```      SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
\$                   ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
\$                   IFAIL, 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, LWORK, M, N
DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
INTEGER            IFAIL( * ), IWORK( * )
DOUBLE PRECISION   A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  DSYEVX 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.
*
*  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.
*
*  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.
*
*          Eigenvalues will be computed most accurately when ABSTOL is
*          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*          If this routine returns with INFO>0, indicating that some
*          eigenvectors did not converge, try setting ABSTOL to
*          2*DLAMCH('S').
*
*          See "Computing Small Singular Values of Bidiagonal Matrices
*          with Guaranteed High Relative Accuracy," by Demmel and
*          Kahan, LAPACK Working Note #3.
*
*  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)
*          On normal exit, 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 an eigenvector fails to converge, then that column of Z
*          contains the latest approximation to the eigenvector, and the
*          index of the eigenvector is returned in IFAIL.
*          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.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.  LDZ >= 1, and if
*          JOBZ = 'V', LDZ >= max(1,N).
*
*  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 length of the array WORK.  LWORK >= 1, when N <= 1;
*          otherwise 8*N.
*          For optimal efficiency, LWORK >= (NB+3)*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) INTEGER array, dimension (5*N)
*
*  IFAIL   (output) INTEGER array, dimension (N)
*          If JOBZ = 'V', then if INFO = 0, the first M elements of
*          IFAIL are zero.  If INFO > 0, then IFAIL contains the
*          indices of the eigenvectors that failed to converge.
*          If JOBZ = 'N', then IFAIL is not referenced.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = i, then i eigenvectors failed to converge.
*                Their indices are stored in array IFAIL.
*
* =====================================================================
*
*     .. Parameters ..
DOUBLE PRECISION   ZERO, ONE
PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
LOGICAL            ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
\$                   WANTZ
CHARACTER          ORDER
INTEGER            I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
\$                   INDISP, INDIWO, INDTAU, INDWKN, INDWRK, ISCALE,
\$                   ITMP1, J, JJ, LLWORK, LLWRKN, LWKMIN,
\$                   LWKOPT, 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, DLACPY, DORGTR, DORMTR, DSCAL, DSTEBZ,
\$                   DSTEIN, DSTEQR, DSTERF, DSWAP, DSYTRD, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
LOWER = LSAME( UPLO, 'L' )
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
LQUERY = ( LWORK.EQ.-1 )
*
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
END IF
END IF
*
IF( INFO.EQ.0 ) THEN
IF( N.LE.1 ) THEN
LWKMIN = 1
WORK( 1 ) = LWKMIN
ELSE
LWKMIN = 8*N
NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) )
LWKOPT = MAX( LWKMIN, ( NB + 3 )*N )
WORK( 1 ) = LWKOPT
END IF
*
IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
\$      INFO = -17
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYEVX', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
*     Quick return if possible
*
M = 0
IF( N.EQ.0 ) THEN
RETURN
END IF
*
IF( N.EQ.1 ) THEN
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
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
*
*     Call DSYTRD to reduce symmetric matrix to tridiagonal form.
*
INDTAU = 1
INDE = INDTAU + N
INDD = INDE + N
INDWRK = INDD + N
LLWORK = LWORK - INDWRK + 1
CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ),
\$             WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
*
*     If all eigenvalues are desired and ABSTOL is less than or equal to
*     zero, then call DSTERF or DORGTR and SSTEQR.  If this fails for
*     some eigenvalue, then try DSTEBZ.
*
TEST = .FALSE.
IF( INDEIG ) THEN
IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
TEST = .TRUE.
END IF
END IF
IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
INDEE = INDWRK + 2*N
IF( .NOT.WANTZ ) THEN
CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
CALL DSTERF( N, W, WORK( INDEE ), INFO )
ELSE
CALL DLACPY( 'A', N, N, A, LDA, Z, LDZ )
CALL DORGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
\$                   WORK( INDWRK ), LLWORK, IINFO )
CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
\$                   WORK( INDWRK ), INFO )
IF( INFO.EQ.0 ) THEN
DO 30 I = 1, N
IFAIL( I ) = 0
30          CONTINUE
END IF
END IF
IF( INFO.EQ.0 ) THEN
M = N
GO TO 40
END IF
INFO = 0
END IF
*
*     Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
INDIBL = 1
INDISP = INDIBL + N
INDIWO = INDISP + N
CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
\$             WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
\$             IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
\$             IWORK( INDIWO ), INFO )
*
IF( WANTZ ) THEN
CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
\$                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
\$                WORK( INDWRK ), IWORK( INDIWO ), IFAIL, 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.
*
40 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.
*
IF( WANTZ ) THEN
DO 60 J = 1, M - 1
I = 0
TMP1 = W( J )
DO 50 JJ = J + 1, M
IF( W( JJ ).LT.TMP1 ) THEN
I = JJ
TMP1 = W( JJ )
END IF
50       CONTINUE
*
IF( I.NE.0 ) THEN
ITMP1 = IWORK( INDIBL+I-1 )
W( I ) = W( J )
IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
W( J ) = TMP1
IWORK( INDIBL+J-1 ) = ITMP1
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
IF( INFO.NE.0 ) THEN
ITMP1 = IFAIL( I )
IFAIL( I ) = IFAIL( J )
IFAIL( J ) = ITMP1
END IF
END IF
60    CONTINUE
END IF
*
*     Set WORK(1) to optimal workspace size.
*
WORK( 1 ) = LWKOPT
*
RETURN
*
*     End of DSYEVX
*
END

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