Bessel functions of Large Orders and Arguments

Graphical Analysis of  J1000000(x)

D. Baruth,

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], 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)).[6]  The behavior of the Bessel function for large arguments with index 1,000,000 is displayed below.

J1000000(x)  for  1,000,000 ≤ x ≤ 100,000,000

  1. 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.
  2. 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.
  3. Calculations were performed on an AMD Athlon XP 1700+, 1.47 GHz, microprocessor.
  4. Handbook of Mathematical Functions, Abramowitz & Stegun, Dover Publications, fifth printing, pp366, 9.3.7.
  5. Table of Integrals Series and Products, Gradshteyn & Rizhik, Academic Press, 1965, p963, 8.452.
  6. Bessel Function of the First Kind  J10000(x) - Debye's asymptotic expansion for x ≤ 9990, Graphical Error Analysis, D. Baruth,

Copyright Dan Baruth © 2006.  All rights reserved.