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[Octave-cvsupdate] SF.net SVN: octave:[10279] trunk/octave-forge/main/linear-algebra/inst
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From: <nir...@us...> - 2012-04-18 20:42:30
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Revision: 10279
http://octave.svn.sourceforge.net/octave/?rev=10279&view=rev
Author: nir-krakauer
Date: 2012-04-18 20:42:20 +0000 (Wed, 18 Apr 2012)
Log Message:
-----------
Added new functions for working with circulant matrices
Added Paths:
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trunk/octave-forge/main/linear-algebra/inst/circulant_eig.m
trunk/octave-forge/main/linear-algebra/inst/circulant_inv.m
trunk/octave-forge/main/linear-algebra/inst/circulant_make_matrix.m
trunk/octave-forge/main/linear-algebra/inst/circulant_matrix_vector_product.m
Added: trunk/octave-forge/main/linear-algebra/inst/circulant_eig.m
===================================================================
--- trunk/octave-forge/main/linear-algebra/inst/circulant_eig.m (rev 0)
+++ trunk/octave-forge/main/linear-algebra/inst/circulant_eig.m 2012-04-18 20:42:20 UTC (rev 10279)
@@ -0,0 +1,60 @@
+## Copyright (C) 2012 Nir Krakauer
+##
+## 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 2 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
+##
+## @end deftypefn
+## @seealso{circulant_make_matrix, circulant_matrix_vector_product, circulant_inv}
+
+## Author: Nir Krakauer <nkr...@cc...>
+
+function [a, b] = circulant_eig(v)
+
+
+warning ("off", "Octave:broadcast"); #the code below uses broadcasting
+
+#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);
+
+
+endfunction
+
+
+%!shared v,C,vs,lambda
+%! v = [1 2 3]';
+%! C = circulant_make_matrix(v);
+%! [vs lambda] = circulant_eig(v);
+%!assert (vs*lambda, C*vs, 100*eps);
Added: trunk/octave-forge/main/linear-algebra/inst/circulant_inv.m
===================================================================
--- trunk/octave-forge/main/linear-algebra/inst/circulant_inv.m (rev 0)
+++ trunk/octave-forge/main/linear-algebra/inst/circulant_inv.m 2012-04-18 20:42:20 UTC (rev 10279)
@@ -0,0 +1,49 @@
+## Copyright (C) 2012 Nir Krakauer
+##
+## 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 2 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
+##
+## @end deftypefn
+## @seealso{circulant_make_matrix, circulant_matrix_vector_product, circulant_eig}
+
+## Author: Nir Krakauer <nkr...@cc...>
+
+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 (make_circulant_matrix(circulant_inv(v)), inv(make_circulant_matrix(v)), 10*eps);
Added: trunk/octave-forge/main/linear-algebra/inst/circulant_make_matrix.m
===================================================================
--- trunk/octave-forge/main/linear-algebra/inst/circulant_make_matrix.m (rev 0)
+++ trunk/octave-forge/main/linear-algebra/inst/circulant_make_matrix.m 2012-04-18 20:42:20 UTC (rev 10279)
@@ -0,0 +1,46 @@
+## Copyright (C) 2012 Nir Krakauer
+##
+## 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 2 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_make_matrix(@var{v})
+##
+## Produce a full circulant matrix given the first column@*
+## Given an @var{n}*1 vector @var{v}, returns the @var{n}*@var{n} circulant matrix @var{C} where @var{v} is the left column and all other columns are downshifted versions of @var{v}
+##
+## Note: If the first row @var{r} of a circulant matrix is given, the first column @var{v} can be obtained as @code{v = r([1 end:-1:2])}
+##
+## Reference: Gene H. Golub and Charles F. Van Loan, Matrix Computations, 3rd Ed., Section 4.7.7
+##
+## @end deftypefn
+## @seealso{circulant_matrix_vector_product, circulant_eig, circulant_inv}
+
+## Author: Nir Krakauer <nkr...@cc...>
+
+
+function C = circulant_make_matrix(v)
+
+n = numel(v);
+
+C = ones(n, n);
+
+for i = 1:n
+ C(:, i) = v([(end-i+2):end 1:(end-i+1)]); #or circshift(v, i-1)
+endfor
+
+endfunction
+
+%!shared v,C
+%! v = [1 2 3]'; C = [1 3 2; 2 1 3; 3 2 1];
+%!assert (circulant_make_matrix(v), C);
Added: trunk/octave-forge/main/linear-algebra/inst/circulant_matrix_vector_product.m
===================================================================
--- trunk/octave-forge/main/linear-algebra/inst/circulant_matrix_vector_product.m (rev 0)
+++ trunk/octave-forge/main/linear-algebra/inst/circulant_matrix_vector_product.m 2012-04-18 20:42:20 UTC (rev 10279)
@@ -0,0 +1,48 @@
+## Copyright (C) 2012 Nir Krakauer
+##
+## 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 2 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{y}] =} circulant_matrix_vector_product(@var{v}, @var{x})
+##
+## Fast, compact calculation of the product of a circulant matrix with a vector@*
+## Given @var{n}*1 vectors @var{v} and @var{x}, return the matrix-vector product @var{y} = @var{C}@var{x}, where @var{C} is the @var{n}*@var{n} circulant matrix that has @var{v} as its first column
+##
+## Theoretically the same as @code{make_circulant_matrix(x) * v}, but does not form @var{C} explicitly; uses the discrete Fourier transform
+##
+## Because of roundoff, the returned @var{y} may have a small imaginary component even if @var{v} and @var{x} are real (use @code{real(y)} to remedy this)
+##
+## Reference: Gene H. Golub and Charles F. Van Loan, Matrix Computations, 3rd Ed., Section 4.7.7
+##
+## @end deftypefn
+## @seealso{circulant_make_matrix, circulant_eig, circulant_inv}
+
+## Author: Nir Krakauer <nkr...@cc...>
+
+function y = circulant_matrix_vector_product(v, x)
+
+
+xf = fft(x);
+vf = fft(v);
+z = vf .* xf;
+y = ifft(z);
+
+
+endfunction
+
+
+
+%!shared v,x
+%! v = [1 2 3]'; x = [2 5 6]';
+%!assert (circulant_matrix_vector_product(v, x), circulant_make_matrix(v)*x, eps);
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