Relative error plot for Besseli(nu, x)

[inkydragon]

I came across a new implementation of the Bessel I function (Zaghloul, M. R., & Johnson, S. G. (2025))
that reports improvements in both accuracy and speed compared to Algorithm 644.
The work has not yet been formally published, so the source code is currently unavailable.

The paper provide a figure showing an improved domain partitioning for the function.

Algo 644: Amos1983-SAND83-0643

I analyzed the implementation in Bessels.jl and produced a plot of its current algorithmic domain partitioning.
I then selected several parameter combinations and plotted the resulting relative error distribution. These results suggest that the current computational approach has room for further improvement.

Rel error summary

Function	Max Error	P99 Error	Mean Error	Median Error	Time (s)
besseli0(x)	7.060e-16	4.252e-16	1.136e-16	8.803e-17	8.83
besseli1(x)	7.426e-16	4.183e-16	1.092e-16	8.262e-17	8.17
besseli9(x)	2.381e-14	1.032e-14	7.853e-16	1.640e-16	9.09
besseli24(x)	3.928e-14	1.562e-14	1.399e-15	3.042e-16	8.42
besseli30(x)	1.000e+00	3.737e-14	2.760e-06	2.926e-15	8.36
besseli90(x)	2.703e-02	6.557e-14	9.086e-08	9.027e-15	7.97

Rel error plots

besseli0

besseli1

besseli9

NOTE: error↑↑ where x = 35

besseli24

besseli30

NOTE: There are some underflow issues near x = 0

besseli90

When x start from 1.0

Underflow around 0?

besseli24

besseli30

besseli90

env

julia> versioninfo()
Julia Version 1.12.3
Commit 966d0af0fd (2025-12-15 11:20 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: 32 × 13th Gen Intel(R) Core(TM) i9-13900HX
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, alderlake)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 32 virtual cores)

(@v1.12) pkg> st
Status `D:\env\julia\.julia\environments\v1.12\Project.toml`
  [7e558dbc] ArbNumerics v1.6.3
  [6e4b80f9] BenchmarkTools v1.6.3
  [0e736298] Bessels v0.2.8
  [92d709cd] IrrationalConstants v0.2.6
  [98e50ef6] JuliaFormatter v2.2.0
⌃ [14b8a8f1] PkgTemplates v0.7.58
⌃ [91a5bcdd] Plots v1.41.3
  [276daf66] SpecialFunctions v2.6.1
  [10745b16] Statistics v1.11.1

ref

• Bessels.jl Acc test - Gist

• Zaghloul, M. R., & Johnson, S. G. (2025).
  Efficient Calculation of Modified Bessel Functions of the First Kind, I_{\nu}(z), for Real Orders and Complex Arguments:
  Fortran Implementation with Double and Quadruple Precision.
  arXiv preprint arXiv:2505.09770.

• Amos, D E (1983). Computation of Bessel functions of complex argument and large order. SAND-83-0643. https://doi.org/10.2172/5903937