Graphical Analysis of J1000000(x)
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:"... 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  has been used. Utilizing the x87 FPU's 80-bit registers, J1000000(x) is evaluated and analyzed at x ≈ 1,000,000 and graphically mapped for 1,000,000 ≤ x ≤ 100,000,000. At these ranges, (the less accurate) Debye's asymptotic expansions are not needed. For x ≤ 999,990 , however, Debye's asymptotic expansion[4,5] is implemented (using the first three terms only!).
J1000000(x) for 999,500 ≤ x ≤ 1,000,700
J1000000(999995) — Error Analysis
From this graph, one finds that the relative error of J1000000(999995) is ε < 2*10-13. 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 for J10000(9990)). 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 J10000(x) - Debye's asymptotic expansion for x ≤ 9990, Graphical Error Analysis, D. Baruth, http://www.iging.com/Transcendental.
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