*> \brief DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DSYEVR + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, * ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, * IWORK, LIWORK, INFO ) * * .. 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, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> 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. *> \endverbatim * * Arguments: * ========== * *> \param[in] JOBZ *> \verbatim *> JOBZ is CHARACTER*1 *> = 'N': Compute eigenvalues only; *> = 'V': Compute eigenvalues and eigenvectors. *> \endverbatim *> *> \param[in] RANGE *> \verbatim *> RANGE is 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 *> \endverbatim *> *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> = 'U': Upper triangle of A is stored; *> = 'L': Lower triangle of A is stored. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is 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. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[in] VL *> \verbatim *> VL is DOUBLE PRECISION *> \endverbatim *> *> \param[in] VU *> \verbatim *> VU is 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'. *> \endverbatim *> *> \param[in] IL *> \verbatim *> IL is INTEGER *> \endverbatim *> *> \param[in] IU *> \verbatim *> IU is 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'. *> \endverbatim *> *> \param[in] ABSTOL *> \verbatim *> ABSTOL is 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. *> \endverbatim *> *> \param[out] M *> \verbatim *> M is INTEGER *> The total number of eigenvalues found. 0 <= M <= N. *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. *> \endverbatim *> *> \param[out] W *> \verbatim *> W is DOUBLE PRECISION array, dimension (N) *> The first M elements contain the selected eigenvalues in *> ascending order. *> \endverbatim *> *> \param[out] Z *> \verbatim *> Z is 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. *> \endverbatim *> *> \param[in] LDZ *> \verbatim *> LDZ is INTEGER *> The leading dimension of the array Z. LDZ >= 1, and if *> JOBZ = 'V', LDZ >= max(1,N). *> \endverbatim *> *> \param[out] ISUPPZ *> \verbatim *> ISUPPZ is 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 *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is 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. *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) *> On exit, if INFO = 0, IWORK(1) returns the optimal LWORK. *> \endverbatim *> *> \param[in] LIWORK *> \verbatim *> LIWORK is 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. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> > 0: Internal error *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date September 2012 * *> \ingroup doubleSYeigen * *> \par Contributors: * ================== *> *> Inderjit Dhillon, IBM Almaden, USA \n *> Osni Marques, LBNL/NERSC, USA \n *> Ken Stanley, Computer Science Division, University of *> California at Berkeley, USA \n *> Jason Riedy, Computer Science Division, University of *> California at Berkeley, USA \n *> * ===================================================================== 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 3.4.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * September 2012 * * .. 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, * ) * .. * * ===================================================================== * * .. 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 ) THEN Z( 1, 1 ) = ONE ISUPPZ( 1 ) = 1 ISUPPZ( 2 ) = 1 END IF 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 IF (VALEIG) THEN VLL = VL VUU = VU END IF 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 = INDIFL + 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