LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
ssyevx.f
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00001 *> \brief <b> SSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY 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 SSYEVX + dependencies 
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00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ssyevx.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ssyevx.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
00022 *                          ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
00023 *                          IFAIL, INFO )
00024 * 
00025 *       .. Scalar Arguments ..
00026 *       CHARACTER          JOBZ, RANGE, UPLO
00027 *       INTEGER            IL, INFO, IU, LDA, LDZ, LWORK, M, N
00028 *       REAL               ABSTOL, VL, VU
00029 *       ..
00030 *       .. Array Arguments ..
00031 *       INTEGER            IFAIL( * ), IWORK( * )
00032 *       REAL               A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
00033 *       ..
00034 *  
00035 *
00036 *> \par Purpose:
00037 *  =============
00038 *>
00039 *> \verbatim
00040 *>
00041 *> SSYEVX computes selected eigenvalues and, optionally, eigenvectors
00042 *> of a real symmetric matrix A.  Eigenvalues and eigenvectors can be
00043 *> selected by specifying either a range of values or a range of indices
00044 *> for the desired eigenvalues.
00045 *> \endverbatim
00046 *
00047 *  Arguments:
00048 *  ==========
00049 *
00050 *> \param[in] JOBZ
00051 *> \verbatim
00052 *>          JOBZ is CHARACTER*1
00053 *>          = 'N':  Compute eigenvalues only;
00054 *>          = 'V':  Compute eigenvalues and eigenvectors.
00055 *> \endverbatim
00056 *>
00057 *> \param[in] RANGE
00058 *> \verbatim
00059 *>          RANGE is CHARACTER*1
00060 *>          = 'A': all eigenvalues will be found.
00061 *>          = 'V': all eigenvalues in the half-open interval (VL,VU]
00062 *>                 will be found.
00063 *>          = 'I': the IL-th through IU-th eigenvalues will be found.
00064 *> \endverbatim
00065 *>
00066 *> \param[in] UPLO
00067 *> \verbatim
00068 *>          UPLO is CHARACTER*1
00069 *>          = 'U':  Upper triangle of A is stored;
00070 *>          = 'L':  Lower triangle of A is stored.
00071 *> \endverbatim
00072 *>
00073 *> \param[in] N
00074 *> \verbatim
00075 *>          N is INTEGER
00076 *>          The order of the matrix A.  N >= 0.
00077 *> \endverbatim
00078 *>
00079 *> \param[in,out] A
00080 *> \verbatim
00081 *>          A is REAL array, dimension (LDA, N)
00082 *>          On entry, the symmetric matrix A.  If UPLO = 'U', the
00083 *>          leading N-by-N upper triangular part of A contains the
00084 *>          upper triangular part of the matrix A.  If UPLO = 'L',
00085 *>          the leading N-by-N lower triangular part of A contains
00086 *>          the lower triangular part of the matrix A.
00087 *>          On exit, the lower triangle (if UPLO='L') or the upper
00088 *>          triangle (if UPLO='U') of A, including the diagonal, is
00089 *>          destroyed.
00090 *> \endverbatim
00091 *>
00092 *> \param[in] LDA
00093 *> \verbatim
00094 *>          LDA is INTEGER
00095 *>          The leading dimension of the array A.  LDA >= max(1,N).
00096 *> \endverbatim
00097 *>
00098 *> \param[in] VL
00099 *> \verbatim
00100 *>          VL is REAL
00101 *> \endverbatim
00102 *>
00103 *> \param[in] VU
00104 *> \verbatim
00105 *>          VU is REAL
00106 *>          If RANGE='V', the lower and upper bounds of the interval to
00107 *>          be searched for eigenvalues. VL < VU.
00108 *>          Not referenced if RANGE = 'A' or 'I'.
00109 *> \endverbatim
00110 *>
00111 *> \param[in] IL
00112 *> \verbatim
00113 *>          IL is INTEGER
00114 *> \endverbatim
00115 *>
00116 *> \param[in] IU
00117 *> \verbatim
00118 *>          IU is INTEGER
00119 *>          If RANGE='I', the indices (in ascending order) of the
00120 *>          smallest and largest eigenvalues to be returned.
00121 *>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
00122 *>          Not referenced if RANGE = 'A' or 'V'.
00123 *> \endverbatim
00124 *>
00125 *> \param[in] ABSTOL
00126 *> \verbatim
00127 *>          ABSTOL is REAL
00128 *>          The absolute error tolerance for the eigenvalues.
00129 *>          An approximate eigenvalue is accepted as converged
00130 *>          when it is determined to lie in an interval [a,b]
00131 *>          of width less than or equal to
00132 *>
00133 *>                  ABSTOL + EPS *   max( |a|,|b| ) ,
00134 *>
00135 *>          where EPS is the machine precision.  If ABSTOL is less than
00136 *>          or equal to zero, then  EPS*|T|  will be used in its place,
00137 *>          where |T| is the 1-norm of the tridiagonal matrix obtained
00138 *>          by reducing A to tridiagonal form.
00139 *>
00140 *>          Eigenvalues will be computed most accurately when ABSTOL is
00141 *>          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
00142 *>          If this routine returns with INFO>0, indicating that some
00143 *>          eigenvectors did not converge, try setting ABSTOL to
00144 *>          2*SLAMCH('S').
00145 *>
00146 *>          See "Computing Small Singular Values of Bidiagonal Matrices
00147 *>          with Guaranteed High Relative Accuracy," by Demmel and
00148 *>          Kahan, LAPACK Working Note #3.
00149 *> \endverbatim
00150 *>
00151 *> \param[out] M
00152 *> \verbatim
00153 *>          M is INTEGER
00154 *>          The total number of eigenvalues found.  0 <= M <= N.
00155 *>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00156 *> \endverbatim
00157 *>
00158 *> \param[out] W
00159 *> \verbatim
00160 *>          W is REAL array, dimension (N)
00161 *>          On normal exit, the first M elements contain the selected
00162 *>          eigenvalues in ascending order.
00163 *> \endverbatim
00164 *>
00165 *> \param[out] Z
00166 *> \verbatim
00167 *>          Z is REAL array, dimension (LDZ, max(1,M))
00168 *>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
00169 *>          contain the orthonormal eigenvectors of the matrix A
00170 *>          corresponding to the selected eigenvalues, with the i-th
00171 *>          column of Z holding the eigenvector associated with W(i).
00172 *>          If an eigenvector fails to converge, then that column of Z
00173 *>          contains the latest approximation to the eigenvector, and the
00174 *>          index of the eigenvector is returned in IFAIL.
00175 *>          If JOBZ = 'N', then Z is not referenced.
00176 *>          Note: the user must ensure that at least max(1,M) columns are
00177 *>          supplied in the array Z; if RANGE = 'V', the exact value of M
00178 *>          is not known in advance and an upper bound must be used.
00179 *> \endverbatim
00180 *>
00181 *> \param[in] LDZ
00182 *> \verbatim
00183 *>          LDZ is INTEGER
00184 *>          The leading dimension of the array Z.  LDZ >= 1, and if
00185 *>          JOBZ = 'V', LDZ >= max(1,N).
00186 *> \endverbatim
00187 *>
00188 *> \param[out] WORK
00189 *> \verbatim
00190 *>          WORK is REAL array, dimension (MAX(1,LWORK))
00191 *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00192 *> \endverbatim
00193 *>
00194 *> \param[in] LWORK
00195 *> \verbatim
00196 *>          LWORK is INTEGER
00197 *>          The length of the array WORK.  LWORK >= 1, when N <= 1;
00198 *>          otherwise 8*N.
00199 *>          For optimal efficiency, LWORK >= (NB+3)*N,
00200 *>          where NB is the max of the blocksize for SSYTRD and SORMTR
00201 *>          returned by ILAENV.
00202 *>
00203 *>          If LWORK = -1, then a workspace query is assumed; the routine
00204 *>          only calculates the optimal size of the WORK array, returns
00205 *>          this value as the first entry of the WORK array, and no error
00206 *>          message related to LWORK is issued by XERBLA.
00207 *> \endverbatim
00208 *>
00209 *> \param[out] IWORK
00210 *> \verbatim
00211 *>          IWORK is INTEGER array, dimension (5*N)
00212 *> \endverbatim
00213 *>
00214 *> \param[out] IFAIL
00215 *> \verbatim
00216 *>          IFAIL is INTEGER array, dimension (N)
00217 *>          If JOBZ = 'V', then if INFO = 0, the first M elements of
00218 *>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
00219 *>          indices of the eigenvectors that failed to converge.
00220 *>          If JOBZ = 'N', then IFAIL is not referenced.
00221 *> \endverbatim
00222 *>
00223 *> \param[out] INFO
00224 *> \verbatim
00225 *>          INFO is INTEGER
00226 *>          = 0:  successful exit
00227 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00228 *>          > 0:  if INFO = i, then i eigenvectors failed to converge.
00229 *>                Their indices are stored in array IFAIL.
00230 *> \endverbatim
00231 *
00232 *  Authors:
00233 *  ========
00234 *
00235 *> \author Univ. of Tennessee 
00236 *> \author Univ. of California Berkeley 
00237 *> \author Univ. of Colorado Denver 
00238 *> \author NAG Ltd. 
00239 *
00240 *> \date November 2011
00241 *
00242 *> \ingroup realSYeigen
00243 *
00244 *  =====================================================================
00245       SUBROUTINE SSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
00246      $                   ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
00247      $                   IFAIL, INFO )
00248 *
00249 *  -- LAPACK driver routine (version 3.4.0) --
00250 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00251 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00252 *     November 2011
00253 *
00254 *     .. Scalar Arguments ..
00255       CHARACTER          JOBZ, RANGE, UPLO
00256       INTEGER            IL, INFO, IU, LDA, LDZ, LWORK, M, N
00257       REAL               ABSTOL, VL, VU
00258 *     ..
00259 *     .. Array Arguments ..
00260       INTEGER            IFAIL( * ), IWORK( * )
00261       REAL               A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
00262 *     ..
00263 *
00264 * =====================================================================
00265 *
00266 *     .. Parameters ..
00267       REAL               ZERO, ONE
00268       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00269 *     ..
00270 *     .. Local Scalars ..
00271       LOGICAL            ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
00272      $                   WANTZ
00273       CHARACTER          ORDER
00274       INTEGER            I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
00275      $                   INDISP, INDIWO, INDTAU, INDWKN, INDWRK, ISCALE,
00276      $                   ITMP1, J, JJ, LLWORK, LLWRKN, LWKMIN,
00277      $                   LWKOPT, NB, NSPLIT
00278       REAL               ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
00279      $                   SIGMA, SMLNUM, TMP1, VLL, VUU
00280 *     ..
00281 *     .. External Functions ..
00282       LOGICAL            LSAME
00283       INTEGER            ILAENV
00284       REAL               SLAMCH, SLANSY
00285       EXTERNAL           LSAME, ILAENV, SLAMCH, SLANSY
00286 *     ..
00287 *     .. External Subroutines ..
00288       EXTERNAL           SCOPY, SLACPY, SORGTR, SORMTR, SSCAL, SSTEBZ,
00289      $                   SSTEIN, SSTEQR, SSTERF, SSWAP, SSYTRD, XERBLA
00290 *     ..
00291 *     .. Intrinsic Functions ..
00292       INTRINSIC          MAX, MIN, SQRT
00293 *     ..
00294 *     .. Executable Statements ..
00295 *
00296 *     Test the input parameters.
00297 *
00298       LOWER = LSAME( UPLO, 'L' )
00299       WANTZ = LSAME( JOBZ, 'V' )
00300       ALLEIG = LSAME( RANGE, 'A' )
00301       VALEIG = LSAME( RANGE, 'V' )
00302       INDEIG = LSAME( RANGE, 'I' )
00303       LQUERY = ( LWORK.EQ.-1 )
00304 *
00305       INFO = 0
00306       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00307          INFO = -1
00308       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00309          INFO = -2
00310       ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
00311          INFO = -3
00312       ELSE IF( N.LT.0 ) THEN
00313          INFO = -4
00314       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00315          INFO = -6
00316       ELSE
00317          IF( VALEIG ) THEN
00318             IF( N.GT.0 .AND. VU.LE.VL )
00319      $         INFO = -8
00320          ELSE IF( INDEIG ) THEN
00321             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
00322                INFO = -9
00323             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
00324                INFO = -10
00325             END IF
00326          END IF
00327       END IF
00328       IF( INFO.EQ.0 ) THEN
00329          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
00330             INFO = -15
00331          END IF
00332       END IF
00333 *
00334       IF( INFO.EQ.0 ) THEN
00335          IF( N.LE.1 ) THEN
00336             LWKMIN = 1
00337             WORK( 1 ) = LWKMIN
00338          ELSE
00339             LWKMIN = 8*N
00340             NB = ILAENV( 1, 'SSYTRD', UPLO, N, -1, -1, -1 )
00341             NB = MAX( NB, ILAENV( 1, 'SORMTR', UPLO, N, -1, -1, -1 ) )
00342             LWKOPT = MAX( LWKMIN, ( NB + 3 )*N )
00343             WORK( 1 ) = LWKOPT
00344          END IF
00345 *
00346          IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
00347      $      INFO = -17
00348       END IF
00349 *
00350       IF( INFO.NE.0 ) THEN
00351          CALL XERBLA( 'SSYEVX', -INFO )
00352          RETURN
00353       ELSE IF( LQUERY ) THEN
00354          RETURN
00355       END IF
00356 *
00357 *     Quick return if possible
00358 *
00359       M = 0
00360       IF( N.EQ.0 ) THEN
00361          RETURN
00362       END IF
00363 *
00364       IF( N.EQ.1 ) THEN
00365          IF( ALLEIG .OR. INDEIG ) THEN
00366             M = 1
00367             W( 1 ) = A( 1, 1 )
00368          ELSE
00369             IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN
00370                M = 1
00371                W( 1 ) = A( 1, 1 )
00372             END IF
00373          END IF
00374          IF( WANTZ )
00375      $      Z( 1, 1 ) = ONE
00376          RETURN
00377       END IF
00378 *
00379 *     Get machine constants.
00380 *
00381       SAFMIN = SLAMCH( 'Safe minimum' )
00382       EPS = SLAMCH( 'Precision' )
00383       SMLNUM = SAFMIN / EPS
00384       BIGNUM = ONE / SMLNUM
00385       RMIN = SQRT( SMLNUM )
00386       RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
00387 *
00388 *     Scale matrix to allowable range, if necessary.
00389 *
00390       ISCALE = 0
00391       ABSTLL = ABSTOL
00392       IF( VALEIG ) THEN
00393          VLL = VL
00394          VUU = VU
00395       END IF
00396       ANRM = SLANSY( 'M', UPLO, N, A, LDA, WORK )
00397       IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
00398          ISCALE = 1
00399          SIGMA = RMIN / ANRM
00400       ELSE IF( ANRM.GT.RMAX ) THEN
00401          ISCALE = 1
00402          SIGMA = RMAX / ANRM
00403       END IF
00404       IF( ISCALE.EQ.1 ) THEN
00405          IF( LOWER ) THEN
00406             DO 10 J = 1, N
00407                CALL SSCAL( N-J+1, SIGMA, A( J, J ), 1 )
00408    10       CONTINUE
00409          ELSE
00410             DO 20 J = 1, N
00411                CALL SSCAL( J, SIGMA, A( 1, J ), 1 )
00412    20       CONTINUE
00413          END IF
00414          IF( ABSTOL.GT.0 )
00415      $      ABSTLL = ABSTOL*SIGMA
00416          IF( VALEIG ) THEN
00417             VLL = VL*SIGMA
00418             VUU = VU*SIGMA
00419          END IF
00420       END IF
00421 *
00422 *     Call SSYTRD to reduce symmetric matrix to tridiagonal form.
00423 *
00424       INDTAU = 1
00425       INDE = INDTAU + N
00426       INDD = INDE + N
00427       INDWRK = INDD + N
00428       LLWORK = LWORK - INDWRK + 1
00429       CALL SSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ),
00430      $             WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
00431 *
00432 *     If all eigenvalues are desired and ABSTOL is less than or equal to
00433 *     zero, then call SSTERF or SORGTR and SSTEQR.  If this fails for
00434 *     some eigenvalue, then try SSTEBZ.
00435 *
00436       TEST = .FALSE.
00437       IF( INDEIG ) THEN
00438          IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
00439             TEST = .TRUE.
00440          END IF
00441       END IF
00442       IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
00443          CALL SCOPY( N, WORK( INDD ), 1, W, 1 )
00444          INDEE = INDWRK + 2*N
00445          IF( .NOT.WANTZ ) THEN
00446             CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
00447             CALL SSTERF( N, W, WORK( INDEE ), INFO )
00448          ELSE
00449             CALL SLACPY( 'A', N, N, A, LDA, Z, LDZ )
00450             CALL SORGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
00451      $                   WORK( INDWRK ), LLWORK, IINFO )
00452             CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
00453             CALL SSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
00454      $                   WORK( INDWRK ), INFO )
00455             IF( INFO.EQ.0 ) THEN
00456                DO 30 I = 1, N
00457                   IFAIL( I ) = 0
00458    30          CONTINUE
00459             END IF
00460          END IF
00461          IF( INFO.EQ.0 ) THEN
00462             M = N
00463             GO TO 40
00464          END IF
00465          INFO = 0
00466       END IF
00467 *
00468 *     Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
00469 *
00470       IF( WANTZ ) THEN
00471          ORDER = 'B'
00472       ELSE
00473          ORDER = 'E'
00474       END IF
00475       INDIBL = 1
00476       INDISP = INDIBL + N
00477       INDIWO = INDISP + N
00478       CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
00479      $             WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
00480      $             IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
00481      $             IWORK( INDIWO ), INFO )
00482 *
00483       IF( WANTZ ) THEN
00484          CALL SSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
00485      $                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
00486      $                WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
00487 *
00488 *        Apply orthogonal matrix used in reduction to tridiagonal
00489 *        form to eigenvectors returned by SSTEIN.
00490 *
00491          INDWKN = INDE
00492          LLWRKN = LWORK - INDWKN + 1
00493          CALL SORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
00494      $                LDZ, WORK( INDWKN ), LLWRKN, IINFO )
00495       END IF
00496 *
00497 *     If matrix was scaled, then rescale eigenvalues appropriately.
00498 *
00499    40 CONTINUE
00500       IF( ISCALE.EQ.1 ) THEN
00501          IF( INFO.EQ.0 ) THEN
00502             IMAX = M
00503          ELSE
00504             IMAX = INFO - 1
00505          END IF
00506          CALL SSCAL( IMAX, ONE / SIGMA, W, 1 )
00507       END IF
00508 *
00509 *     If eigenvalues are not in order, then sort them, along with
00510 *     eigenvectors.
00511 *
00512       IF( WANTZ ) THEN
00513          DO 60 J = 1, M - 1
00514             I = 0
00515             TMP1 = W( J )
00516             DO 50 JJ = J + 1, M
00517                IF( W( JJ ).LT.TMP1 ) THEN
00518                   I = JJ
00519                   TMP1 = W( JJ )
00520                END IF
00521    50       CONTINUE
00522 *
00523             IF( I.NE.0 ) THEN
00524                ITMP1 = IWORK( INDIBL+I-1 )
00525                W( I ) = W( J )
00526                IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
00527                W( J ) = TMP1
00528                IWORK( INDIBL+J-1 ) = ITMP1
00529                CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
00530                IF( INFO.NE.0 ) THEN
00531                   ITMP1 = IFAIL( I )
00532                   IFAIL( I ) = IFAIL( J )
00533                   IFAIL( J ) = ITMP1
00534                END IF
00535             END IF
00536    60    CONTINUE
00537       END IF
00538 *
00539 *     Set WORK(1) to optimal workspace size.
00540 *
00541       WORK( 1 ) = LWKOPT
00542 *
00543       RETURN
00544 *
00545 *     End of SSYEVX
00546 *
00547       END
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