SSBGVX  -  compute  selected  eigenvalues, and optionally,
       eigenvectors  of  a  real  generalized  symmetric-definite
       banded eigenproblem, of the form A*x=(lambda)*B*x


SYNOPSIS

       SUBROUTINE SSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB,
                          BB, LDBB,  Q,  LDQ,  VL,  VU,  IL,  IU,
                          ABSTOL,  M,  W,  Z,  LDZ,  WORK, IWORK,
                          IFAIL, INFO )

           CHARACTER      JOBZ, RANGE, UPLO

           INTEGER        IL, INFO, IU, KA, KB, LDAB, LDBB,  LDQ,
                          LDZ, M, N

           REAL           ABSTOL, VL, VU

           INTEGER        IFAIL( * ), IWORK( * )

           REAL           AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, *
                          ), W( * ), WORK( * ), Z( LDZ, * )


PURPOSE

       SSBGVX  computes  selected  eigenvalues,  and  optionally,
       eigenvectors  of  a  real  generalized  symmetric-definite
       banded eigenproblem, of the form A*x=(lambda)*B*x. Here  A
       and  B  are  assumed  to be symmetric and banded, and B is
       also positive definite.  Eigenvalues and eigenvectors  can
       be  selected by specifying either all eigenvalues, a range
       of values or a range of indices for the desired  eigenval­
       ues.


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 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.

       KA      (input) INTEGER
               The  number  of  superdiagonals of the matrix A if

               = 'L'.  KA >= 0.

       KB      (input) INTEGER
               The  number  of  superdiagonals of the matrix B if
               UPLO = 'U', or the number of subdiagonals if  UPLO
               = 'L'.  KB >= 0.

       AB      (input/output) REAL array, dimension (LDAB, N)
               On  entry, the upper or lower triangle of the sym­
               metric band matrix A, stored  in  the  first  ka+1
               rows of the array.  The j-th column of A is stored
               in the j-th column of the array AB as follows:  if
               UPLO  =  'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-
               ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j)    =  A(i,j)
               for j<=i<=min(n,j+ka).

               On exit, the contents of AB are destroyed.

       LDAB    (input) INTEGER
               The  leading  dimension  of the array AB.  LDAB >=
               KA+1.

       BB      (input/output) REAL array, dimension (LDBB, N)
               On entry, the upper or lower triangle of the  sym­
               metric  band  matrix  B,  stored in the first kb+1
               rows of the array.  The j-th column of B is stored
               in  the j-th column of the array BB as follows: if
               UPLO = 'U', BB(ka+1+i-j,j) = B(i,j)  for  max(1,j-
               kb)<=i<=j;  if UPLO = 'L', BB(1+i-j,j)    = B(i,j)
               for j<=i<=min(n,j+kb).

               On exit, the factor S from the split Cholesky fac­
               torization B = S**T*S, as returned by SPBSTF.

       LDBB    (input) INTEGER
               The  leading  dimension  of the array BB.  LDBB >=
               KB+1.

       Q       (output) REAL array, dimension (LDQ, N)
               If JOBZ = 'V',  the  n-by-n  matrix  used  in  the
               reduction  of A*x = (lambda)*B*x to standard form,
               i.e. C*x  =  (lambda)*x,  and  consequently  C  to
               tridiagonal  form.   If JOBZ = 'N', the array Q is
               not referenced.

       LDQ     (input) INTEGER
               The leading dimension of the array Q.  If  JOBZ  =
               'N', LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).

       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 =


       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
               2*SLAMCH('S'),  not zero.  If this routine returns
               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)
               If INFO = 0, the eigenvalues in ascending order.

       Z       (output) REAL array, dimension (LDZ, N)
               If JOBZ = 'V', then if INFO = 0,  Z  contains  the
               matrix  Z of eigenvectors, with the i-th column of
               Z holding the eigenvector  associated  with  W(i).
               The  eigenvectors  are normalized so Z**T*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/output) REAL array, dimension (7N)

       IWORK   (workspace/output) INTEGER array, dimension (5N)

       IFAIL   (input) INTEGER array, dimension (M)
               If  JOBZ  =  'V',  then  if  INFO = 0, the first M

               IFAIL contains the indices of the eigenvalues 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
               <= N: if INFO = i, then i eigenvectors  failed  to
               converge.  Their indices are stored in IFAIL.  > N
               : SPBSTF returned an error code; i.e., if INFO = N
               +  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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