DSPGV  -  compute all the eigenvalues and, optionally, the
       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


SYNOPSIS

       SUBROUTINE DSPGV( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ,
                         WORK, INFO )

           CHARACTER     JOBZ, UPLO

           INTEGER       INFO, ITYPE, LDZ, N

           DOUBLE        PRECISION  AP(  *  ),  BP(  * ), W( * ),
                         WORK( * ), Z( LDZ, * )


PURPOSE

       DSPGV computes all the eigenvalues  and,  optionally,  the
       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 format, and B is
       also positive definite.


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.

       UPLO    (input) CHARACTER*1
               = 'U':  Upper triangles of A and B are stored;
               = 'L':  Lower triangles of A and B are stored.

       N       (input) INTEGER
               The order of the matrices A and 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.

               dimension (N*(N+1)/2)
               On entry, the upper or lower triangle of the  sym­
               metric  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.

       W       (output) DOUBLE PRECISION array, dimension (N)
               If INFO = 0, the eigenvalues in ascending order.

       Z       (output) DOUBLE PRECISION array, dimension (LDZ,
               N)
               If JOBZ = 'V', then if INFO = 0,  Z  contains  the
               matrix  Z  of  eigenvectors.  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 JOBZ =
               'N', then Z is not referenced.

       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
               (3*N)

       INFO    (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value
               > 0:  DPPTRF or DSPEV returned an error code:
               <=  N:   if  INFO = i, DSPEV failed to converge; i
               off-diagonal elements of an intermediate tridiago­
               nal form did not converge to zero.  > 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.


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