Timeline for Calculating Debye functions to high accuracy (hundreds of bits), is it possible to be faster than generic quadrature?
Current License: CC BY-SA 4.0
11 events
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| Mar 16 at 9:44 | comment | added | user2373145 | The reason I wanted more than a hundred bits of accuracy is to find some minimax polynomials for faster Debye function evaluation for binary64 floating-point afterwards. This is the result, currently being registered in Julia's General registry: gitlab.com/nsajko/DebyeFunctions.jl | |
| Mar 8 at 9:27 | comment | added | njuffa | @user2373145 I am not familiar with the Debye functions (encountered them here for the first time), but I have implemented elementary functions and special functions in a professional capacity. Generally speaking, when it comes to performance (your question), quadrature is rarely the best choice. | |
| Mar 8 at 7:31 | comment | added | Federico Poloni | Did you mean underflow rather than overflow? The integrand is bounded when $t\to 0$. | |
| Mar 8 at 7:08 | comment | added | user2373145 |
Tentatively, Wolfgang's suggestion to use expm1 seems to solve all my problems so far.
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| Mar 7 at 20:58 | comment | added | njuffa | There is a question (with answer) about a continued fraction expansion for $D_{3}$ on Mathematics Stackexchange which looks worthy of exploration. The paper by Guseinov and Mamedov linked on the Wikipedia page likewise looks worthy of followup, as it relates the Debye functions to the incomplete gamma functions, and various arbitrary-precision packages (e.g. ARB) have support for those. | |
| Mar 7 at 20:33 | comment | added | Wolfgang Bangerth |
Separately, the underflow of $e^t-1$ should be addressed by using the function expm1 instead of computing exp(t)-1.
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| Mar 7 at 20:32 | comment | added | Wolfgang Bangerth | What application requires you to have hundreds of bits of accuracy? | |
| Mar 7 at 19:37 | comment | added | njuffa | The Wikipedia page linked to in the question lists multiple publications dealing with implementations of the Debye functions. Have you consulted any of those? If so, what did you find? ACM TOMS is a highly-reputed journal, so I took a brief look at the publication by MacLeod and it seems that he does not use quadrature. Whether it is practical to extend the method used for hundreds of bits of accuracy I cannot determine on the double. | |
| Mar 7 at 19:26 | comment | added | Maxim Umansky | That Wikipedia page shows series expansion for $D_n(x)$. Why not try it instead of integration, if the series converges fast then that would be the way to go. | |
| S Mar 7 at 14:16 | review | First questions | |||
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| S Mar 7 at 14:16 | history | asked | user2373145 | CC BY-SA 4.0 |