Add trigamma function

[agribov_no_s_p_a_m@…]

Boost has implementation of digamma function. It would be nice to have implementation of trigamma function (second derivative of ln(Gamma())). ​http://en.wikipedia.org/wiki/Trigamma_function

[anonymous]

Yes, trigamma would be appreciated! Please.

[anonymous]

FYI There's a Google summer of code student working on polygamma so hopefully this will come to pass...

[anonymous]

Do you mean this? ​http://code.google.com/p/pmtk3/source/browse/trunk/misc/fastfit/trigamma.m?spec=svn1197&r=1197

Looks interesting. Thank you. But I trust boost MUCH more than google, taking into account eg google translator which is an bottomless source of inspiration for demotivators. When I use something from boost I can assume that it is perfect (almost). When I use something from google it is like mine field.

[John Maddock]

No, I meant: ​http://www.google-melange.com/gsoc/project/google/gsoc2013/nikhar/52001 with code at: ​https://github.com/BoostGSoC/boost.math

But please note it's very early days yet, nothing there ready for Boost primetime yet...

[anonymous]

Looks like a great beginning! Good luck. And I implemented trigamma using ​http://upload.wikimedia.org/math/a/8/0/a80acd50152662cd7bcd41659fe489cc.png ​http://upload.wikimedia.org/math/6/6/9/66906e84ea81f06b3dbad5a8bf24558c.png LoL. Very slow working but quick to implement. :-) Proper implementation would use Remez and asymptotic expansions I suppose. Have no time for it.

[anonymous]

Is there any progress on that? Really looking forward to use the trigamma function!

[John Maddock]

Well it only took a year!

There's now a complete polygamma implementation in the bernoulli-tgamma branch of Boost math, available as a zip here ​https://github.com/boostorg/math/archive/bernoulli-tgamma.zip.

There are still docs to write, so it won't get officially released just yet, but in the mean time if you have any feedback or concrete use cases they would be much appreciated.

[anonymous]

Hi,

thanks for the fine work! I stumbled on this by chance today! I am currently struggling with extracting the minimal subset required to test it with my software, but so far no real success as the polygamma function relies on quite a large new subset of things.

But may I ask why the multiplication theorem is not used in the formulas? It seems to be a nice way to reduce the problem for large arguments instead of heaving to rely on Bernoulli coefficients.

As you asked for concrete use cases: I want to use it for the gradient and Hessian of the Dirichlet-Function and in this case I need the bigamma and trigamma function at the same time. Therefore I miss a way to just compute both at the same time as they should share some computations...

[John Maddock]

Many thanks for the response,

But may I ask why the multiplication theorem is not used in the formulas? It seems to be a nice way to reduce the problem for large arguments instead of heaving to rely on Bernoulli coefficients.

The Bernoulli expansion is pretty efficient, for example at double precision it requires recursion to x > 9, then just 9 Bernoulli terms to converge.

I don't immediately see how the ​argument multiplication formula helps in this case - as each reduction in x results in 2 new polygamma functions.

As you asked for concrete use cases: I want to use it for the gradient and Hessian of the Dirichlet-Function and in this case I need the bigamma and trigamma function at the same time. Therefore I miss a way to just compute both at the same time as they should share some computations...

I don't think there much overlap in terms of computation as digamma uses rational approximations for the evaluation.

[John Maddock]

Fixed in develop