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I'm trying to implement a numeric solver of quantum 3-body problems, which would be able to calculate the wavefunctions (not only energies!) to high precision (15 decimal places) on a simple home PC. Currently I'm studying papers like this and this ones. They describe a nonvariational approach, where rotational degrees of freedom are separated away, leaving us with coupled 3D eigenproblems. Then there are various ways of calculating eigenvalues from there, and some ways to calculate eigenfunctions.

This approach does work for me, but it appears that for excited states I have to wait really long to get high-precision eigenvalues, needless to say that eigenfunctions are even slower since an additional calculation is needed to get them from intermediate results for eigenvalues.

I also read some papers describing variational approaches, like this one, but they were mostly concerned about getting eigenvalues, so I'm not sure if I'd get comparable precision for eigenfunctions (a caveat of variational methods) if I try it.

So, before I actually do finish my implementation of a new square wheel, I'd like to learn if there are some much better suited approaches at solving general quantum 3-body Coulombic problems with emphasis on high-precision calculation of eigenfunctions. If a method only allows for 12 decimal places when working with binary64 floating-point numbers, I'd also be interested in it.

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    Most scientific computation is done with double precision floating point numbers. These are capable of representing real numbers with a precision of around 1e-15. But as soon a you start doing significant calculations, you'll get round-off error (and other types of error), and your results will not have 15 good digits. And, of course, if your numbers are larger than 1, then 15 good digits doesn't give you 15 good decimal places, anyway. So, you'll need something other than double-precision arithmetic, which will be hard to find, and probably very slow. – 2017-02-25
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    @bubba I'm doing my final calculations (after getting the relevant matrices using arbitrary precision arithmetic) in machine precision, and getting eigenvalues precise to about $5\times10^{-16}$ (compared to reference variational calculations). Of course, eigenfunctions might not get that precise. But at least 14 decimal places with usable speed would already be good enough for an answer here. – 2017-02-25

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