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Chebfun is an open-source package for computing
with functions to about 15-digit accuracy. Most Chebfun commands
are overloads of familiar MATLAB commands — for
example sum(f) computes an integral,
roots(f) finds zeros, and u =
L\f solves a differential equation.
% Create operator for Ginzburg-Landau problem d = 20*[-1.2 3.2 -1 1]; tt = [0 46.5]; S = spinop2(d,tt); S.linearPart = @(u) lap(u); S.nonlinearPart = @(u) u-(1+1.5i)*u.*abs(u).^2; % Set an initial condition S.init = chebfun2(@(x,y) ... (1i*x+y).*exp(-.03*(x.^2+y.^2)), d); % Solve the PDE and plot u = spin2(S,'plot','off'); plot(real(u))
% Create a chebfun on the interval [-3,3] x = chebfun('x', [-3 3]); % Define a potential function V = abs(x); % Plot the first 8 eigenstates of % the Schrodinger operator quantumstates(V, 8)
% Create a chebfun f x = chebfun('x'); f = exp(-1./(x+1)); % Plot abs vals of Chebyshev coeffs of f plotcoeffs(f,'.')
% Define a rectangular domain d = pi*[-2.2 2.2 -1 1]/2; % Create a complex-valued chebfun2 f = chebfun2(@(z) ... cos((z-1).^2)+exp((z+1).^2), d); % Plot its phase portrait plot(f)
% Construct a pair of 2D chebfuns d = [-3 10 -3 3]; f = chebfun2('y.*cos(y.^2+x)-.1',d); g = chebfun2('cos(x.^2/2).*sin(y.^2)-.1',d); % Plot zero contours of f & g plot(roots(f)), hold on, plot(roots(g)) % Plot their common roots r = roots(f, g, 'resultant'); plot(r(:,1), r(:,2), '.')
% Define two functions f = chebfun(@(x) sin(x.^2)+sin(x).^2, [0,10]); g = chebfun(@(x) exp(-(x-5).^2/10), [0,10]); % Compute their intersections rr = roots(f - g); % Plot the functions plot([f g]), hold on % Plot the intersections plot(rr, f(rr), 'o')
% Airy operator op = @(x,u) 0.01*diff(u,2) - x.*u; % Create a chebop L = chebop(op, [-5,5]); % Apply boundary conditions L.bc = 'dirichlet'; % Solve the differential equation u = L \ 1; plot(u)
% The Dixon-Szego function f = @(x,y) (4-2.1*x.^2+ x.^4/3).*x.^2 ... + x.*y + 4*(y.^2-1).*y.^2; % Create a chebfun2 F = chebfun2(f, [-2,2,-1.25,1.25]); % Find the minimum and mark it [minf,minx] = min2(F); contour(F,30), hold on plot(minx(1),minx(2),'.w')
