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## Copyright (C) 2012 Nir Krakauer <nkrakauer@ccny.cuny.edu>
##
## This program is free software; you can redistribute it and/or modify it under
## the terms of the GNU General Public License as published by the Free Software
## Foundation; either version 3 of the License, or (at your option) any later
## version.
##
## This program is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
## details.
##
## You should have received a copy of the GNU General Public License along with
## this program; if not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn{Function File} {@var{lambda} =} circulant_eig (@var{v})
## @deftypefnx{Function File} {[@var{vs}, @var{lambda}] =} circulant_eig (@var{v})
##
## Fast, compact calculation of eigenvalues and eigenvectors of a circulant matrix@*
## Given an @var{n}*1 vector @var{v}, return the eigenvalues @var{lambda} and optionally eigenvectors @var{vs} of the @var{n}*@var{n} circulant matrix @var{C} that has @var{v} as its first column
##
## Theoretically same as @code{eig(make_circulant_matrix(v))}, but many fewer computations; does not form @var{C} explicitly
##
## Reference: Robert M. Gray, Toeplitz and Circulant Matrices: A Review, Now Publishers, http://ee.stanford.edu/~gray/toeplitz.pdf, Chapter 3
##
## @seealso{gallery, circulant_matrix_vector_product, circulant_inv}
## @end deftypefn
function [a, b] = circulant_eig (v)
## FIXME when warning for broadcastin is turned off by default, this
## unwind_protect block could be removed
## we are using broadcasting on the code below so we turn off the
## warnings but will restore to previous state at the end
bc_warn = warning ("query", "Octave:broadcast");
unwind_protect
warning ("off", "Octave:broadcast");
#find the eigenvalues
n = numel(v);
lambda = ones(n, 1);
s = (0:(n-1));
lambda = sum(v .* exp(-2*pi*i*s'*s/n))';
if nargout < 2
a = lambda;
return
endif
#find the eigenvectors (which in fact do not depend on v)
a = exp(-2*i*pi*s'*s/n) / sqrt(n);
b = diag(lambda);
unwind_protect_cleanup
## restore broadcats warning status
warning (bc_warn.state, "Octave:broadcast");
end_unwind_protect
endfunction
%!shared v,C,vs,lambda
%! v = [1 2 3]';
%! C = gallery("circul", v)';
%! [vs lambda] = circulant_eig(v);
%!assert (vs*lambda, C*vs, 100*eps);

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