Assume that a rational generating function f(z)g(z), with
- f(z) and g(z) relatively prime and
- g(0)≠0,
- has a unique pole 1β of smallest modulus (that is, g(1α)=0 and α≠β implies that |1α|>|1β|, or |α|<|β|).
Then, if the multiplicity of
1β is
ν, we have a powerful theorem for precise approximation
|
[zn]f(z)g(z)∼Cβnnν−1whereC=ν(−β)νf(1/β)g(ν)(1/β)! |
Example:
|
an=5n−1−6an−2for n≥2 with a0=0 and a1=1an=5n−1−6an−2+δn1make recurrence work for all nA(z)=5zA(z)−6z2A(z)+zmultiply by zn and sum on nA(z)=z1−5z+6z2=z(z−12)(z−13) |
The root closest to zero in the denominator is
13, which is therefore the "smallest modulus" with a multiplicity of
1 (ie as a root
13 appears only once).
|
1β=13v=1g′(z)=−5+12zg′(13)=−5+123=−1C=(−3)1f(13)−1=(−3)⋅13−1=1[zn]A(z)≡[zn]z1−5z+6z2≡an∼3n |
Source: Analysis of Algorithm
# posted by rot13(Unafba Pune) @ 8:27 PM
