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Hi I'm doing a computation where the modified bessel function of the third kind is the main source of computational strain, we are using a 10,000 dimension's for our distribution, is there any easier way to compute it besides the standard approach, any estimations?

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I think you are using a now obsolete terminology, see en.wikipedia.org/wiki/Bessel_function Can you tell us what range of parameters you're interested in? –  Stopple Oct 10 '11 at 23:20
www.fmpm.ch/docs/11th/papers_2008_web/B2b.pdf page 6 –  redhatuser Oct 11 '11 at 0:01
What you've termed "the third kind" is what would now be called "the second kind": functions.wolfram.com/Bessel-TypeFunctions/BesselK ; echoing Stopple's question, how big/small are $\nu$ and $z$ in $K_\nu (z)$? How accurate an approximation are you expecting? –  J. M. Oct 11 '11 at 8:42
thank you both for clearing that up , z is a 10,000x10,000 matrix, v =5 which is small i was looking for an approximation that would be less computationally complex while still maintaining a high order of accuracy –  redhatuser Oct 11 '11 at 20:55
So you have Bessel functions of a matrix argument; the Digital Library of Mathematical Functions has links to papers on asymptotics, see dlmf.nist.gov/35.5 –  Stopple Oct 11 '11 at 23:00

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