```      SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
\$                   IL, IU, ABSTOL, M, W, Z, LDZ, WORK, 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, ITYPE, IU, LDZ, M, N
DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
INTEGER            IFAIL( * ), IWORK( * )
DOUBLE PRECISION   AP( * ), BP( * ), W( * ), WORK( * ),
\$                   Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  DSPGVX computes selected eigenvalues, and optionally, eigenvectors
*  of a real generalized symmetric-definite eigenproblem, of the form
*  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
*  and B are assumed to be symmetric, stored in packed storage, and B
*  is also positive definite.  Eigenvalues and eigenvectors can be
*  selected by specifying either a range of values or a range of indices
*  for the desired eigenvalues.
*
*  Arguments
*  =========
*
*  ITYPE   (input) INTEGER
*          Specifies the problem type to be solved:
*          = 1:  A*x = (lambda)*B*x
*          = 2:  A*B*x = (lambda)*x
*          = 3:  B*A*x = (lambda)*x
*
*  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 and B are stored;
*          = 'L':  Lower triangle of A and B are stored.
*
*  N       (input) INTEGER
*          The order of the matrix pencil (A,B).  N >= 0.
*
*  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          A, packed columnwise in a linear array.  The j-th column of A
*          is stored in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*
*          On exit, the contents of AP are destroyed.
*
*  BP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          B, packed columnwise in a linear array.  The j-th column of B
*          is stored in the array BP as follows:
*          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
*          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
*
*          On exit, the triangular factor U or L from the Cholesky
*          factorization B = U**T*U or B = L*L**T, in the same storage
*          format as B.
*
*  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').
*
*  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 = 'N', then Z is not referenced.
*          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).
*          The eigenvectors are normalized as follows:
*          if ITYPE = 1 or 2, Z**T*B*Z = I;
*          if ITYPE = 3, Z**T*inv(B)*Z = 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.
*          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) DOUBLE PRECISION array, dimension (8*N)
*
*  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:  DPPTRF or DSPEVX returned an error code:
*             <= N:  if INFO = i, DSPEVX failed to converge;
*                    i eigenvectors failed to converge.  Their indices
*                    are stored in array IFAIL.
*             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
*                    minor of order i of B is not positive definite.
*                    The factorization of B could not be completed and
*                    no eigenvalues or eigenvectors were computed.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
* =====================================================================
*
*     .. Local Scalars ..
LOGICAL            ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
CHARACTER          TRANS
INTEGER            J
*     ..
*     .. External Functions ..
LOGICAL            LSAME
EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
EXTERNAL           DPPTRF, DSPEVX, DSPGST, DTPMV, DTPSV, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
UPPER = LSAME( UPLO, 'U' )
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
*
INFO = 0
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -3
ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL ) THEN
INFO = -9
END IF
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 ) THEN
INFO = -10
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -11
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -16
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPGVX', -INFO )
RETURN
END IF
*
*     Quick return if possible
*
M = 0
IF( N.EQ.0 )
\$   RETURN
*
*     Form a Cholesky factorization of B.
*
CALL DPPTRF( UPLO, N, BP, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
*     Transform problem to standard eigenvalue problem and solve.
*
CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
CALL DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M,
\$             W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
*
IF( WANTZ ) THEN
*
*        Backtransform eigenvectors to the original problem.
*
IF( INFO.GT.0 )
\$      M = INFO - 1
IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
*           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
*           backtransform eigenvectors: x = inv(L)'*y or inv(U)*y
*
IF( UPPER ) THEN
TRANS = 'N'
ELSE
TRANS = 'T'
END IF
*
DO 10 J = 1, M
CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
\$                     1 )
10       CONTINUE
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
*           For B*A*x=(lambda)*x;
*           backtransform eigenvectors: x = L*y or U'*y
*
IF( UPPER ) THEN
TRANS = 'T'
ELSE
TRANS = 'N'
END IF
*
DO 20 J = 1, M
CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
\$                     1 )
20       CONTINUE
END IF
END IF
*
RETURN
*
*     End of DSPGVX
*
END

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