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[CreaPhase.git] / octave_packages / optim-1.2.0 / cauchy.m

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+## Copyright (C) 2011 Fernando Damian Nieuwveldt <fdnieuwveldt@gmail.com>
+## 2012 Adapted by Juan Pablo Carbajal <carbajal@ifi.uzh.ch>
+##
+## 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,
+## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+## GNU General Public License for more details.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {} cauchy (@var{N}, @var{r}, @var{x}, @var{f} )
+## Return the Taylor coefficients and numerical differentiation of a function
+## @var{f} for the first @var{N-1} coefficients or derivatives using the fft.
+## @var{N} is the number of points to evaluate,
+## @var{r} is the radius of convergence, needs to be chosen less then the smallest singularity,
+## @var{x} is point to evaluate the Taylor expansion or differentiation. For example,
+##
+## If @var{x} is a scalar, the function @var{f} is evaluated in a row vector
+## of length @var{N}. If @var{x} is a column vector, @var{f} is evaluated in a
+## matrix of length(x)-by-N elements and must return a matrix of the same size.
+##
+## @example
+## @group
+## d = cauchy(16, 1.5, 0, @@(x) exp(x));
+## @result{} d(2) = 1.0000 # first (2-1) derivative of function f (index starts from zero)
+## @end group
+## @end example
+## @end deftypefn
+
+function deriv = cauchy(N, r, x, f)
+
+  if nargin != 4
+    print_usage ();
+  end
+
+  [nx m]   = size (x);
+  if m > 1
+    error('cauchy:InvalidArgument', 'The 3rd argument must be a column vector');
+  end
+
+  n        = 0:N-1;
+  th       = 2*pi*n/N;
+
+  f_p = f (bsxfun (@plus, x, r * exp (i * th) ) );
+
+  evalfft  = real(fft (f_p, [], 2));
+
+  deriv    = bsxfun (@times, evalfft, 1./(N*(r.^n)).* factorial(n)) ;
+
+endfunction
+
+function g = hermite(order,x)
+   ## N should be bigger than order+1
+   N      = 32;
+   r      = 0.5;
+   Hnx    = @(t) exp ( bsxfun (@minus, kron(t(:).', x(:)) , t(:).'.^2/2) );
+   Hnxfft = cauchy(N, r, 0, Hnx);
+   g      = Hnxfft(:, order+1);
+endfunction
+
+%!demo
+%! # Cauchy integral formula: Application to Hermite polynomials
+%! # Author: Fernando Damian Nieuwveldt
+%! # Edited by: Juan Pablo Carbajal
+%!
+%! Hnx     = @(t,x) exp ( bsxfun (@minus, kron(t(:).', x(:)) , t(:).'.^2/2) );
+%! hermite = @(order,x) cauchy(32, 0.5, 0, @(t)Hnx(t,x))(:, order+1);
+%!
+%! t    = linspace(-1,1,30);
+%! he2  = hermite(2,t);
+%! he2_ = t.^2-1;
+%!
+%! figure(1)
+%! clf
+%! plot(t,he2,'bo;Contour integral representation;', t,he2_,'r;Exact;');
+%! grid
+%! clear all
+%!
+%! % --------------------------------------------------------------------------
+%! % The plots compares the approximation of the Hermite polynomial using the
+%! % Cauchy integral (circles) and the corresposind polynomial H_2(x) = x.^2 - 1.
+%! % See http://en.wikipedia.org/wiki/Hermite_polynomials#Contour_integral_representation
+
+%!demo
+%! # Cauchy integral formula: Application to Hermite polynomials
+%! # Author: Fernando Damian Nieuwveldt
+%! # Edited by: Juan Pablo Carbajal
+%!
+%!  xx = sort (rand (100,1));
+%!  yy = sin (3*2*pi*xx);
+%!
+%!  # Exact first derivative derivative
+%!  diffy = 6*pi*cos (3*2*pi*xx);
+%!
+%!  np = [10 15 30 100];
+%!
+%!  for i =1:4
+%!    idx = sort(randperm (100,np(i)));
+%!    x   = xx(idx);
+%!    y   = yy(idx);
+%!
+%!    p     = spline (x,y);
+%!    yval  = ppval (ppder(p),x);
+%!    # Use the cauchy formula for computing the derivatives
+%!    deriv =  cauchy (fix (np(i)/4), .1, x, @(x) sin (3*2*pi*x));
+%!
+%!    subplot(2,2,i)
+%!    h = plot(xx,diffy,'-b;Exact;',...
+%!             x,yval,'-or;ppder solution;',...
+%!             x,deriv(:,2),'-og;Cauchy formula;');
+%!    set(h(1),'linewidth',2);
+%!    set(h(2:3),'markersize',3);
+%!
+%!    legend(h, 'Location','Northoutside','Orientation','horizontal');
+%!    if i!=1
+%!      legend('hide');
+%!    end
+%!  end
+%!
+%! % --------------------------------------------------------------------------
+%! % The plots compares the derivatives calculated with Cauchy and with ppder.
+%! % Each subplot shows the results with increasing number of samples.