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## Copyright (C) 2006-2013 David Bateman and Marco Caliari
## Copyright (C) 2009 VZLU Prague
##
## This file is part of Octave.
##
## Octave 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.
##
## Octave 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 Octave; see the file COPYING. If not, see
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {@var{n} =} normest (@var{A})
## @deftypefnx {Function File} {@var{n} =} normest (@var{A}, @var{tol})
## @deftypefnx {Function File} {[@var{n}, @var{c}] =} normest (@dots{})
## Estimate the 2-norm of the matrix @var{A} using a power series
## analysis. This is typically used for large matrices, where the cost
## of calculating @code{norm (@var{A})} is prohibitive and an approximation
## to the 2-norm is acceptable.
##
## @var{tol} is the tolerance to which the 2-norm is calculated. By default
## @var{tol} is 1e-6. @var{c} returns the number of iterations needed for
## @code{normest} to converge.
## @end deftypefn
function [n, c] = normest (A, tol = 1e-6)
if (nargin != 1 && nargin != 2)
print_usage ();
endif
if (! (isnumeric (A) && ndims (A) == 2))
error ("normest: A must be a numeric 2-D matrix");
endif
if (! (isscalar (tol) && isreal (tol)))
error ("normest: TOL must be a real scalar");
endif
if (! isfloat (A))
A = double (A);
endif
tol = max (tol, eps (class (A)));
## Set random number generator to depend on target matrix
v = rand ("state");
rand ("state", trace (A));
ncols = columns (A);
## Randomize y to avoid bad guesses for important matrices.
y = rand (ncols, 1);
c = 0;
n = 0;
do
n0 = n;
x = A * y;
normx = norm (x);
if (normx == 0)
x = rand (ncols, 1);
else
x = x / normx;
endif
y = A' * x;
n = norm (y);
c += 1;
until (abs (n - n0) <= tol * n)
rand ("state", v); # restore state of random number generator
endfunction
%!test
%! A = toeplitz ([-2,1,0,0]);
%! assert (normest (A), norm (A), 1e-6);
%!test
%! A = rand (10);
%! assert (normest (A), norm (A), 1e-6);
%% Test input validation
%!error normest ()
%!error normest (1, 2, 3)
%!error normest ([true true])
%!error normest (ones (3,3,3))
%!error normest (1, [1, 2])
%!error normest (1, 1+1i)
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