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


SYNOPSIS

       SUBROUTINE DSBGVX( 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

           DOUBLE         PRECISION ABSTOL, VL, VU

           INTEGER        IFAIL( * ), IWORK( * )

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


PURPOSE

       DSBGVX  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.

               The number of superdiagonals of the  matrix  A  if
               UPLO  = 'U', or the number of subdiagonals if UPLO
               = '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) DOUBLE PRECISION 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)  DOUBLE PRECISION 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 DPBSTF.

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

       Q       (output) DOUBLE PRECISION 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).

               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 refer­
               enced 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)
               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, 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).

               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  ele­
               ments  of IFAIL are zero.  If INFO > 0, then 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
               : DPBSTF 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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