WeierstrassFunction.m

%% A pathological function of Weierstrass
% Hrothgar, October 2013

%%
% (Chebfun example approx/WeierstrassFunction.m)
% [Tags: #Weierstrass, #approximation, #infinite series]

%%
format compact

%%
% In the late nineteenth century, Karl Weierstrass rocked the analysis community
% when he constructed an example of a function that is everywhere continuous but
% nowhere differentiable. His now eponymous function, also one of the first
% appearances of fractal geometry, is defined as the sum
% $$ \sum_{k=0}^{\infty} a^k \cos(b^k \pi x), $$
% where $0 < a < 1$ and $b$ is a positive odd integer with $ab < 1 + \frac32
% \pi$. Since its publication, Weierstrass's work has been generalized in many
% directions.

%%
% Chebfun is designed for work with functions with a bit of smoothness, but in
% this example we will see how Chebfun fares against a pathological function
% lying on the edge of discontinuity.

%%
% Let us consider the Weierstrass-type function
% $$ F(x) = \sum_{k=0}^{\infty} 2^{-k} \cos\left( \frac{\pi}{2} 4^k x \right) $$
% on the interval $[-1, 1]$.
% With its default parameter settings, Chebfun resolves the first eight
% iterates to machine precision, but not the ninth.
f_k = @(k) @(x) 2^-k * cos(pi/2*x*4^k);
F{1} = chebfun(f_k(0));
for k = 1:8,
    F{k+1} = F{k} + chebfun(f_k(k));
end

%%
% Here is what the unresolved ninth iterate looks like.
plot(F{9}, 'k')
title('A pathological function of Weierstrass')

%%
% We must zoom in 400 times to see that Chebfun is in fact plotting a
% smooth function.
plot(F{9}, 'k', 'interval', [0 0.005])
title('Close-up of Weierstrass approximant')

%%
% The function $F(x)$ is not differentiable, but it is integrable. For this
% particular Weierstrass function, the exact value of the integral can be found
% easily. We begin with
%
% $$ \int_{-1}^{1} F(x)\mathrm{d}x = \int_{-1}^{1}
% \sum_{k=0}^{\infty} f_k(x) \mathrm{d}x = \int_{-1}^{1} \sum_{k=0}^{\infty}
% 2^{-k} \cos\left( \frac{\pi}{2}4^k x \right) \mathrm{d}x. $$
%
% Because $\int
% \sum |f_k| < \infty$, we can move the integral inside the sum and evaluate
% each term as
%
% $$ \sum_{k=0}^{\infty} \int_{-1}^{1} 2^{-k} \cos\left(
% \frac{\pi}{2}4^k x \right) \mathrm{d}x = \sum_{k=0}^{\infty} \frac{1}{8^k}
% \frac{4}{\pi} \sin\left( \frac{\pi}{2} 4^k \right). $$
%
% However, $\sin(
% \frac{\pi}{2} 4^k ) = 0$ for all $k > 0$, so the sum is equal to its first
% term, $\frac{4}{\pi}$.

%%
% Let's check our answer against Chebfun's.
error = sum(F{9}) - 4/pi

%%
% A more difficult problem is to find the global minimum of $F(x)$ on the
% interval $[-1, 1]$. Even if it were possible to differentiate $F$ to find
% where $F'(x) = 0$, we would discover infinitely many local extrema. Of course,
% Chebfun's representation of $F$ is a polynomial approximant, so we can locate
% the roots of the derivative for any iterate. As we may expect, performance
% rapidly gets worse as we take more terms.
tt = []; xx = []; mm = [];
for k = 1:2:7
    tic
    kk = (k+1)/2;
    [mm(kk), xx(kk)] = min(F{k});
    tt(kk) = toc;
end
str = [sprintf('%2s %11s %16s %19s\n', 'k', 'x_min', 'F_k(x_min)', 'computation time') ...
       repmat('-',1,52) sprintf('\n') ...
       sprintf('%2d %12.7f %+15.7f %11.2f sec\n', [(2:2:8); xx; mm; tt])];
disp(str)

%%
% Chebfun is slowly converging to the actual solution given by $F_{min} =
% \sin(\frac{\pi}{5}) - \cos(\frac{\pi}{5}) = -0.2212317420...$ at the points $x
% = \pm \frac35$. Chebfun's difficulty is not with accurately locating the
% minima: the `x_min` iterates are geometrically converging to the correct
% solution as they should. The problem is that the iterates' global minima so
% slowly converge to the global minimum of $F$ while Chebfun must deal with
% polynomials of geometrically increasing degree.

%% References
%
% 1. K. Weierstrass, _Abhandlungen aus der Functionenlehre_. J. Springer, 1886.
%
% 2. G. H. Hardy, "Weierstrass's non-differentiable function."
%    _Transactions of the American Mathematical Society_, 17, no. 3 (1916), 301-325.
%
% 3. L. N. Trefethen, _Approximation Theory and Approximation Practice, 
% Extended Edition_, SIAM, 2019._