```      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 = 0.0D+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

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