Singular Fourier-Pade approximation 1.0

Partial sums of Fourier terms for a function with jumps in value or derivative converge poorly, because of the Gibbs phenomenon. This file uses the Fourier coefficients, and locations of the singularities, to construct a different approximation that converges spectrally. For details, see T. A. Driscoll and B. Fornberg, Numerical Algorithms 26 (2001), pp. 77-92.Example for f(x)=|x|, using 7 Fourier coefficients:c = [pi/4 zeros(1,11)];c(2:2:12) = -(2/pi)*(1:2:11).^(-2);z0 = exp(1i*[-pi 0]);[p,q,r] = padelog(c,z0);% Make a plot:x = linspace(-pi+10*eps,pi-10*eps,200); z = exp(1i*x);pz = polyval(p(end:-1:1),z);qz = polyval(q(end:-1:1),z);rz{1} = polyval(r{1}(end:-1:1),z);rz{2} = polyval(r{2}(end:-1:1),z);fplus = ( pz + rz{1}.*log(1-z/z0(1)) + rz{2}.*log(1-z/z0(2)) ) ./ qz;plot(x,abs(x),x,2*real(fplus),'k.')

Requirements:	No special requirements
Platforms:	Matlab
Keyword:	2pi12112,  Algorithms,  C2212,  Exp Ipi,  Fx3d X,  Numerical,  Padelogcz,  Zeros
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