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], J_1000000(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!).

J_1000000(x)  for  999,500 ≤ x ≤ 1,000,700

J_1000000(999995)  —  Error Analysis

From this graph, one finds that the relative error of  J_1000000(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.

J_1000000(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, http://www.iging.com/Transcendental.