Computing terms of a sequence generated by a nonlinear difference equation

I would like to compute the first 19 terms of a sequence. I find it takes a very long time. Is there something I am missing? Anyway to speed it up?

It takes 78 secs. Using Excel would be instantaneous.

tic

syms alpha

T=19;

X = sym(1:T);

X(1)=0.1;

for i=2:T

X(i)=(1-alpha - X(i-1)).*X(i-1);

end

disp(X)

toc

Answer 1

[Alan Stevens]

Must be the fact that you are using symbolic maths. Without that it's fast (as long as you specify a value for alpha of course):

tic
T=19;
alpha = 0.1;  
X = zeros(1,T);
X(1) = 0.1;
for i = 2:T
X(i) = (1-alpha-X(i-1)).*X(i-1);
end
disp(X);
toc

[Hassan Benchekroun]

Thanks Alan but I need to keep vector X as a function of alpha. I use them to maximize a function of X wrt alpha.

The maximization step also takes a very long time, after checking I found that just computing X is time consuming. So I want to try and fix that before dealing with the maximization part.

Thanks again.

[Alan Stevens]

You could still just make X a function of alpha along the following lines:

tic
T = 19;

% Example calculating X rapidly for many values of alpha.
alpha = 0.01:0.01:1;
X = zeros(numel(alpha),T);
for k = 1:numel(alpha)
    X(k,:) = fn(alpha(k),T);
end
toc

function x = fn(alpha,T)

        x=zeros(1,T);
        x(1) = 0.1;
        for i = 2:T
            x(i)=(1-alpha-x(i-1))*x(i-1);
        end

end

The above calculates 100 sets of 19 values of X in about 0.002 seconds on my laptop.

[Hassan Benchekroun]

Thanks a lot Alan. Indeed it's quite fast. I will use this for the purpose of my example.