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POFinding Binet form in Mathematica
 

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    <p>Suppose I have a linear recurrence* and I want to find its closed-form 'Binet' representation. Is there a good way to do this in Mathematica?</p> <p>It seems like a very basic request, and there are certainly many natural ways to ask Mathematica to do it for me. But so far everything I've tried has failed: it churns until its memory use is so high the operating system is obliged to close it, or it gives warnings that it does not know how to simplify simple expressions†, or the like. I could understand this if the question was hard, but it's not&mdash;factor the characteristic equation, find the roots, and solve a linear system. The most recent time I tried this (and had the program crash) was on a degree-9 example, and I just don't think a 9-by-9 linear system should be that hard to solve.</p> <p>Surely I'm not the only one who need to do this from time to time! What is the right way to do this?</p> <p>I lost my session so I don't have the exact code I tried. One solution created a List with the recurrence and its initial points and used RSolve. Another found and factored the characteristic equation and took appropriate roots to the n-th power multiplied by polynomials of degree corresponding to the multiplicity with coefficients generated from C[i]. I also tried Solve and Reduce in various ways.</p> <p><em>*</em> Or a rational generating function. Actually I'll start from a <code>List</code> of numbers which are described by a recurrence of less than half its length, and <code>FindLinearRecurrence</code> or <code>FindGeneratingFunction</code> can do the not-too-difficult conversion.</p> <p>† For example, when I asked it to solve one recurrence it choked on sin^2 (3pi/14) + cos^2(3pi/14) in the course of the calculation, saying that ran out of precision. You'd think it could symbolically simplify something like that, but no.</p>
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    1. CODo you mean this? http://mathworld.wolfram.com/BinetForms.html If so, there isn't going to be a function to do it for you, but it doesn't look like too bad a of a program to write.
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    2. CO@Searke: Yes. Any linear recurrence has a formula in the form `a[n_] := Sum[polynomial[[i]][n] * base[[i]]^n, {i, 1, k}]` for some Lists polynomial and base of length k. I'd like to find these lists. The four programs I wrote all failed to give an answer: the programs were correct but Mathematica was unable to solve the equations for reasons I outlined above. I'm looking for a way around Mathemastica's limitations.
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    3. COPush come to shove I can write a program from the ground-up to do this, not using Mathematica's symbolic capabilities, but in that case I won't be using Mathematica, either -- why pay the performance cost if you can't use any of the advanced features?
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