Weird difference between matlab's evaluation of elliptic integral to my implementation
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I am evaluating the complete ellipitic integral of the first kind and I am comparing my implementation results with matlab's.
The matlab version, I simply call ellipke. For example,
% matlabs
display(ellipke(0.5))
It returns 1.8541.
My integral method is this
b = integral(@(t) integrando(t,0.5),0,pi/2,'RelTol',0,'AbsTol',1e-12);
function r = integrando(t,k)
r = 1./sqrt(1-k.^2.*sin(t).^2);
end
The result is b = 1.6858. Quite different from matlab's.
The agm method is this
c = pi/2/agm(1,sqrt(1-0.5^2))
function r = agm(x0,y0,rel,maxtol)
arguments
x0 (1,1) double
y0 (1,1) double
rel (1,1) double = 1e-6
maxtol (1,1) double = 1000
end
a = x0;
g = y0;
count = 1;
while (a-g > rel && count < maxtol)
anext = (a+g)/2;
gnext = sqrt(a*g);
a = anext;
g = gnext;
end
r = (a+g)/2;
end
It also computes c = 1.6858. Exactly like the integral.
Why is there a differente between the results?
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