add exponential scaling support to Bessel function family

[jdrugo]

Would the Julia development team consider supporting the option to return exponentially scaled versions of the Bessel function? This option already exists in the used Amos library through setting the KODE argument, but is currently not used in Julia.

Supporting this option would allow evaluating the Bessel function family for larger arguments, for which the standard return values would be out of bounds. This is, for example, useful for computing various properties of the von Mises distribution (see JuliaStats/Distributions.jl#223 ).

I could imagine it being either implemented as a flag to the Bessel functions, for example as

function besseli(nu::Float64, z::Complex128, scaled::Bool = false)

as in R, or by a set of separate methods

besseliscaled
besselhscaled
...

I could provide the required code if you are happy with this request.

[simonbyrne]

I think this should be in Base. Matlab also makes it available as an option to the bessel* function, but I think a more julian approach would be to create separate functions.

There is already erfcx which scales erfc by exp(x^2): we could call them bessel*x?

[stevengj]

bessel[jyikh]x sounds good to me. These kinds of functions are quite useful in practice, by allowing you to deal separately (analytically) with the exponential factors in algorithms where they arise.

Note that this is needed for besselk too, to avoid underflow, and for bessel{jy} in the complex plane.

[jiahao]

The core Julia team have tried very hard to not introduce more function names, and these functions, while arguably important, would introduce a large number of one-off functions.

I think we need a better nomenclature for these things.

[stevengj]

@jiahao, we could always use besselk(ν,x, scaled=true).

[JeffBezanson]

If these names are reasonably standard it could be ok to add them.

[jdrugo]

@JeffBezanson, both R and MATLAB support scaling through optional arguments rather than separate function names.

[jiahao]

I would much prefer the scaled=true form. The ___x names are unfamiliar to me.

[JeffBezanson]

Sounds good.

[kmsquire]

Is it worth calling the argument "logscaled" yo make it perfectly clear?
To the uninformed (me), scaled implies linear scaling. (Of course, I don't
use these functions.)

On Tuesday, April 29, 2014, Jeff Bezanson notifications@github.com wrote:

Sounds good.

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Reply to this email directly or view it on GitHubhttps://github.com//issues/6665#issuecomment-41724269
.P

[stevengj]

I don't think scaled implies any particular type of scaling; you will have to read the documentation to find out precisely what it means (but it is easy to guess if you use these kinds of functions).

[StefanKarpinski]

How about log=true as the keyword name?

[jiahao]

logscaled or log suggests that the log of the function (or something) is being computed, which isn't the case.

[StefanKarpinski]

I'm unclear on how this scaling is done, which makes it hard to come up with an apt name.

[jiahao]

The scale factor is there to damp the exponentially growing asymptotics of the special function. See, for example, the comment in amos:besseli and compare that with the asymptotic formula from DLMF. Note that it isn't fully accounting for the zeroth order asymptotics, just the exponential part.