Add a useful package (from Source forge) for octave

[CreaPhase.git] / octave_packages / linear-algebra-2.2.0 / circulant_inv.m

diff --git a/octave_packages/linear-algebra-2.2.0/circulant_inv.m b/octave_packages/linear-algebra-2.2.0/circulant_inv.m
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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{c} =} circulant_inv (@var{v})
+##
+## Fast, compact calculation of inverse of a circulant matrix@*
+## Given an @var{n}*1 vector @var{v}, return the inverse @var{c} of the @var{n}*@var{n} circulant matrix @var{C} that has @var{v} as its first column
+## The returned @var{c} is the first column of the inverse, which is also circulant -- to get the full matrix, use `circulant_make_matrix(c)'
+##
+## Theoretically same as @code{inv(make_circulant_matrix(v))(:, 1)}, but requires many fewer computations and does not form matrices explicitly
+##
+## Roundoff may induce a small imaginary component in @var{c} even if @var{v} is real -- use @code{real(c)} to remedy this
+##
+## Reference: Robert M. Gray, Toeplitz and Circulant Matrices: A Review, Now Publishers, http://ee.stanford.edu/~gray/toeplitz.pdf, Chapter 3
+##
+## @seealso{circulant_make_matrix, circulant_matrix_vector_product, circulant_eig}
+## @end deftypefn
+
+function c = circulant_inv(v)
+
+  ## Find the eigenvalues and eigenvectors
+  [vs, lambda] = circulant_eig(v);
+
+  ## Find the first column of the inverse
+  c = vs * diag(1 ./ diag(lambda)) * conj(vs(:, 1));
+
+endfunction
+
+%!shared v
+%! v = [1 2 3]';
+%!assert (circulant_make_matrix(circulant_inv(v)), inv(circulant_make_matrix(v)), 10*eps);