Increasing accuracy of Whittaker functions

Hello.

I know that there were a lot questions devoted to accuracy of fslove procedure.

I'm trying to solve a transcendental equation with so-called Whittaker functions and the gamma function

where k is the integer number, and are some parameters (see below) and E is the unknown variable.

Below is my code

function z=dispersion

mm=linspace(0,20,21);
for j=1:21
 
[y,fval]=fzero(@(E) self(E,mm(j)),0.51,optimset('Display','off','TolFun', 1.0e-15, 'TolX',1.0e-15)); 
Energy(j)=y;
value(j)=fval;
end
Energy
value
end

function z=self(E,m)
    
r1=0.25;
r2=5; 
    
z=gamma(1/2+floor(abs(m))/2-E-floor(m)/2).*(whittakerM(E+floor(m)/2,floor(abs(m))/2,r2^2/2).*whittakerW(E+floor(m)/2,floor(abs(m))/2,r1^2/2)-...
    whittakerM(E+floor(m)/2,floor(abs(m))/2,r1^2/2).*whittakerW(E+floor(m)/2,floor(abs(m))/2,r2^2/2));

end

I have a problem with accuracy of Whittaker functions in the case of large parameter m . When for instance I obtain and value = -3.048609148470981*1.0e+22 that is inappropriate accuracy. Is it possible to resolve such a kind of problem?

P.S. Actually I understand the origin of the problem. Abnormal values of fval is due to behaviour of Whittaker functions for large values of m. Nevertheless I would greatly appreciate for any suggestions.

[Joshua Brown]

Hey! I've had the same issue. A quick fix is the folowing:

m = sym(m);
double(whittakerM(E+floor(m)/2,floor(abs(m))/2,r1^2/2));

(I.e. going via symbolic algebra to deal with things like (e^(pi*m))*e^-(pi*m) better) then should be accurate even for large m.