`leg2cheb`/`cheb2leg` bug in "A fast Cheb-Leg transform"

[trefethen]

Something is wrong in examples/cheb/FastChebyshevLegendreTransform.m. The errors that are supposed to be near-zero are actually equal to about 1/sqrt(2).

[nickhale]

I can't reproduce this:

No. of evaluation points = 23945
Evaluation time = 0.01s
N =
       31709
Elapsed time is 1.328169 seconds.
Elapsed time is 0.659965 seconds.
ans =
     4.828082378338650e-14
ans =
     0

[trefethen]

I get the these numbers too, but if I look at the plot labeled "Absolute error", it is blank. That seems to stem from errors of size 0.7 plotted on axes expecting smaller than 1e-12.

[nickhale]

Oh. c_leg should be a column vector. This probably worked before because leg2cheb() wasn't vectorized.

The following now works:

t = .999i; f = chebfun(@(x) 1./sqrt(1 - 2*x.*t +t.^2)); % generating function
N = length(f);
ns = sprintf('No. of evaluation points = %u\n',N);
s = tic;                                                % evaluate f
c_leg = t.^(N-1:-1:0).';                                  % via Legendre coeffs
cheb_vals = chebtech2.coeffs2vals(leg2cheb(c_leg));     % and time it...
tt = toc(s);
ts = sprintf('Evaluation time = %1.2fs\n', tt);
fprintf([ns, ts])
semilogy(chebpts(length(f)), abs(f.values - cheb_vals))

[trefethen]

OK, I will make a change in the repo accordingly.