:

## faster numeric Bessel functions

Maple 18
This post in reply to the Question, Definite Double Integral = Definite Integral

Some years ago member William Fish started a long discussion in part about a numeric integral involving high parameter (high oscillation) Bessel J0. That numeric integration task appeared in a Bitwise Magazine article.

At that time even obtaining numeric results involved extra effort such as handling real and imaginary components of the integrand separately, and requesting particular methods (sometimes hacked, to bump up the subinterval limit, for very high parameter values).

That led to a post where I showed that the result could be obtained quickly by using a fast compiled BesselJ (J0) from an external library along with a modified low-level call to a particular evalf/Int solver.

And sometime after that a numeric result for the real & imaginary split integrand became much more readily (if not quickly) available by using a new `maxintervals` option of evalf/Int to specify the maximal number of subintervals for the particular solver.

Maple 18 has its own compiled implementations of the Bessel functions for "hardware" (double) precision arguments. So now the numeric evaluations of the integrand are computed much faster.

Using Maple 18.00 on 64bit Windows 7 the same numeric results obtain in under a second, in a simple, single call to evalf,Int.

```restart:

CodeTools:-Usage(
evalf(Int(BesselJ(0, 50001*x)*x*exp(I*(355*x^2*1/2)), x = .35 .. 1))
);
memory used=9.28MiB, alloc change=32.00MiB, cpu time=437.00ms, real time=441.00ms, gc time=0ns

-8                 -8
3.181753502 10   - 7.798301124 10   I

restart:

CodeTools:-Usage(
evalf(Int(BesselJ(0, 10000*x)*x*exp(I*(355*x^2*1/2)), x = .35 .. 1))
);
memory used=6.83MiB, alloc change=32.00MiB, cpu time=218.00ms, real time=211.00ms, gc time=15.60ms

-7                 -7
-2.007752340 10   + 4.275388462 10   I
```

Of course the ramifications of fast, compiled Bessel functions at double precision extend much farther than just this one example. But I like seeing the speed improvement in terms of a concrete example.

acer ﻿