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CHPGVD(l)			       )			     CHPGVD(l)

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

SYNOPSIS
       SUBROUTINE CHPGVD( ITYPE,  JOBZ,	 UPLO,	N,  AP,	 BP,  W, Z, LDZ, WORK,
			  LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )

	   CHARACTER	  JOBZ, UPLO

	   INTEGER	  INFO, ITYPE, LDZ, LIWORK, LRWORK, LWORK, N

	   INTEGER	  IWORK( * )

	   REAL		  RWORK( * ), W( * )

	   COMPLEX	  AP( * ), BP( * ), WORK( * ), Z( LDZ, * )

PURPOSE
       CHPGVD computes all the eigenvalues and, optionally,  the  eigenvectors
       of  a  complex generalized Hermitian-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  Hermitian, stored in packed format, and B is also
       positive definite.
       If eigenvectors are desired, it uses a divide and conquer algorithm.

       The divide and conquer algorithm	 makes	very  mild  assumptions	 about
       floating	 point arithmetic. It will work on machines with a guard digit
       in add/subtract, or on those binary machines without guard digits which
       subtract	 like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
       conceivably fail on hexadecimal or decimal machines without guard  dig‐
       its, but we know of none.

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.

       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.

       AP      (input/output) COMPLEX array, dimension (N*(N+1)/2)
	       On  entry,  the upper or lower triangle of the Hermitian 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) COMPLEX array, dimension (N*(N+1)/2)
	       On  entry,  the upper or lower triangle of the Hermitian 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 factor‐
	       ization B = U**H*U or B = L*L**H, in the same storage format as
	       B.

       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.  The eigenvectors are normalized as  follows:  if
	       ITYPE  = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(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) COMPLEX array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The dimension of array WORK.  If N <= 1,		      LWORK >=
	       1.  If JOBZ = 'N' and N > 1, LWORK >= N.	 If JOBZ = 'V' and N >
	       1, LWORK >= 2*N.

	       If  LWORK  = -1, then a workspace query is assumed; the routine
	       only calculates the optimal size of  the	 WORK  array,  returns
	       this  value  as the first entry of the WORK array, and no error
	       message related to LWORK is issued by XERBLA.

       RWORK   (workspace) REAL array, dimension (LRWORK)
	       On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.

       LRWORK  (input) INTEGER
	       The dimension of array RWORK.  If N <= 1,		LRWORK
	       >= 1.  If JOBZ = 'N' and N > 1, LRWORK >= N.  If JOBZ = 'V' and
	       N > 1, LRWORK >= 1 + 5*N + 2*N**2.

	       If LRWORK = -1, then a workspace query is assumed; the  routine
	       only  calculates	 the  optimal size of the RWORK array, returns
	       this value as the first entry of the RWORK array, and no	 error
	       message related to LRWORK is issued by XERBLA.

       IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
	       On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

       LIWORK  (input) INTEGER
	       The dimension of array IWORK.  If JOBZ  = 'N' or N <= 1, LIWORK
	       >= 1.  If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.

	       If LIWORK = -1, then a workspace query is assumed; the  routine
	       only  calculates	 the  optimal size of the IWORK array, returns
	       this value as the first entry of the IWORK array, and no	 error
	       message related to LIWORK is issued by XERBLA.

       INFO    (output) INTEGER
	       = 0:  successful exit
	       < 0:  if INFO = -i, the i-th argument had an illegal value
	       > 0:  CPPTRF or CHPEVD returned an error code:
	       <=  N:	if INFO = i, CHPEVD failed to converge; i off-diagonal
	       elements of an intermediate tridiagonal form did not convergeto
	       zero; > N:   if INFO = N + i, for 1 <= i <= n, then the leading
	       minor of order i of 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			     CHPGVD(l)
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