!>
!> 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 eigenvalues.
!> 

JOBZ

!>          JOBZ is CHARACTER*1
!>          = 'N':  Compute eigenvalues only;
!>          = 'V':  Compute eigenvalues and eigenvectors.
!> 

RANGE

!>          RANGE is 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

!>          UPLO is CHARACTER*1
!>          = 'U':  Upper triangles of A and B are stored;
!>          = 'L':  Lower triangles of A and B are stored.
!> 

N

!>          N is INTEGER
!>          The order of the matrices A and B.  N >= 0.
!> 

KA

!>          KA is INTEGER
!>          The number of superdiagonals of the matrix A if UPLO = 'U',
!>          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.
!> 

KB

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

AB

!>          AB is DOUBLE PRECISION array, dimension (LDAB, N)
!>          On entry, the upper or lower triangle of the symmetric 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

!>          LDAB is INTEGER
!>          The leading dimension of the array AB.  LDAB >= KA+1.
!> 

BB

!>          BB is DOUBLE PRECISION array, dimension (LDBB, N)
!>          On entry, the upper or lower triangle of the symmetric 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 factorization
!>          B = S**T*S, as returned by DPBSTF.
!> 

LDBB

!>          LDBB is INTEGER
!>          The leading dimension of the array BB.  LDBB >= KB+1.
!> 

Q

!>          Q is 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

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

VL

!>          VL is DOUBLE PRECISION
!>
!>          If RANGE='V', the lower bound of the interval to
!>          be searched for eigenvalues. VL < VU.
!>          Not referenced if RANGE = 'A' or 'I'.
!> 

VU

!>          VU is DOUBLE PRECISION
!>
!>          If RANGE='V', the upper bound of the interval to
!>          be searched for eigenvalues. VL < VU.
!>          Not referenced if RANGE = 'A' or 'I'.
!> 

IL

!>          IL is INTEGER
!>
!>          If RANGE='I', the index of the
!>          smallest eigenvalue 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'.
!> 

IU

!>          IU is INTEGER
!>
!>          If RANGE='I', the index of the
!>          largest eigenvalue 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

!>          ABSTOL is 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

!>          M is INTEGER
!>          The total number of eigenvalues found.  0 <= M <= N.
!>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
!> 

W

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

Z

!>          Z is 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

!>          LDZ is INTEGER
!>          The leading dimension of the array Z.  LDZ >= 1, and if
!>          JOBZ = 'V', LDZ >= max(1,N).
!> 

WORK

!>          WORK is DOUBLE PRECISION array, dimension (7*N)
!> 

IWORK

!>          IWORK is INTEGER array, dimension (5*N)
!> 

IFAIL

!>          IFAIL is INTEGER array, dimension (M)
!>          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 eigenvalues that failed to converge.
!>          If JOBZ = 'N', then IFAIL is not referenced.
!> 

INFO

!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal 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
!>                principal minor of order i of B is not positive.
!>                The factorization of B could not be completed and
!>                no eigenvalues or eigenvectors were computed.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA