How to quickly calculate the following function？

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ma Jack
ma Jack 2022-9-2
回答： Jan 2022-9-4
Hi all,I want to calculate the function shown below:
This is Lerch transcendent function( https://en.wikipedia.org/wiki/Lerch_zeta_function ).
I use the following formula to calculate it, but this is too slow (I need to calculate it many times)：
There is also no official function here to calculate it.
So I would like to know how to calculate it quickly, thank you all in advance！
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采纳的回答

Jan
Jan 2022-9-4
See also: https://fredrikj.net/blog/2022/02/computing-the-lerch-transcendent/
Calculating the sum seems to be more stable. For cases, in which the integral method is successful, the sum is 10 to 200 times faster for the given test data:
function test_Lerch
% See: https://people.math.sc.edu/Burkardt/py_src/polpak/lerch_values.py
% and: https://people.sc.fsu.edu/~jburkardt/m_src/test_values/test_values.html
a_vec = [0.0E+00, ...
0.0E+00, ...
0.0E+00, ...
1.0E+00, ...
1.0E+00, ...
1.0E+00, ...
2.0E+00, ...
2.0E+00, ...
2.0E+00, ...
3.0E+00, ...
3.0E+00, ...
3.0E+00];
s_vec = [2, 3, 10, ...
2, 3, 10, ...
2, 3, 10, ...
2, 3, 10];
z_vec = [0.1000000000000000E+01, ...
0.1000000000000000E+01, ...
0.1000000000000000E+01, ...
0.5000000000000000E+00, ...
0.5000000000000000E+00, ...
0.5000000000000000E+00, ...
0.3333333333333333E+00, ...
0.3333333333333333E+00, ...
0.3333333333333333E+00, ...
0.1000000000000000E+00, ...
0.1000000000000000E+00, ...
0.1000000000000000E+00];
f_vec = [ ...
0.1644934066848226E+01, ...
0.1202056903159594E+01, ...
0.1000994575127818E+01, ...
0.1164481052930025E+01, ...
0.1074426387216080E+01, ...
0.1000492641212014E+01, ...
0.2959190697935714E+00, ...
0.1394507503935608E+00, ...
0.9823175058446061E-03, ...
0.1177910993911311E+00, ...
0.3868447922298962E-01, ...
0.1703149614186634E-04];
for k = 1:numel(z_vec)
z = z_vec(k);
s = s_vec(k);
a = a_vec(k);
f = f_vec(k);
fprintf('\nk: %d\n', k)
tic
for k = 1:1
y1 = Lerch_fun_integral(z,s,a);
end
toc
tic
for k = 1:1
y2 = Lerch_fun_sum(z,s,a);
end
toc
fprintf('Ref: %.16g\n', f);
fprintf('Int: %.16g delta: %16g\n', y1, abs(y1 - f));
fprintf('Sum: %.16g delta: %16g\n', y2, abs(y2 - f));
end
end
function out = Lerch_fun_integral(z,s,a)
f=@(t)((t.^(s-1)).*exp(-a*t))./(1-z.*exp(-t));
out=(1/gamma(s))*(integral(f,0,Inf));
end
function out = Lerch_fun_sum(z, s, a)
out = 0;
if z <= 0
return
end
lim = 1e-16;
k = 0;
z_k = 1;
term = Inf;
while abs(term) > lim * (1.0 + abs(out))
if a + k ~= 0
term = z_k / (a + k)^s;
out = out + term;
end
k = k + 1;
z_k = z_k * z;
end
end

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