How to solve this equation

Hi,everyone,there is a equation as followed:

f(x)=h0*f(x)+h1*f(x-1)+h2*f(x-2)+h3*f(x-3);

The values of the h0~h3 are given,then how to solve this equation?

thank you!

[Geoff]

This equation looks intractably recursive. Are those f(x) terms on the right-hand side in fact supposed to be some other function/dataset, such as g(x)?

Answer 1

[Walter Roberson]

You need three values of f(x) in order to completely solve the equation.

Or it can be expressed in terms of the known constants together with the unknown f(0), f(1), f(2)

sum(-((h0 - 1 + R .* h1 + h2 * R.^2) .* f(0) + R .* ((h0 - 1 + R .* h1) .* f(1) + R .* f(2) .* (h0 - 1))) .* (1./R)^x ./ (3 * h3 .* R.^3 + 2 * h2 .* R.^2 + R * h1)

where

R = roots([h3, h2, h1, h0-1]);

Note: all three roots of R must be included, even if imaginary!

Answer 2

[Stefan Wehmeier]

This is a recurrence equation and thus has to declared as such using rec(...).

   eq = evalin(symengine, 'rec(f(x)=h0*f(x)+h1*f(x-1)+h2*f(x-2)+h3*f(x-3), f(x))');
   solve(eq)