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Solving Recurrence equation for n

asked 10 years ago

student gravatar image

updated 10 years ago

I have a recurrence equation tn which is defined as

tn=2.5tn11.5tn4,for n5

Where t1=5, t2=10.5, t3=26.25, and t4=62.625.

I need to solve my recurrence equation for n when tn=13×1010. How can I achieve this in Sage?

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answered 10 years ago

kcrisman gravatar image

Nothing native, but sympy has rsolve and Maxima has solve_rec (see e.g. this sage-support thread). Unless this ticket has something that helped?

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answered 10 years ago

rws gravatar image

updated 10 years ago

Is the exact answer needed? I guess not, since the coefficients are given in floating point. Then an estimate with five significant digits should do. This is usually done symbolically by constructing the (rational) generating function of the recurrence, computing the roots of the denominator polynomial, and from the largest root r the asymptotics which is of form c*r^n for linear recurrences. This can be done manually, with Sage performing these step by step, or (when you have the g.f.) using code from ticket #10519. For examples see my paper.

However, even more simple, but without getting any asymptotics as byproduct, would be to use matrix exponentiation. See for example http://fusharblog.com/solving-linear-.... This looks like a Project Euler question anyway.

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Asked: 10 years ago

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Last updated: Feb 16 '15