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

DSBGVX(l)					)					DSBGVX(l)

NAME
       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 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 sub-
	       diagonals if UPLO = 'L'.  KA >= 0.

       KB      (input) INTEGER
	       The number of superdiagonals of the matrix B if UPLO = 'U', or the number of  sub-
	       diagonals if UPLO = 'L'.  KB >= 0.

       AB      (input/output) 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 col-
	       umn  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 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 col-
	       umn 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    (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).

       VL      (input) DOUBLE PRECISION
	       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 referenced 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).

       WORK    (workspace/output) DOUBLE PRECISION array, 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 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    (output) 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 minor of order i of B is not positive definite.  The
	       factorization  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 				DSBGVX(l)


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