See Section 2.2 in our paper.
In the table below we give information on which maximum interval
length L and cutoff N we used in the formula (11),
and which cutoff N and internal precision we used in the formula (12).
These parameter choices were made using trial and error,
and tested by computing long series
of K-Bessel values and checking against the result obtained using
(12) with extra precision and longer cutoff,
for various values of r in the interval 2.5 <= r <= 15
(the case D=1050 was also tested for 15 <= r <= 20).
The formulae (11) and (12) were only used for values of u
lying in a certain interval 0 < u < U, as specified in the second
column in the table below.
The value U is chosen so that for u >= U, the normalized K-Bessel
function exp(Pi r/2) K_ir(u) is MUCH smaller than 10^(-D),
so that its contribution in our equation system (5) is in any case
dominated by the error term in (5).
For values u > U we simply used the crude approximation
K_ir(u) ~ sqrt(Pi/2u) exp(-u)
(which is asymptotically correct as u tends to infinity for fixed r).
| D |
Interval for u |
Parameters in (11)
| Parameters in (12)
|
| 50 |
0 < u < 150 |
| If 0 < u < 30: | N=85, L=1 |
If 30 <= u < 150: | N=85, L=5 |
|
| If 0 < u < 20: | N=75, digits=65 |
If 20 <= u < 100: | N=500, digits=130 |
If 100 <= u < 150: | N=520, digits=180 |
|
| 100 |
0 < u < 300 |
| If 0 < u < 35: | N=150, L=2 |
If 35 <= u < 300: | N=150, L=6 |
|
| If 0 < u < 20: | N=100, digits=120 |
If 20 <= u < 100: | N=270, digits=190 |
If 100 <= u < 300: | N=630, digits=360 |
|
| 215 |
0 < u < 520 |
| If 0 < u < 100: | N=245, L=2 |
If 100 <= u < 520: | N=245, L=6 |
|
| If 0 < u < 20: | N=160, digits=240 |
If 20 <= u < 100: | N=340, digits=300 |
If 100 <= u < 520: | N=1200, digits=670 |
|
| 525 |
0 < u < 1300 |
| If 0 < u < 150: | N=600, L=3 |
If 150 <= u < 1300: | N=600, L=15 |
|
| If 0 < u < 40: | N=350, digits=560 |
If 40 <= u < 100: | N=510, digits=610 |
If 100 <= u < 500: | N=1320, digits=1020 |
If 500 <= u < 1300: | N=2800, digits=1700 |
|
| 1050 |
0 < u < 3400 |
| If 0 < u < 500: | N=900, L=6 |
If 500 <= u < 3400: | N=900, L=20 |
|
| If 0 < u < 100: | N=800, digits=1200 |
If 100 <= u < 1000: | N=3000, digits=2000 |
If 1000 <= u < 3400: | N=8500, digits=5500 |
|