Approximation of Faddeeva function with Laplace Continued Fraction

Dear Community,

I am attempting to approximate the Faddeeva function, , with complex argument (z) using the Laplace Continued Fraction shown below:

I wish to understand the variation in as additional "terms" i.e. continued fractions are used hence want to do a forward calculation starting with 1, 1/2, 2/2, 3/2...... n/2. I can compute the reverse calculation starting from n/2 and working backwards but wrap my head around how to do this the opposite way.

Thanks in advance!

[Martin Doherty]

My current solution is to create an outer loop which increments the number of additional continued fractions whilst still solving "backwards" in an inner loop as shown below. It doesn't feel like the most efficient way of doing things mind you.

%% Laplace Continued Fraction

z = 20.0*exp(1i*5*pi/8)
r = 1.0;
nterms = 10;
i = 1;

for j = 1:nterms                                                % Outer loop - iterate # of additional continued fractions
    for k = i:-1:1                                              % Inner loop - backwards last term to first 
        if k == i                                               % Last term only divided by z;
            b = k/2/z;
        else
            b = k/2;
        end                                              
        r = 1/(z-b*r); 
        F4_store(j) = 1i/sqrt(pi)*r;      % Store each iteration  
    end    
        i = i+1;
        r = 1.0;                                                % Reset r for additional continued fraction
end