Compute first partial derivatives for a recursive function.

I would like to ask if there is a way to compute the first derivatives of a recursive function.

For example, let say that we have a function f_{i} = w + a * f_{i-1} and we want to compute the first partial derivatives of a function f_{i} with respect to the variables w and a.

It is a simple example and it is cumputed easily. But I would like to know if there is a method when I have a more complicated example.

Thank you in advance.

Answer 1

[Roger Stafford]

I don't see any particular difficulty here. By taking the derivatives of each f_(i) with respect to a or w you obtain another recursive formula. In your particular example you get

 df_(i)/da = f_(i-1)+a*df_(i-1)/da

and

 df_(i)/dw = 1 + a*df_(i-1)/dw

Each of these can be evaluated by a recursive procedure, (preferably using ordinary for-loops.) Presumably the initial values of df_(1)/da and df_(1)/dw are regarded as zero. If not, you will need to know what they are.

[Fio]

Thanks for your quick response. As I have said this is a simple example and it is computed easily. Let say than f_i = W + A x_i + B f_i-1, where f is a vector kx1, A matrix kxk and B matrix kxk. And you want the first partial derivatives of each parameter. I have found the form of partial derivatives analytically and I have also write a program. What I want to know is if there is a way to do it with a mfile of matlab. I have found that using this http://www.mathworks.com/help/symbolic/jacobian.html you can compute the Jacobian matrix of a vector f when f has not a recursive form. I wondered if there is something similar to this.

[Walter Roberson]

Would that be a Markov chain ?