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CHBGVX(l)			       )			     CHBGVX(l)

NAME
       CHBGVX  - compute all the eigenvalues, and optionally, the eigenvectors
       of a complex generalized Hermitian-definite banded eigenproblem, of the
       form A*x=(lambda)*B*x

SYNOPSIS
       SUBROUTINE CHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q,
			  LDQ, VL, VU, IL, IU, ABSTOL, M,  W,  Z,  LDZ,	 WORK,
			  RWORK, 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		  RWORK( * ), W( * )

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

PURPOSE
       CHBGVX computes all the eigenvalues, and optionally,  the  eigenvectors
       of a complex generalized Hermitian-definite banded eigenproblem, of the
       form A*x=(lambda)*B*x. Here A and B are assumed	to  be	Hermitian  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.

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 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) COMPLEX array, dimension (LDAB, N)
	       On  entry,  the	upper  or lower triangle of the Hermitian 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) COMPLEX array, dimension (LDBB, N)
	       On entry, the upper or lower triangle  of  the  Hermitian  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(kb+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**H*S, as returned by CPBSTF.

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

       Q       (output) COMPLEX 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 con‐
	       sequently 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 = '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
	       referenced if RANGE = 'A' or 'V'.

       ABSTOL  (input) REAL
	       The  absolute error tolerance for the eigenvalues.  An approxi‐
	       mate 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 reduc‐
	       ing AP 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	eigen‐
	       vectors 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) COMPLEX 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  that
	       Z**H*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 >= N.

       WORK    (workspace) COMPLEX array, dimension (N)

       RWORK   (workspace) REAL array, dimension (7*N)

       IWORK   (workspace) INTEGER array, dimension (5*N)

       IFAIL   (output) INTEGER array, dimension (N)
	       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
	       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 illegal value
	       > 0:  if INFO = i, and i is:
	       <=  N:	then i eigenvectors failed to converge.	 Their indices
	       are stored in array IFAIL.  > N:	  if INFO = N + i, for 1 <=  i
	       <= N, then CPBSTF
	       returned	 INFO = i: B is not positive definite.	The factoriza‐
	       tion of B could not be completed and no eigenvalues  or	eigen‐
	       vectors were computed.

FURTHER DETAILS
       Based on contributions by
	  Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

LAPACK version 3.0		 15 June 2000			     CHBGVX(l)
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