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authorjason <jason@8a072113-8704-0410-8d35-dd094bca7971>2008-10-28 01:38:50 +0000
committerjason <jason@8a072113-8704-0410-8d35-dd094bca7971>2008-10-28 01:38:50 +0000
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+ SUBROUTINE DSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
+ $ ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
+ $ IWORK, LIWORK, INFO )
+*
+* -- LAPACK driver routine (version 3.1) --
+* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
+* November 2006
+*
+* .. Scalar Arguments ..
+ CHARACTER JOBZ, RANGE, UPLO
+ INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
+ DOUBLE PRECISION ABSTOL, VL, VU
+* ..
+* .. Array Arguments ..
+ INTEGER ISUPPZ( * ), IWORK( * )
+ DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
+* ..
+*
+* Purpose
+* =======
+*
+* DSYEVR computes selected eigenvalues and, optionally, eigenvectors
+* of a real symmetric matrix A. Eigenvalues and eigenvectors can be
+* selected by specifying either a range of values or a range of
+* indices for the desired eigenvalues.
+*
+* DSYEVR first reduces the matrix A to tridiagonal form T with a call
+* to DSYTRD. Then, whenever possible, DSYEVR calls DSTEMR to compute
+* the eigenspectrum using Relatively Robust Representations. DSTEMR
+* computes eigenvalues by the dqds algorithm, while orthogonal
+* eigenvectors are computed from various "good" L D L^T representations
+* (also known as Relatively Robust Representations). Gram-Schmidt
+* orthogonalization is avoided as far as possible. More specifically,
+* the various steps of the algorithm are as follows.
+*
+* For each unreduced block (submatrix) of T,
+* (a) Compute T - sigma I = L D L^T, so that L and D
+* define all the wanted eigenvalues to high relative accuracy.
+* This means that small relative changes in the entries of D and L
+* cause only small relative changes in the eigenvalues and
+* eigenvectors. The standard (unfactored) representation of the
+* tridiagonal matrix T does not have this property in general.
+* (b) Compute the eigenvalues to suitable accuracy.
+* If the eigenvectors are desired, the algorithm attains full
+* accuracy of the computed eigenvalues only right before
+* the corresponding vectors have to be computed, see steps c) and d).
+* (c) For each cluster of close eigenvalues, select a new
+* shift close to the cluster, find a new factorization, and refine
+* the shifted eigenvalues to suitable accuracy.
+* (d) For each eigenvalue with a large enough relative separation compute
+* the corresponding eigenvector by forming a rank revealing twisted
+* factorization. Go back to (c) for any clusters that remain.
+*
+* The desired accuracy of the output can be specified by the input
+* parameter ABSTOL.
+*
+* For more details, see DSTEMR's documentation and:
+* - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
+* to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
+* Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
+* - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
+* Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
+* 2004. Also LAPACK Working Note 154.
+* - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
+* tridiagonal eigenvalue/eigenvector problem",
+* Computer Science Division Technical Report No. UCB/CSD-97-971,
+* UC Berkeley, May 1997.
+*
+*
+* Note 1 : DSYEVR calls DSTEMR when the full spectrum is requested
+* on machines which conform to the ieee-754 floating point standard.
+* DSYEVR calls DSTEBZ and SSTEIN on non-ieee machines and
+* when partial spectrum requests are made.
+*
+* Normal execution of DSTEMR may create NaNs and infinities and
+* hence may abort due to a floating point exception in environments
+* which do not handle NaNs and infinities in the ieee standard default
+* manner.
+*
+* Arguments
+* =========
+*
+* JOBZ (input) CHARACTER*1
+* = 'N': Compute eigenvalues only;
+* = 'V': Compute eigenvalues and eigenvectors.
+*
+* RANGE (input) CHARACTER*1
+* = 'A': all eigenvalues will be found.
+* = 'V': all eigenvalues in the half-open interval (VL,VU]
+* will be found.
+* = 'I': the IL-th through IU-th eigenvalues will be found.
+********** For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
+********** DSTEIN are called
+*
+* UPLO (input) CHARACTER*1
+* = 'U': Upper triangle of A is stored;
+* = 'L': Lower triangle of A is stored.
+*
+* N (input) INTEGER
+* The order of the matrix A. N >= 0.
+*
+* A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
+* On entry, the symmetric matrix A. If UPLO = 'U', the
+* leading N-by-N upper triangular part of A contains the
+* upper triangular part of the matrix A. If UPLO = 'L',
+* the leading N-by-N lower triangular part of A contains
+* the lower triangular part of the matrix A.
+* On exit, the lower triangle (if UPLO='L') or the upper
+* triangle (if UPLO='U') of A, including the diagonal, is
+* destroyed.
+*
+* LDA (input) INTEGER
+* The leading dimension of the array A. LDA >= max(1,N).
+*
+* VL (input) DOUBLE PRECISION
+* VU (input) DOUBLE PRECISION
+* If RANGE='V', the lower and upper bounds of the interval to
+* be searched for eigenvalues. VL < VU.
+* Not referenced if RANGE = 'A' or 'I'.
+*
+* IL (input) INTEGER
+* IU (input) INTEGER
+* If RANGE='I', the indices (in ascending order) of the
+* smallest and largest eigenvalues to be returned.
+* 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
+* Not referenced if RANGE = 'A' or 'V'.
+*
+* ABSTOL (input) DOUBLE PRECISION
+* The absolute error tolerance for the eigenvalues.
+* An approximate eigenvalue is accepted as converged
+* when it is determined to lie in an interval [a,b]
+* of width less than or equal to
+*
+* ABSTOL + EPS * max( |a|,|b| ) ,
+*
+* where EPS is the machine precision. If ABSTOL is less than
+* or equal to zero, then EPS*|T| will be used in its place,
+* where |T| is the 1-norm of the tridiagonal matrix obtained
+* by reducing A to tridiagonal form.
+*
+* See "Computing Small Singular Values of Bidiagonal Matrices
+* with Guaranteed High Relative Accuracy," by Demmel and
+* Kahan, LAPACK Working Note #3.
+*
+* If high relative accuracy is important, set ABSTOL to
+* DLAMCH( 'Safe minimum' ). Doing so will guarantee that
+* eigenvalues are computed to high relative accuracy when
+* possible in future releases. The current code does not
+* make any guarantees about high relative accuracy, but
+* future releases will. See J. Barlow and J. Demmel,
+* "Computing Accurate Eigensystems of Scaled Diagonally
+* Dominant Matrices", LAPACK Working Note #7, for a discussion
+* of which matrices define their eigenvalues to high relative
+* accuracy.
+*
+* M (output) INTEGER
+* The total number of eigenvalues found. 0 <= M <= N.
+* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
+*
+* W (output) DOUBLE PRECISION array, dimension (N)
+* The first M elements contain the selected eigenvalues in
+* ascending order.
+*
+* Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
+* If JOBZ = 'V', then if INFO = 0, the first M columns of Z
+* contain the orthonormal eigenvectors of the matrix A
+* corresponding to the selected eigenvalues, with the i-th
+* column of Z holding the eigenvector associated with W(i).
+* If JOBZ = 'N', then Z is not referenced.
+* Note: the user must ensure that at least max(1,M) columns are
+* supplied in the array Z; if RANGE = 'V', the exact value of M
+* is not known in advance and an upper bound must be used.
+* Supplying N columns is always safe.
+*
+* LDZ (input) INTEGER
+* The leading dimension of the array Z. LDZ >= 1, and if
+* JOBZ = 'V', LDZ >= max(1,N).
+*
+* ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
+* The support of the eigenvectors in Z, i.e., the indices
+* indicating the nonzero elements in Z. The i-th eigenvector
+* is nonzero only in elements ISUPPZ( 2*i-1 ) through
+* ISUPPZ( 2*i ).
+********** Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
+*
+* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
+* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
+*
+* LWORK (input) INTEGER
+* The dimension of the array WORK. LWORK >= max(1,26*N).
+* For optimal efficiency, LWORK >= (NB+6)*N,
+* where NB is the max of the blocksize for DSYTRD and DORMTR
+* returned by ILAENV.
+*
+* If LWORK = -1, then a workspace query is assumed; the routine
+* only calculates the optimal size of the WORK array, returns
+* this value as the first entry of the WORK array, and no error
+* message related to LWORK is issued by XERBLA.
+*
+* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
+* On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.
+*
+* LIWORK (input) INTEGER
+* The dimension of the array IWORK. LIWORK >= max(1,10*N).
+*
+* If LIWORK = -1, then a workspace query is assumed; the
+* routine only calculates the optimal size of the IWORK array,
+* returns this value as the first entry of the IWORK array, and
+* no error message related to LIWORK is issued by XERBLA.
+*
+* INFO (output) INTEGER
+* = 0: successful exit
+* < 0: if INFO = -i, the i-th argument had an illegal value
+* > 0: Internal error
+*
+* Further Details
+* ===============
+*
+* Based on contributions by
+* Inderjit Dhillon, IBM Almaden, USA
+* Osni Marques, LBNL/NERSC, USA
+* Ken Stanley, Computer Science Division, University of
+* California at Berkeley, USA
+* Jason Riedy, Computer Science Division, University of
+* California at Berkeley, USA
+*
+* =====================================================================
+*
+* .. Parameters ..
+ DOUBLE PRECISION ZERO, ONE, TWO
+ PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
+* ..
+* .. Local Scalars ..
+ LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ,
+ $ TRYRAC
+ CHARACTER ORDER
+ INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE,
+ $ INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU,
+ $ INDWK, INDWKN, ISCALE, J, JJ, LIWMIN,
+ $ LLWORK, LLWRKN, LWKOPT, LWMIN, NB, NSPLIT
+ DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
+ $ SIGMA, SMLNUM, TMP1, VLL, VUU
+* ..
+* .. External Functions ..
+ LOGICAL LSAME
+ INTEGER ILAENV
+ DOUBLE PRECISION DLAMCH, DLANSY
+ EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY
+* ..
+* .. External Subroutines ..
+ EXTERNAL DCOPY, DORMTR, DSCAL, DSTEBZ, DSTEMR, DSTEIN,
+ $ DSTERF, DSWAP, DSYTRD, XERBLA
+* ..
+* .. Intrinsic Functions ..
+ INTRINSIC MAX, MIN, SQRT
+* ..
+* .. Executable Statements ..
+*
+* Test the input parameters.
+*
+ IEEEOK = ILAENV( 10, 'DSYEVR', 'N', 1, 2, 3, 4 )
+*
+ LOWER = LSAME( UPLO, 'L' )
+ WANTZ = LSAME( JOBZ, 'V' )
+ ALLEIG = LSAME( RANGE, 'A' )
+ VALEIG = LSAME( RANGE, 'V' )
+ INDEIG = LSAME( RANGE, 'I' )
+*
+ LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
+*
+ LWMIN = MAX( 1, 26*N )
+ LIWMIN = MAX( 1, 10*N )
+*
+ INFO = 0
+ IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
+ INFO = -1
+ ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
+ INFO = -2
+ ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
+ INFO = -3
+ ELSE IF( N.LT.0 ) THEN
+ INFO = -4
+ ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
+ INFO = -6
+ ELSE
+ IF( VALEIG ) THEN
+ IF( N.GT.0 .AND. VU.LE.VL )
+ $ INFO = -8
+ ELSE IF( INDEIG ) THEN
+ IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
+ INFO = -9
+ ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
+ INFO = -10
+ END IF
+ END IF
+ END IF
+ IF( INFO.EQ.0 ) THEN
+ IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
+ INFO = -15
+ ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
+ INFO = -18
+ ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
+ INFO = -20
+ END IF
+ END IF
+*
+ IF( INFO.EQ.0 ) THEN
+ NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
+ NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) )
+ LWKOPT = MAX( ( NB+1 )*N, LWMIN )
+ WORK( 1 ) = LWKOPT
+ IWORK( 1 ) = LIWMIN
+ END IF
+*
+ IF( INFO.NE.0 ) THEN
+ CALL XERBLA( 'DSYEVR', -INFO )
+ RETURN
+ ELSE IF( LQUERY ) THEN
+ RETURN
+ END IF
+*
+* Quick return if possible
+*
+ M = 0
+ IF( N.EQ.0 ) THEN
+ WORK( 1 ) = 1
+ RETURN
+ END IF
+*
+ IF( N.EQ.1 ) THEN
+ WORK( 1 ) = 7
+ IF( ALLEIG .OR. INDEIG ) THEN
+ M = 1
+ W( 1 ) = A( 1, 1 )
+ ELSE
+ IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN
+ M = 1
+ W( 1 ) = A( 1, 1 )
+ END IF
+ END IF
+ IF( WANTZ )
+ $ Z( 1, 1 ) = ONE
+ RETURN
+ END IF
+*
+* Get machine constants.
+*
+ SAFMIN = DLAMCH( 'Safe minimum' )
+ EPS = DLAMCH( 'Precision' )
+ SMLNUM = SAFMIN / EPS
+ BIGNUM = ONE / SMLNUM
+ RMIN = SQRT( SMLNUM )
+ RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
+*
+* Scale matrix to allowable range, if necessary.
+*
+ ISCALE = 0
+ ABSTLL = ABSTOL
+ VLL = VL
+ VUU = VU
+ ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
+ IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
+ ISCALE = 1
+ SIGMA = RMIN / ANRM
+ ELSE IF( ANRM.GT.RMAX ) THEN
+ ISCALE = 1
+ SIGMA = RMAX / ANRM
+ END IF
+ IF( ISCALE.EQ.1 ) THEN
+ IF( LOWER ) THEN
+ DO 10 J = 1, N
+ CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 )
+ 10 CONTINUE
+ ELSE
+ DO 20 J = 1, N
+ CALL DSCAL( J, SIGMA, A( 1, J ), 1 )
+ 20 CONTINUE
+ END IF
+ IF( ABSTOL.GT.0 )
+ $ ABSTLL = ABSTOL*SIGMA
+ IF( VALEIG ) THEN
+ VLL = VL*SIGMA
+ VUU = VU*SIGMA
+ END IF
+ END IF
+
+* Initialize indices into workspaces. Note: The IWORK indices are
+* used only if DSTERF or DSTEMR fail.
+
+* WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the
+* elementary reflectors used in DSYTRD.
+ INDTAU = 1
+* WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries.
+ INDD = INDTAU + N
+* WORK(INDE:INDE+N-1) stores the off-diagonal entries of the
+* tridiagonal matrix from DSYTRD.
+ INDE = INDD + N
+* WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over
+* -written by DSTEMR (the DSTERF path copies the diagonal to W).
+ INDDD = INDE + N
+* WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over
+* -written while computing the eigenvalues in DSTERF and DSTEMR.
+ INDEE = INDDD + N
+* INDWK is the starting offset of the left-over workspace, and
+* LLWORK is the remaining workspace size.
+ INDWK = INDEE + N
+ LLWORK = LWORK - INDWK + 1
+
+* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
+* stores the block indices of each of the M<=N eigenvalues.
+ INDIBL = 1
+* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
+* stores the starting and finishing indices of each block.
+ INDISP = INDIBL + N
+* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
+* that corresponding to eigenvectors that fail to converge in
+* DSTEIN. This information is discarded; if any fail, the driver
+* returns INFO > 0.
+ INDIFL = INDISP + N
+* INDIWO is the offset of the remaining integer workspace.
+ INDIWO = INDISP + N
+
+*
+* Call DSYTRD to reduce symmetric matrix to tridiagonal form.
+*
+ CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ),
+ $ WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO )
+*
+* If all eigenvalues are desired
+* then call DSTERF or DSTEMR and DORMTR.
+*
+ IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND.
+ $ IEEEOK.EQ.1 ) THEN
+ IF( .NOT.WANTZ ) THEN
+ CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
+ CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
+ CALL DSTERF( N, W, WORK( INDEE ), INFO )
+ ELSE
+ CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
+ CALL DCOPY( N, WORK( INDD ), 1, WORK( INDDD ), 1 )
+*
+ IF (ABSTOL .LE. TWO*N*EPS) THEN
+ TRYRAC = .TRUE.
+ ELSE
+ TRYRAC = .FALSE.
+ END IF
+ CALL DSTEMR( JOBZ, 'A', N, WORK( INDDD ), WORK( INDEE ),
+ $ VL, VU, IL, IU, M, W, Z, LDZ, N, ISUPPZ,
+ $ TRYRAC, WORK( INDWK ), LWORK, IWORK, LIWORK,
+ $ INFO )
+*
+*
+*
+* Apply orthogonal matrix used in reduction to tridiagonal
+* form to eigenvectors returned by DSTEIN.
+*
+ IF( WANTZ .AND. INFO.EQ.0 ) THEN
+ INDWKN = INDE
+ LLWRKN = LWORK - INDWKN + 1
+ CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA,
+ $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ),
+ $ LLWRKN, IINFO )
+ END IF
+ END IF
+*
+*
+ IF( INFO.EQ.0 ) THEN
+* Everything worked. Skip DSTEBZ/DSTEIN. IWORK(:) are
+* undefined.
+ M = N
+ GO TO 30
+ END IF
+ INFO = 0
+ END IF
+*
+* Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN.
+* Also call DSTEBZ and DSTEIN if DSTEMR fails.
+*
+ IF( WANTZ ) THEN
+ ORDER = 'B'
+ ELSE
+ ORDER = 'E'
+ END IF
+
+ CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
+ $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
+ $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWK ),
+ $ IWORK( INDIWO ), INFO )
+*
+ IF( WANTZ ) THEN
+ CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
+ $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
+ $ WORK( INDWK ), IWORK( INDIWO ), IWORK( INDIFL ),
+ $ INFO )
+*
+* Apply orthogonal matrix used in reduction to tridiagonal
+* form to eigenvectors returned by DSTEIN.
+*
+ INDWKN = INDE
+ LLWRKN = LWORK - INDWKN + 1
+ CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
+ $ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
+ END IF
+*
+* If matrix was scaled, then rescale eigenvalues appropriately.
+*
+* Jump here if DSTEMR/DSTEIN succeeded.
+ 30 CONTINUE
+ IF( ISCALE.EQ.1 ) THEN
+ IF( INFO.EQ.0 ) THEN
+ IMAX = M
+ ELSE
+ IMAX = INFO - 1
+ END IF
+ CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
+ END IF
+*
+* If eigenvalues are not in order, then sort them, along with
+* eigenvectors. Note: We do not sort the IFAIL portion of IWORK.
+* It may not be initialized (if DSTEMR/DSTEIN succeeded), and we do
+* not return this detailed information to the user.
+*
+ IF( WANTZ ) THEN
+ DO 50 J = 1, M - 1
+ I = 0
+ TMP1 = W( J )
+ DO 40 JJ = J + 1, M
+ IF( W( JJ ).LT.TMP1 ) THEN
+ I = JJ
+ TMP1 = W( JJ )
+ END IF
+ 40 CONTINUE
+*
+ IF( I.NE.0 ) THEN
+ W( I ) = W( J )
+ W( J ) = TMP1
+ CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
+ END IF
+ 50 CONTINUE
+ END IF
+*
+* Set WORK(1) to optimal workspace size.
+*
+ WORK( 1 ) = LWKOPT
+ IWORK( 1 ) = LIWMIN
+*
+ RETURN
+*
+* End of DSYEVR
+*
+ END