How to create recursive function handle efficiently

Assume the following:

u=[1 2 3];

W_in=[4 5 6]';

W=[1 2 3;4 5 6;7 8 9];

x=zeros(3,4); %initialization

x(:,1)=[1 2 3]';

How can I create a function handle such that:

x(:,i)=@(gamma) (1-gamma)*x(:,i-1)+gamma*(W*x(:,i-1)+W_in*u(i-1))

where gamma is a scalar between 0 and 1.

Since I CANNOT use double array to store function handle, I tried to use cell array to store function handles generated in the loop:

x=cell(1,4);

x{1}=[1 2 3]';

for i=2:4,

x{i}=@(gamma) (1-gamma)*x{i-1}(gamma)+gamma*(W*x{i-1}(gamma)+W_in*u(i-1))

end

However, I get the error with dimensions of W*x{i-1}(gamma),

also if I wish to set gamma to 0.4 and call x{2}(0.4), this cannot be done since I am only allowed to use integer index like x{2}(1),x{2}(2)...

Please help me on this, thank you so much!

[John D'Errico]

Your question asks about how to define this for a non-integer index. However you have not defined how the recurrence would start for a non-integer "index". A recurrence makes mathematical sense only if you can start it in some way. Anyway, MATLAB does not allow non-integer indexes for arrays of any class. That includes cell arrays.

Your idea of a cell array of function handles is pretty much un-workable for what you are asking to do. Sorry. It may look pretty, but in practice the idea is worthless.

Instead, use a loop. I know it seems so terrible. But loops are fine in terms of efficiency. This silly cell array of function handles is still an implicit loop anyway. Using a loop allows you to start the recurrence simply. It allows you to deal with the initial value for that recurrence.

Finally, if you have a current MATLAB release, you might be able to use the memoize function to help you make your code more efficient.

[Maple]

Thanks for your answer! if you use x{i}=@(gamma) (1-gamma)*x{i-1}+gamma*(W*x{i-1}+W_in*u(i-1)), and try to access x{2}(0.4) you will get [7.8 16 24.2]', which is what I want. However if you try x{3}(0.4), you get an error,that's the problem....