Weird difference between matlab's evaluation of elliptic integral to my implementation

I am evaluating the complete ellipitic integral of the first kind and I am comparing my implementation results with matlab's.

Following Elliptic Integrals, developed two method to compute the integral, by integral and by agm.

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?

[Dyuman Joshi]

Reference - https://in.mathworks.com/help/matlab/ref/ellipke.html#buiokid-5

[Lucas Ruela]

thank you. I looked at that page but missed that.

Answer 1

[Torsten]

You have to supply k^2, not k:

ellipke(0.25)

ans = 1.6858

[Lucas Ruela]

thank you for answer!