMIN,
00382      $                   LWKOPT, LWMIN, NB, NSPLIT
00383       DOUBLE PRECISION   ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
00384      $                   SIGMA, SMLNUM, TMP1, VLL, VUU
00385 *     ..
00386 *     .. External Functions ..
00387       LOGICAL            LSAME
00388       INTEGER            ILAENV
00389       DOUBLE PRECISION   DLAMCH, ZLANSY
00390       EXTERNAL           LSAME, ILAENV, DLAMCH, ZLANSY
00391 *     ..
00392 *     .. External Subroutines ..
00393       EXTERNAL           DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
00394      $                   ZHETRD, ZSTEMR, ZSTEIN, ZSWAP, ZUNMTR
00395 *     ..
00396 *     .. Intrinsic Functions ..
00397       INTRINSIC          DBLE, MAX, MIN, SQRT
00398 *     ..
00399 *     .. Executable Statements ..
00400 *
00401 *     Test the input parameters.
00402 *
00403       IEEEOK = ILAENV( 10, 'ZHEEVR', 'N', 1, 2, 3, 4 )
00404 *
00405       LOWER = LSAME( UPLO, 'L' )
00406       WANTZ = LSAME( JOBZ, 'V' )
00407       ALLEIG = LSAME( RANGE, 'A' )
00408       VALEIG = LSAME( RANGE, 'V' )
00409       INDEIG = LSAME( RANGE, 'I' )
00410 *
00411       LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LRWORK.EQ.-1 ) .OR.
00412      $         ( LIWORK.EQ.-1 ) )
00413 *
00414       LRWMIN = MAX( 1, 24*N )
00415       LIWMIN = MAX( 1, 10*N )
00416       LWMIN = MAX( 1, 2*N )
00417 *
00418       INFO = 0
00419       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00420          INFO = -1
00421       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00422          INFO = -2
00423       ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
00424          INFO = -3
00425       ELSE IF( N.LT.0 ) THEN
00426          INFO = -4
00427       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00428          INFO = -6
00429       ELSE
00430          IF( VALEIG ) THEN
00431             IF( N.GT.0 .AND. VU.LE.VL )
00432      $         INFO = -8
00433          ELSE IF( INDEIG ) THEN
00434             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
00435                INFO = -9
00436             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
00437                INFO = -10
00438             END IF
00439          END IF
00440       END IF
00441       IF( INFO.EQ.0 ) THEN
00442          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
00443             INFO = -15
00444          END IF
00445       END IF
00446 *
00447       IF( INFO.EQ.0 ) THEN
00448          NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
00449          NB = MAX( NB, ILAENV( 1, 'ZUNMTR', UPLO, N, -1, -1, -1 ) )
00450          LWKOPT = MAX( ( NB+1 )*N, LWMIN )
00451          WORK( 1 ) = LWKOPT
00452          RWORK( 1 ) = LRWMIN
00453          IWORK( 1 ) = LIWMIN
00454 *
00455          IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
00456             INFO = -18
00457          ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
00458             INFO = -20
00459          ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
00460             INFO = -22
00461          END IF
00462       END IF
00463 *
00464       IF( INFO.NE.0 ) THEN
00465          CALL XERBLA( 'ZHEEVR', -INFO )
00466          RETURN
00467       ELSE IF( LQUERY ) THEN
00468          RETURN
00469       END IF
00470 *
00471 *     Quick return if possible
00472 *
00473       M = 0
00474       IF( N.EQ.0 ) THEN
00475          WORK( 1 ) = 1
00476          RETURN
00477       END IF
00478 *
00479       IF( N.EQ.1 ) THEN
00480          WORK( 1 ) = 2
00481          IF( ALLEIG .OR. INDEIG ) THEN
00482             M = 1
00483             W( 1 ) = DBLE( A( 1, 1 ) )
00484          ELSE
00485             IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) )
00486      $           THEN
00487                M = 1
00488                W( 1 ) = DBLE( A( 1, 1 ) )
00489             END IF
00490          END IF
00491          IF( WANTZ ) THEN
00492             Z( 1, 1 ) = ONE
00493             ISUPPZ( 1 ) = 1
00494             ISUPPZ( 2 ) = 1
00495          END IF
00496          RETURN
00497       END IF
00498 *
00499 *     Get machine constants.
00500 *
00501       SAFMIN = DLAMCH( 'Safe minimum' )
00502       EPS = DLAMCH( 'Precision' )
00503       SMLNUM = SAFMIN / EPS
00504       BIGNUM = ONE / SMLNUM
00505       RMIN = SQRT( SMLNUM )
00506       RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
00507 *
00508 *     Scale matrix to allowable range, if necessary.
00509 *
00510       ISCALE = 0
00511       ABSTLL = ABSTOL
00512       IF (VALEIG) THEN
00513          VLL = VL
00514          VUU = VU
00515       END IF
00516       ANRM = ZLANSY( 'M', UPLO, N, A, LDA, RWORK )
00517       IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
00518          ISCALE = 1
00519          SIGMA = RMIN / ANRM
00520       ELSE IF( ANRM.GT.RMAX ) THEN
00521          ISCALE = 1
00522          SIGMA = RMAX / ANRM
00523       END IF
00524       IF( ISCALE.EQ.1 ) THEN
00525          IF( LOWER ) THEN
00526             DO 10 J = 1, N
00527                CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 )
00528    10       CONTINUE
00529          ELSE
00530             DO 20 J = 1, N
00531                CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 )
00532    20       CONTINUE
00533          END IF
00534          IF( ABSTOL.GT.0 )
00535      $      ABSTLL = ABSTOL*SIGMA
00536          IF( VALEIG ) THEN
00537             VLL = VL*SIGMA
00538             VUU = VU*SIGMA
00539          END IF
00540       END IF
00541 
00542 *     Initialize indices into workspaces.  Note: The IWORK indices are
00543 *     used only if DSTERF or ZSTEMR fail.
00544 
00545 *     WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the
00546 *     elementary reflectors used in ZHETRD.
00547       INDTAU = 1
00548 *     INDWK is the starting offset of the remaining complex workspace,
00549 *     and LLWORK is the remaining complex workspace size.
00550       INDWK = INDTAU + N
00551       LLWORK = LWORK - INDWK + 1
00552 
00553 *     RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal
00554 *     entries.
00555       INDRD = 1
00556 *     RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the
00557 *     tridiagonal matrix from ZHETRD.
00558       INDRE = INDRD + N
00559 *     RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over
00560 *     -written by ZSTEMR (the DSTERF path copies the diagonal to W).
00561       INDRDD = INDRE + N
00562 *     RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over
00563 *     -written while computing the eigenvalues in DSTERF and ZSTEMR.
00564       INDREE = INDRDD + N
00565 *     INDRWK is the starting offset of the left-over real workspace, and
00566 *     LLRWORK is the remaining workspace size.
00567       INDRWK = INDREE + N
00568       LLRWORK = LRWORK - INDRWK + 1
00569 
00570 *     IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
00571 *     stores the block indices of each of the M<=N eigenvalues.
00572       INDIBL = 1
00573 *     IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
00574 *     stores the starting and finishing indices of each block.
00575       INDISP = INDIBL + N
00576 *     IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
00577 *     that corresponding to eigenvectors that fail to converge in
00578 *     DSTEIN.  This information is discarded; if any fail, the driver
00579 *     returns INFO > 0.
00580       INDIFL = INDISP + N
00581 *     INDIWO is the offset of the remaining integer workspace.
00582       INDIWO = INDISP + N
00583 
00584 *
00585 *     Call ZHETRD to reduce Hermitian matrix to tridiagonal form.
00586 *
00587       CALL ZHETRD( UPLO, N, A, LDA, RWORK( INDRD ), RWORK( INDRE ),
00588      $             WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO )
00589 *
00590 *     If all eigenvalues are desired
00591 *     then call DSTERF or ZSTEMR and ZUNMTR.
00592 *
00593       TEST = .FALSE.
00594       IF( INDEIG ) THEN
00595          IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
00596             TEST = .TRUE.
00597          END IF
00598       END IF
00599       IF( ( ALLEIG.OR.TEST ) .AND. ( IEEEOK.EQ.1 ) ) THEN
00600          IF( .NOT.WANTZ ) THEN
00601             CALL DCOPY( N, RWORK( INDRD ), 1, W, 1 )
00602             CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
00603             CALL DSTERF( N, W, RWORK( INDREE ), INFO )
00604          ELSE
00605             CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
00606             CALL DCOPY( N, RWORK( INDRD ), 1, RWORK( INDRDD ), 1 )
00607 *
00608             IF (ABSTOL .LE. TWO*N*EPS) THEN
00609                TRYRAC = .TRUE.
00610             ELSE
00611                TRYRAC = .FALSE.
00612             END IF
00613             CALL ZSTEMR( JOBZ, 'A', N, RWORK( INDRDD ),
00614      $                   RWORK( INDREE ), VL, VU, IL, IU, M, W,
00615      $                   Z, LDZ, N, ISUPPZ, TRYRAC,
00616      $                   RWORK( INDRWK ), LLRWORK,
00617      $                   IWORK, LIWORK, INFO )
00618 *
00619 *           Apply unitary matrix used in reduction to tridiagonal
00620 *           form to eigenvectors returned by ZSTEIN.
00621 *
00622             IF( WANTZ .AND. INFO.EQ.0 ) THEN
00623                INDWKN = INDWK
00624                LLWRKN = LWORK - INDWKN + 1
00625                CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA,
00626      $                      WORK( INDTAU ), Z, LDZ, WORK( INDWKN ),
00627      $                      LLWRKN, IINFO )
00628             END IF
00629          END IF
00630 *
00631 *
00632          IF( INFO.EQ.0 ) THEN
00633             M = N
00634             GO TO 30
00635          END IF
00636          INFO = 0
00637       END IF
00638 *
00639 *     Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
00640 *     Also call DSTEBZ and ZSTEIN if ZSTEMR fails.
00641 *
00642       IF( WANTZ ) THEN
00643          ORDER = 'B'
00644       ELSE
00645          ORDER = 'E'
00646       END IF
00647 
00648       CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
00649      $             RWORK( INDRD ), RWORK( INDRE ), M, NSPLIT, W,
00650      $             IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
00651      $             IWORK( INDIWO ), INFO )
00652 *
00653       IF( WANTZ ) THEN
00654          CALL ZSTEIN( N, RWORK( INDRD ), RWORK( INDRE ), M, W,
00655      $                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
00656      $                RWORK( INDRWK ), IWORK( INDIWO ), IWORK( INDIFL ),
00657      $                INFO )
00658 *
00659 *        Apply unitary matrix used in reduction to tridiagonal
00660 *        form to eigenvectors returned by ZSTEIN.
00661 *
00662          INDWKN = INDWK
00663          LLWRKN = LWORK - INDWKN + 1
00664          CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
00665      $                LDZ, WORK( INDWKN ), LLWRKN, IINFO )
00666       END IF
00667 *
00668 *     If matrix was scaled, then rescale eigenvalues appropriately.
00669 *
00670    30 CONTINUE
00671       IF( ISCALE.EQ.1 ) THEN
00672          IF( INFO.EQ.0 ) THEN
00673             IMAX = M
00674          ELSE
00675             IMAX = INFO - 1
00676          END IF
00677          CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
00678       END IF
00679 *
00680 *     If eigenvalues are not in order, then sort them, along with
00681 *     eigenvectors.
00682 *
00683       IF( WANTZ ) THEN
00684          DO 50 J = 1, M - 1
00685             I = 0
00686             TMP1 = W( J )
00687             DO 40 JJ = J + 1, M
00688                IF( W( JJ ).LT.TMP1 ) THEN
00689                   I = JJ
00690                   TMP1 = W( JJ )
00691                END IF
00692    40       CONTINUE
00693 *
00694             IF( I.NE.0 ) THEN
00695                ITMP1 = IWORK( INDIBL+I-1 )
00696                W( I ) = W( J )
00697                IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
00698                W( J ) = TMP1
00699                IWORK( INDIBL+J-1 ) = ITMP1
00700                CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
00701             END IF
00702    50    CONTINUE
00703       END IF
00704 *
00705 *     Set WORK(1) to optimal workspace size.
00706 *
00707       WORK( 1 ) = LWKOPT
00708       RWORK( 1 ) = LRWMIN
00709       IWORK( 1 ) = LIWMIN
00710 *
00711       RETURN
00712 *
00713 *     End of ZHEEVR
00714 *
00715       END
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