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Graphical Analysis of J _{1000000}(x)
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D. Baruth, x87@iging.com

Bessel functions of high orders and arguments occure in some physical problems as summarized by F. A. Chishtie et. al. in Mathematical Physics, abstract math-ph/0503037:^{[1]}"... large order Bessel functions that arise in the Fourier Transform (FT) of the Gravitational Wave (GW) signal from a pulsar. ... Bessel functions which also happen to have large argument. Large order Bessel functions also arise in the Peters-Mathews model of binary inspiralling stars emitting GW and several problems in potential scattering theory. Other applications also arise in a variety of problems in Applied Mathematics as well as in the Natural Sciences and present a challenge for High Performance Computing(HPC)."To evaluate the Bessel function of the First Kind for large orders and arguments, fast and accurately, the procedure _JaX^{[2]}has been used. Utilizing the x87 FPU's 80-bit registers^{[3]},Jis evaluated and analyzed at_{1000000}(x)x ≈ 1,000,000and graphically mapped for1,000,000 ≤ x ≤ 100,000,000. At these ranges, (the less accurate) Debye's asymptotic expansions are not needed. Forx ≤ 999,990, however, Debye's asymptotic expansion^{[4,5]}is implemented (using the first three terms only!).

* J_{1000000}(x) *
for 999,500 ≤ x ≤ 1,000,700

* J_{1000000}(999995) *
— Error Analysis

From this graph, one finds that the relative error ofJis ε < 2*10_{1000000}(999995)^{-13}. Function evaluation time for x > 999,990 is 0.00625 seconds. For x ≤ 999,990 evaluation is much faster, but less accurate. (See also error analysis forJ)._{10000}(9990)^{[6]}The behavior of the Bessel function for large arguments with index 1,000,000 is displayed below.

- The Analysis of Large Order Bessel Functions in Gravitational Wave Signals from Pulsars. Mathematical Physics, abstract math-ph/0503037, F. A. Chishtie, S. R. Valluri, K. M. Rao, D. Sikorski, T. Williams; High Performance Computing Symposium, May 15-18 (HPCS 2005) Guelph, Ontario, Canada.
- The proprietary procedure _JaX was programmed in x86/87 machine language (Assembly MASM 5.1). It can be called, through appropriate interfaces, by FORTRAN, C and other X86/87 compilers.
- Calculations were performed on an AMD Athlon XP 1700+, 1.47 GHz, microprocessor.
- Handbook of Mathematical Functions, Abramowitz & Stegun, Dover Publications, fifth printing, pp366, 9.3.7.
- Table of Integrals Series and Products, Gradshteyn & Rizhik, Academic Press, 1965, p963, 8.452.
- Bessel Function of the First Kind
J- Debye's asymptotic expansion for x ≤ 9990, Graphical Error Analysis, D. Baruth, http://www.iging.com/Transcendental._{10000}(x)

Copyright Dan Baruth © 2006.
All rights reserved.