# Parameters used in the computation of the K-Bessel function K_ir(u) (for 2.5 <= r <= 20).

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