How to solve this equation
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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!
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Geoff
am 26 Mär. 2012
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)?
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Walter Roberson
am 26 Mär. 2012
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!
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Stefan Wehmeier
am 26 Mär. 2012
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)
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