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Wednesday, January 01, 2014

 

Asymptotics for linear recurrences

Assume that a rational generating function f(z)g(z), with

  1. f(z) and g(z) relatively prime and
  2. g(0)0,
  3. has a unique pole 1β of smallest modulus (that is, g(1α)=0 and αβ 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)Cβnnν1whereC=ν(β)νf(1/β)g(ν)(1/β)!
Example:
  an=5n16an2for n2 with a0=0 and a1=1an=5n16an2+δn1make recurrence work for all nA(z)=5zA(z)6z2A(z)+zmultiply by zn and sum on nA(z)=z15z+6z2=z(z12)(z13)
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)131=1[zn]A(z)[zn]z15z+6z2an3n

Source: Analysis of Algorithm


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