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<h1>Chebfun<span class='sep'>—</span><span>numerical computing with functions</span></h1>
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<p>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 <code>sum(f)</code> computes an integral,
<code>roots(f)</code> finds zeros, and <code>u =
L\f</code> solves a differential equation.</p>
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<div class='codewrap'><pre class='code'><span class='comment'>% Create operator for Ginzburg-Landau problem</span>
d = 20*[-1.2 3.2 -1 1]; tspan = [0 46.5];
S = spinop2(d,tspan); S.lin = @(u) lap(u);
S.nonlin = @(u) u - (1+1.5i)*u.*(abs(u).^2);
<span class='comment'>% Set initial condition, solve PDE, plot</span>
S.init = chebfun2(@(x,y) ...
(1i*x+y).*exp(-.03*(x.^2+y.^2)), d);
u = spin2(S, 128, 1e-1, 'plot', 'off');
plot(real(u))
</pre></div><div class='imagewrap'><img src='demo5.png' /></div>
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<div class='codewrap'><pre class='code'><span class='comment'>% Create a chebfun on the interval [-3,3]</span>
x = chebfun('x', [-3 3]);
<span class='comment'>% Define a potential function</span>
V = abs(x);
<span class='comment'>% Plot the first 8 eigenstates of</span>
<span class='comment'>% the Schrodinger operator</span>
quantumstates(V, 8)
</pre></div><div class='imagewrap'><img src='demo4.png' /></div>
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<div class='codewrap'><pre class='code'><span class='comment'>% Create a chebfun f</span>
x = chebfun('x');
f = exp(-1/(x+1));
<span class='comment'>% Plot abs vals of Chebyshev coeffs of f</span>
plotcoeffs(f,'.')
</pre></div><div class='imagewrap'><img src='demo6.png' /></div>
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<div class='codewrap'><pre class='code'><span class='comment'>% Define a rectangular domain</span>
d = pi*[-2.2 2.2 -1 1]/2;
<span class='comment'>% Create a complex-valued chebfun2</span>
f = chebfun2(@(z) ...
cos((z-1).^2)+exp((z+1).^2), d);
<span class='comment'>% Plot its phase portrait</span>
plot(f)
</pre></div><div class='imagewrap'><img src='demo7.png' /></div>
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<div class='codewrap'><pre class='code'><span class='comment'>% Construct a pair of 2D chebfuns</span>
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);
<span class='comment'>% Plot zero contours of f & g</span>
plot(roots(f)), hold on, plot(roots(g))
<span class='comment'>% Plot their common roots</span>
r = roots(f, g, 'resultant');
plot(r(:,1), r(:,2), '.')
</pre></div><div class='imagewrap'><img src='demo9.png' /></div>
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<div class='codewrap'><pre class='code'><span class='comment'>% Define two functions</span>
f = chebfun(@(x) sin(x^2)+sin(x)^2, [0,10]);
g = chebfun(@(x) exp(-(x-5)^2/10), [0,10]);
<span class='comment'>% Compute their intersections</span>
rr = roots(f - g);
<span class='comment'>% Plot the functions</span>
plot([f g]), hold on
<span class='comment'>% Plot the intersections</span>
plot(rr, f(rr), 'o')
</pre></div><div class='imagewrap'><img src='demo1.png' /></div>
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<div class='codewrap'><pre class='code'><span class='comment'>% Airy operator</span>
op = @(x,u) 0.01*diff(u,2) - x*u;
<span class='comment'>% Create a chebop</span>
L = chebop(op, [-5,5]);
<span class='comment'>% Apply boundary conditions</span>
L.bc = 'dirichlet';
<span class='comment'>% Solve the differential equation</span>
u = L \ 1;
plot(u)
</pre></div><div class='imagewrap'><img src='demo2.png' /></div>
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<div class='codewrap'><pre class='code'><span class='comment'>% The Dixon-Szego function</span>
f = @(x,y) (4-2.1*x.^2+ x.^4/3).*x.^2 ...
+ x.*y + 4*(y.^2-1).*y.^2;
<span class='comment'>% Create a chebfun2</span>
F = chebfun2(f, [-2,2,-1.25,1.25]);
<span class='comment'>% Find the minimum and mark it</span>
[minf,minx] = min2(F);
contour(F,30), hold on
plot(minx(1),minx(2),'.w')
</pre></div><div class='imagewrap'><img src='demo3.png' /></div>
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<span class='read-more'>read more »</span>
<h3><span class='date'> 6 December 2021</span><span class='sep'>/</span>20th anniversary of the Chebfun idea</h3>
<p class='oneline'>Chebfun team members gathered Dec. 4 to celebrate the 20th anniversary of the invention of Chebfun.</p>
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<span class='read-more'>read more »</span>
<h3><span class='date'> 4 June 2021</span><span class='sep'>/</span>Chebfun computes 250 million fields of values</h3>
<p class='oneline'>Michael Overton has called FOV about 250 million times in investigating the Crouzeix conjecture.</p>
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