%% Rootfinding with the AAA algorithm
% Stefano Costa, June 2022
%%
% (Chebfun example roots/AAAZeros.m)
% [Tags: #AAA #AAAtrig]
%%
% Recently, much attention has been paid to the AAA algorithm's
% capability of returning poles readily available for subsequent
% approximation tasks. However, we must not forget that it also returns
% zeros, and this example will show it is competitive with other methods
% for rootfinding. Here we tackle some problems previously
% published in the Examples-Rootfinding category.
%%
% To begin with, let's consider the Bessel function $J_0$ on the interval
% $[0,100]$, with its AAA approximation on a set $X$ of just 400 equispaced
% points:
J0 = chebfun(@(x) besselj(0,x),[0 100]);
X = linspace(0,100,400);
tic, [j0,~,~,zer] = aaa(besselj(0,X),X); toc
%%
% We keep only the real zeros in the interval, and measure the
% maximum difference from those computed by Chebfun:
LW = 'linewidth'; MS = 'markersize'; FS = 'fontsize';
zer(imag(zer)~=0) = []; zer = sort(zer(zer>=0 & zer<=100))';
format long e
max_diff = norm(zer-roots(J0)','inf')
figure, plot(J0,'c'), hold on, plot(X,j0(X),'k:'), grid on
title('Roots of Bessel function J_0 in [0,100]')
plot(zer+1i*eps,'r.',MS,15)
%%
% Another interesting test is how the AAA algorithm behaves with random
% real polynomials, intended as a linear combination of Legendre polynomials
% (normalized by 2-norm on $[-1,1]$) with random independent coefficients
% drawn from the standard normal distribution [1].
% We build ten random polynomials of degree 50 on the interval with 600
% evenly distributed points.
X = linspace(-1,1,600); n = 50; diff = []; tic
rng(1)
for k = 1:10
cleg = randn(n+1,1); % Legendre coeffs
ccheb = leg2cheb(cleg,'norm'); % Chebyshev coeffs
p = chebfun(ccheb,'coeffs');
[pa,~,~,zer] = aaa(p(X),X,'tol',1e-9); % AAA approximation
zer(imag(zer)~=0) = []; zer = sort(zer(zer>=-1 & zer<=1));
diff = [diff norm(roots(p)-zer,'inf')];
end, toc
%%
% Here is the maximum difference between roots for each iteration, and how
% the last random polynomial looks like together with its zeros:
diff = diff'
clf, plot(X,p(X),'c-',LW,2), hold on, plot(X,pa(X),'k:',LW,1)
plot(zer+1i*eps,'r.',MS,15), grid on
title('Roots in [-1,1] of random polynomial, degree = 50')
%%
% Let's now find the roots in the unit disk for some complex functions.
% Needless to say, the AAA algorithm catches them instantly.
Z = exp(1i*linspace(0,2*pi,1000)); % 1000 points on the disk boundary
ff = {}; zerff = {}; % cells for functions and zeros
tic
[ff{1},~,~,zer] = aaa((Z-0.5i).*exp(Z),Z);
zerff{1} = zer(abs(zer)<=1); zeros = cell2mat(zerff(1))
%%
[ff{2},~,~,zer] = aaa(cosh(pi*Z),Z);
zerff{2} = zer(abs(zer)<=1); zeros = cell2mat(zerff(2))
%%
[ff{3},~,~,zer] = aaa(cosh(exp(Z)).*(Z-.3).*(1+4*Z.^2),Z);
zerff{3} = zer(abs(zer)<=1); zeros = cell2mat(zerff(3))
%%
[ff{4},~,~,zer] = aaa((Z.^3-1/8).*exp((-1-2i)*Z),Z);
zerff{4} = zer(abs(zer)<=1); zeros = cell2mat(zerff(4))
toc
%%
% Here's a beautiful phaseplot image, with the zeros highlighted in each
% subpicture.
clf, for k = 1:4
subplot(2,2,k), phaseplot(ff{k}), hold on
plot(Z,'k-',LW,1), plot(zerff{k},'r.',MS,20)
end