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## Copyright (C) 2008-2013 N. J. Higham
## Copyright (C) 2010 Richard T. Guy
## Copyright (C) 2010 Marco Caliari
##
## 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{s} =} logm (@var{A})
## @deftypefnx {Function File} {@var{s} =} logm (@var{A}, @var{opt_iters})
## @deftypefnx {Function File} {[@var{s}, @var{iters}] =} logm (@dots{})
## Compute the matrix logarithm of the square matrix @var{A}. The
## implementation utilizes a Pad@'e approximant and the identity
##
## @example
## logm (@var{A}) = 2^k * logm (@var{A}^(1 / 2^k))
## @end example
##
## The optional argument @var{opt_iters} is the maximum number of square roots
## to compute and defaults to 100. The optional output @var{iters} is the
## number of square roots actually computed.
## @seealso{expm, sqrtm}
## @end deftypefn
## Reference: N. J. Higham, Functions of Matrices: Theory and Computation
## (SIAM, 2008.)
##
## Author: N. J. Higham
## Author: Richard T. Guy <guyrt7@wfu.edu>
## Author: Marco Caliari <marco.caliari@univr.it>
function [s, iters] = logm (A, opt_iters = 100)
if (nargin == 0 || nargin > 2)
print_usage ();
endif
if (! issquare (A))
error ("logm: A must be a square matrix");
endif
if (isscalar (A))
s = log (A);
return;
elseif (strfind (typeinfo (A), "diagonal matrix"))
s = diag (log (diag (A)));
return;
endif
[u, s] = schur (A);
if (isreal (A))
[u, s] = rsf2csf (u, s);
endif
eigv = diag (s);
if (any (eigv < 0))
warning ("Octave:logm:non-principal",
"logm: principal matrix logarithm is not defined for matrices with negative eigenvalues; computing non-principal logarithm");
endif
real_eig = all (eigv >= 0);
k = 0;
## Algorithm 11.9 in "Function of matrices", by N. Higham
theta = [0, 0, 1.61e-2, 5.38e-2, 1.13e-1, 1.86e-1, 2.6429608311114350e-1];
p = 0;
m = 7;
while (k < opt_iters)
tau = norm (s - eye (size (s)),1);
if (tau <= theta (7))
p = p + 1;
j(1) = find (tau <= theta, 1);
j(2) = find (tau / 2 <= theta, 1);
if (j(1) - j(2) <= 1 || p == 2)
m = j(1);
break
endif
endif
k = k + 1;
s = sqrtm (s);
endwhile
if (k >= opt_iters)
warning ("logm: maximum number of square roots exceeded; results may still be accurate");
endif
s = s - eye (size (s));
if (m > 1)
s = logm_pade_pf (s, m);
endif
s = 2^k * u * s * u';
## Remove small complex values (O(eps)) which may have entered calculation
if (real_eig && isreal (A))
s = real (s);
endif
if (nargout == 2)
iters = k;
endif
endfunction
################## ANCILLARY FUNCTIONS ################################
###### Taken from the mfttoolbox (GPL 3) by D. Higham.
###### Reference:
###### D. Higham, Functions of Matrices: Theory and Computation
###### (SIAM, 2008.).
#######################################################################
##LOGM_PADE_PF Evaluate Pade approximant to matrix log by partial fractions.
## Y = LOGM_PADE_PF(A,M) evaluates the [M/M] Pade approximation to
## LOG(EYE(SIZE(A))+A) using a partial fraction expansion.
function s = logm_pade_pf (A, m)
[nodes, wts] = gauss_legendre (m);
## Convert from [-1,1] to [0,1].
nodes = (nodes+1)/2;
wts = wts/2;
n = length (A);
s = zeros (n);
for j = 1:m
s += wts(j)*(A/(eye (n) + nodes(j)*A));
endfor
endfunction
######################################################################
## GAUSS_LEGENDRE Nodes and weights for Gauss-Legendre quadrature.
## [X,W] = GAUSS_LEGENDRE(N) computes the nodes X and weights W
## for N-point Gauss-Legendre quadrature.
## Reference:
## G. H. Golub and J. H. Welsch, Calculation of Gauss quadrature
## rules, Math. Comp., 23(106):221-230, 1969.
function [x, w] = gauss_legendre (n)
i = 1:n-1;
v = i./sqrt ((2*i).^2-1);
[V, D] = eig (diag (v, -1) + diag (v, 1));
x = diag (D);
w = 2*(V(1,:)'.^2);
endfunction
%!assert (norm (logm ([1 -1;0 1]) - [0 -1; 0 0]) < 1e-5)
%!assert (norm (expm (logm ([-1 2 ; 4 -1])) - [-1 2 ; 4 -1]) < 1e-5)
%!assert (logm ([1 -1 -1;0 1 -1; 0 0 1]), [0 -1 -1.5; 0 0 -1; 0 0 0], 1e-5)
%!assert (logm (10), log (10))
%!assert (full (logm (eye (3))), logm (full (eye (3))))
%!assert (full (logm (10*eye (3))), logm (full (10*eye (3))), 8*eps)
%!assert (logm (expm ([0 1i; -1i 0])), [0 1i; -1i 0], 10 * eps)
%% Test input validation
%!error logm ()
%!error logm (1, 2, 3)
%!error <logm: A must be a square matrix> logm ([1 0;0 1; 2 2])
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