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RedHat 9 (Linux i386) - man page for chpgvx (redhat section l)

CHPGVX(l)					)					CHPGVX(l)

NAME
       CHPGVX  - compute selected eigenvalues and, optionally, eigenvectors of a complex general-
       ized Hermitian-definite eigenproblem, of the form  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or
       B*A*x=(lambda)*x

SYNOPSIS
       SUBROUTINE CHPGVX( ITYPE,  JOBZ,  RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z,
			  LDZ, WORK, RWORK, IWORK, IFAIL, INFO )

	   CHARACTER	  JOBZ, RANGE, UPLO

	   INTEGER	  IL, INFO, ITYPE, IU, LDZ, M, N

	   REAL 	  ABSTOL, VL, VU

	   INTEGER	  IFAIL( * ), IWORK( * )

	   REAL 	  RWORK( * ), W( * )

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

PURPOSE
       CHPGVX computes selected eigenvalues and, optionally, eigenvectors of a	complex  general-
       ized  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.   Eigenvalues and eigenvectors can be selected by specifying
       either a range of values or a range of indices for the desired eigenvalues.

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.

       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.

       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 factorization B = U**H*U
	       or B = L*L**H, in the same storage format as B.

       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 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 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  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)
	       On normal exit, the first M elements contain the selected eigenvalues in ascending
	       order.

       Z       (output) COMPLEX array, dimension (LDZ, N)
	       If JOBZ = 'N', then Z is not referenced.  If JOBZ = 'V', then if  INFO  =  0,  the
	       first  M  columns of Z contain the orthonormal eigenvectors of the matrix A corre-
	       sponding to the selected eigenvalues, with the i-th column of Z holding the eigen-
	       vector associated with W(i).  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 an eigenvector fails to converge, then that column of  Z  contains  the  latest
	       approximation  to the eigenvector, and the index of the eigenvector is returned in
	       IFAIL.  Note: the user must ensure that at least max(1,M) columns are supplied  in
	       the  array  Z; if RANGE = 'V', the exact value of M is not known in advance and an
	       upper bound must be used.

       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 (2*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:  CPPTRF or CHPEVX returned an error code:
	       <=  N:  if INFO = i, CHPEVX failed to converge; i eigenvectors failed to converge.
	       Their indices are stored in array IFAIL.  > 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 eigenvectors were computed.

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

LAPACK version 3.0			   15 June 2000 				CHPGVX(l)


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