
Re: ellipse points regularly spaced, or inverse of the incomplete elliptic integral of the second (not first) kind
Posted:
Jun 1, 2010 3:45 PM


"David W. Cantrell" <DWCantrell@sigmaxi.net> wrote in message <htmffh$jnc$1@news.acm.uiuc.edu>... > > [By the way, the second link mentioned there, to the sci.math.research > thread "Inverting elliptic integrals", seems to be broken. However, that > thread can now be found at the Math Forum: > <http://mathforum.org/kb/message.jspa?messageID=1672061>.] > > Regards, > David W. Cantrell
Incidentally, I bump to the same problem today, and I try David's method. Unless if I make an error somewhere, the power series seem to have poor accuracy for m ~ 1 and z >> 0.
I hope I make a mistake somewhere:
function E = elliptic12_powerseries(z, m) % Compute EllipticE by power series in z % Provided by David Cantrell % http://mathforum.org/kb/message.jspa?messageID=1672061
% E = z + ... % (m*z^3)/6 + ... % (1/120)*(4*m + 13*m^2)*z^5 + ... % ((16*m  284*m^2 + 493*m^3)*z^7)/5040 + ... % ((64*m + 4944*m^2  31224*m^3 + 37369*m^4)*z^9)/362880 + ... % ((256*m  81088*m^2 + 1406832*m^3  5165224*m^4 + % 4732249*m^5)*z^11)/39916800;
P = {1; (1/6)*[1 0]; (1/120)*[13 4 0]; (1/5040)*[493 284 16 0]; (1/362880)*[37369 31224 4944 64 0]; (1/39916800)*[4732249 5165224 1406832 81088 256 0] }; P = cellfun(@(p) polyval(p,m), P); E = z.*polyval(P(end:1:1),z.^2);
end % elliptic12_powerseries
%% m = 0.9; theta=linspace(0,pi/2); Epowerseries = elliptic12_powerseries(theta, m); % http://www.mathworks.com/matlabcentral/fileexchange/8805 [trash Eanalytic] = elliptic12(theta, m);
plot(theta, Eanalytic, 'r', theta, Epowerseries, 'b') title(sprintf('Ellipictic integral 2nd kind, m = %g', m)); xlabel('z (rad)') legend('Analytic', 'power series','Location','NorthWest');
Any comment?
Bruno

