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Added chebfun library
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‎chebfun/+cheb/README.txt

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The purpose of this package is to provide extra capabilities not found in the
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core Chebfun classes and functions. To be added to this folder, a function must
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have the following properties:
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1. No other functions depend on it.
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2. It depends only on built-in MATLAB and core Chebfun.
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3. It is well documented, following MATLAB conventions for the help
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documentation block. In particular, it should provide at least one example
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of usage that will be printed out by MATLAB help.
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4. It has a comment with the author's name and email, and the month and year
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of original authorship.
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5. It has a notice assigning copyright to The University of Oxford and The
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Chebfun Developers.

‎chebfun/+cheb/bernoulli.m

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function B = bernoulli(N)
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%BERNOULLI Bernoulli polynomials as chebfuns.
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% B = BERNOULLI(N) returns a quasimatrix of the first N+1 Bernoulli
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% polynomials on [0,1].
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%
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% Example (Bernolli numbers):
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% B = cheb.bernoulli(4);
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% format rat, B(0,:)
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% Copyright 2017 by The University of Oxford and The Chebfun Developers.
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% See http://www.chebfun.org/ for Chebfun information.
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x = chebfun('x', [0,1]);
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% Initalize the constant 1 as the first Bernoulli polynomial.
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B = 0*x + 1;
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% Compute the requested polynomials.
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for j = 1:N
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B(:,j+1) = j*cumsum(B(:,j));
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B(:,j+1) = B(:,j+1) - sum(B(:,j+1));
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end
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end

‎chebfun/+cheb/bspline.m

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function B = bspline(m)
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%BSPLINE B-splines as chebfuns.
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% BSPLINE(M) returns the first M B-splines, starting from a constant value 1
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% on [-1/2,1/2]. Each sucessive B-spline is smoother by one derivative and
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% supported on an interval larger by 1/2 in each direction.
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%
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% The returned result is a chebmatrix.
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%
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% Example:
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% B = cheb.bspline(4); hold on
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% for k=1:4, plot(B{k}), end
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% axis([-0.1 1.1 -2 2])
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% Copyright 2017 by The University of Oxford and The Chebfun Developers.
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% See http://www.chebfun.org/ for Chebfun information.
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s = chebfun(1, [-.5 .5]);
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B = chebmatrix(s);
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for k = 1:m-1
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B(k+1) = conv(B{k}, s);
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end
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end

‎chebfun/+cheb/choreo.m

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function choreo
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%CHOREO Compute planar choreographies of the n-body problem.
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% CHEB.CHOREO computes a planar choreography using hand-drawn initial
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% guesses. At the end of the computation, the user is asked to press <1> to
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% display the motion of the planets. To stop the program, press <CTRL>-<C>.
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%
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% Planar choreographies are periodic solutions of the n-body problem in which
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% the bodies share a single orbit. This orbit can be fixed or rotating with some
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% angular velocity relative to an inertial reference frame.
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%
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% The algorithm uses trigonometric interpolation and quasi-Newton methods.
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% See [1] for details.
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%
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% Example: at the prompt, specify 5 bodies, angular rotation 0 or 1.2.
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% Then either draw a curve (if your machine has imfreehand) or click
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% in 10 or 15 points (if it doesn't) roughly along a figure-8. Type
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% <Enter>, and in a few seconds you will see a choreograpy.
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%
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% [1] H. Montanelli and N. I. Gushterov, Computing planar and spherical
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% choreographies, SIAM Journal on Applied Dynamical Systems 15 (2016),
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% Copyright 2017 by The University of Oxford and The Chebfun Developers.
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% See http://www.chebfun.org/ for Chebfun information.
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close all
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LW = 'linewidth'; MS = 'markersize'; FS = 'fontsize';
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lw = 2; ms = 30; fs = 18;
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format long, format compact
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n = input('How many bodies? (e.g. 5) ');
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w = input('Angular velocity? (either 0 or a nonzero noninteger) ');
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k = 15;
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N = k*n;
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dom = [0 2*pi];
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while 1
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% Hand-drawn initial guess:
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xmax = 2.5; ymax = 2.5;
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clf, hold off, axis equal, axis([-xmax xmax -ymax ymax]), box on
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set(gca,FS,fs), title('Draw a curve!')
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if ( exist('imfreehand') == 2 )
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% Try to use IMFREEHAND (need the image processing toolbox):
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h = imfreehand();
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z = getPosition(h);
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delete(h)
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z = z(:,1) + 1i*z(:,2);
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q0 = chebfun(z,dom,'trig');
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q0 = chebfun(q0,dom,N,'trig');
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c0 = trigcoeffs(q0);
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c0(1+floor(N/2)) = 0;
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q0 = chebfun(c0,dom,'coeffs','trig');
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else
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% Otherwise, use GINPUT:
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x = []; y = []; button = 1;
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disp('Input points with mouse, press <enter> for final point.')
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while ( button == 1 )
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[xx,yy,button] = ginput(1);
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x = [x; xx]; y = [y; yy]; %#ok<*AGROW>
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hold on, plot(xx,yy,'xb',MS,ms/2), drawnow
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end
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z = x + 1i*y;
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q0 = chebfun(z,dom,'trig');
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q0 = chebfun(q0,dom,N,'trig');
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end
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% Solve the problem:
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hold on, plot(q0,'.-b',LW,lw), drawnow
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c0 = trigcoeffs(q0);
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c0 = [real(c0);imag(c0)];
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[A0,G0] = actiongradeval(c0,n,w);
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fprintf('\nInitial action: %.6f\n',A0)
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options = optimoptions('fminunc');
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options.Algorithm = 'quasi-newton';
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options.HessUpdate = 'bfgs';
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options.GradObj = 'on';
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options.Display = 'off';
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[c,A,~, ~,G] = fminunc(@(x)actiongradeval(x,n,w),c0,options);
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fprintf('Action after optimization: %.6f\n',A)
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fprintf('Norm of the gradient: %.3e\n',norm(G)/norm(G0))
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% Plot the result:
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c = c(1:N) + 1i*c(N+1:2*N);
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c(1+floor(N/2)) = 0;
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q = chebfun(c,dom,'coeffs','trig');
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hold on, plot(q,'r',LW,lw)
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hold on, plot(q(2*pi*(0:n-1)/n),'.k',MS,ms), title('')
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s = input('Want to see the planets dancing? (Yes=<1>, No=<0>) ');
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if ( s == 1 )
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hold off, plotonplane(q,n,w), pause
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end
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end
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end
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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function [A,G] = actiongradeval(c,n,w)
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%ACTIONGRADEVAL Compute the action and its gradient in the plane.
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% Set up:
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if ( nargin < 3 )
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w = 0;
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end
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N = length(c)/2;
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u = c(1:N);
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v = c(N+1:end);
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c = u + 1i*v;
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if ( mod(N, 2) == 0 )
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k = (-N/2:N/2-1)';
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else
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k = (-(N-1)/2:(N-1)/2)';
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end
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[t,s] = trigpts(N,[0 2*pi]);
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% Evaluate action A:
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c = bsxfun(@times,exp(1i*k*(0:n-1)*2*pi/n),c(:,1));
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vals = ifft(N*c);
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dc = 1i*k.*c(:,1);
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if ( mod(N, 2) == 0 )
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dc(1,1) = 0;
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end
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dvals = ifft(N*dc);
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K = abs(dvals+1i*w*vals(:,1)).^2;
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U = 0;
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for i = 2:n
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U = U - 1./abs(vals(:,1)-vals(:,i));
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end
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A = n/2*s*(K-U);
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if ( nargout > 1 )
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% Initialize gradient G:
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G = zeros(2*N,1);
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% Loop over the bodies for the AU contribution:
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for j = 1:n-1
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a = bsxfun(@times,1-cos(k'*j*2*pi/n),cos(t*k')) + ...
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bsxfun(@times,sin(k'*j*2*pi/n),sin(t*k'));
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b = bsxfun(@times,-1+cos(k'*j*2*pi/n),sin(t*k')) + ...
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bsxfun(@times,sin(k'*j*2*pi/n),cos(t*k'));
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f = (a*u + b*v).^2 + (-b*u + a*v).^2;
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df = 2*(bsxfun(@times,a*u + b*v,a) + bsxfun(@times,-b*u + a*v,-b));
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G(1:N) = G(1:N) - n/4*bsxfun(@rdivide,df,f.^(3/2))'*s';
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df = 2*(bsxfun(@times,a*u + b*v, b) + bsxfun(@times,-b*u + a*v,a));
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G(N+1:2*N) = G(N+1:2*N) - n/4*bsxfun(@rdivide,df,f.^(3/2))'*s';
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end
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% Add the AK contribution:
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if ( mod(N, 2) == 0 )
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G = G + 2*pi*n*[w;(k(2:end)+w);w;(k(2:end)+w)].^2.*[u;v];
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else
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G = G + 2*pi*n*[(k+w);(k+w)].^2.*[u;v];
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end
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end
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end
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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function plotonplane(q,n,w)
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%PLOTONPLANE Plot a planar choreography.
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vscale = max(max(real(q)),max(imag(q)));
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xmax = 1.3*vscale; ymax = xmax;
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LW = 'linewidth'; MS = 'markersize'; FS = 'fontsize';
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lw = 2; ms = 30; fs = 18;
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dom = [0 2*pi]; T = 2*pi; dt = .1;
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xStars = 2*xmax*rand(250,1)-xmax;
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yStars = 2*ymax*rand(250,1)-ymax;
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for t = dt:dt:10*T
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clf, hold off
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fill(xmax*[-1 1 1 -1 -1],ymax*[-1 -1 1 1 -1],'k')
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axis equal, axis([-xmax xmax -ymax ymax]), box on
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set(gca,FS,fs)
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hold on, plot(xStars,yStars,'.w',MS,4)
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if ( w ~= 0 )
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q = chebfun(@(t)exp(1i*w*dt).*q(t),dom,length(q),'trig');
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end
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if ( t < T )
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hold on, plot(q,'r',LW,lw)
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end
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for j = 1:n
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hold on, plot(q(t+2*pi*j/n),'.',MS,ms)
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end
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drawnow
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end
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end

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