SSPGVX  -  compute  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


SYNOPSIS

       SUBROUTINE SSPGVX( ITYPE, JOBZ, RANGE, UPLO,  N,  AP,  BP,
                          VL,  VU,  IL, IU, ABSTOL, M, W, Z, LDZ,
                          WORK, IWORK, IFAIL, INFO )

           CHARACTER      JOBZ, RANGE, UPLO

           INTEGER        IL, INFO, ITYPE, IU, LDZ, M, N

           REAL           ABSTOL, VL, VU

           INTEGER        IFAIL( * ), IWORK( * )

           REAL           AP( * ), BP( * ), W( * ), WORK( * ), Z(
                          LDZ, * )


PURPOSE

       SSPGVX  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.

               The order of the matrix pencil (A,B).  N >= 0.

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

       VL      (input) REAL
               VU      (input) REAL 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  refer­
               enced if RANGE = 'A' or 'V'.

       ABSTOL  (input) REAL
               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

               with INFO>0, indicating that some eigenvectors did
               not converge, try setting ABSTOL to 2*SLAMCH('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) REAL array, dimension (N)
               On  normal  exit, the first M elements contain the
               selected eigenvalues in ascending order.

       Z       (output) REAL 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 asso­
               ciated 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) REAL 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 ele­
               ments 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 ille­
               gal value
               > 0:  SPPTRF or SSPEVX returned an error code:
               <= N:  if INFO = i, SSPEVX failed to  converge;  i
               eigenvectors  failed  to  converge.  Their indices

               i,  for  1  <=  i  <= N, then the leading minor of
               order i of B is not positive definite.   The  fac­
               torization  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  Ken­
       tucky, USA


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