I wrote:
> Especially for large runlengths (and after the first few hundred iterations
> or so), rounding errors tend to be randomly distributed in an approximately
> Gaussian fashion,
Brian Beesley wrote:
> I know perfectly well what you mean, but these two statements tend to
> contradict each other. Gaussian distributions are continuous &
> smooth, we have instead a discrete distribution whose gaps tend to
> increase with size.
I added some code to the Mlucas 2.7a source (my personal copy, not the version
on my website) to better-quantify this. Since fractional errors are defined
in the interval [0, 0.5], I subdivided this interval into 64 equal-sized bins
and counted how many fractional parts fell into each bin on each iteration.
Then I ran around a hundred iterations of M1325011 using an FFT length of 64K,
i.e. an exponent close to the practical roundoff limit for that FFT length on
an IEEE64-compliant machine. Up to iteration 13, all fractional errors fell
into bin 0, i.e. were in [0, 1/128). After that, things looked as follows -
In the table below, the vertical placement corresponds to bins 0 through 64
(bin 64 corresponds to the special case of an error = 0.5), and the table
entries denote the number of fractional errors in the corresponding bin:
iteration
bin 14 15 16 17 18 19 20 21 ... 44
--- ----- ----- ----- ----- ----- ----- ----- ----- ... -----
0 65530 65457 64930 61600 44951 23245 9680 7781 7555
1 6 76 555 3332 15541 19547 10589 7371 7158
2 0 3 44 475 3573 11771 12090 10433 10356
3 0 0 7 96 972 5620 7639 5971 5961
4 0 0 0 30 348 3089 8518 9169 9353
5 0 0 0 3 84 1065 4321 4033 4042
6 0 0 0 0 55 740 4751 5804 5841
7 0 0 0 0 5 164 1923 2462 2489
8 0 0 0 0 6 199 2708 4603 4730
9 0 0 0 0 0 32 743 1262 1290
10 0 0 0 0 1 48 1149 2203 2185
11 0 0 0 0 0 2 266 594 616
12 0 0 0 0 0 10 672 1803 1890
13 0 0 0 0 0 0 62 277 259
14 0 0 0 0 0 3 206 591 606
15 0 0 0 0 0 0 21 110 118
16 0 0 0 0 0 1 137 614 636
17 0 0 0 0 0 0 3 43 49
18 0 0 0 0 0 0 23 146 121
19 0 0 0 0 0 0 0 19 22
20 0 0 0 0 0 0 29 164 162
21 0 0 0 0 0 0 0 5 6
22 0 0 0 0 0 0 2 14 20
23 0 0 0 0 0 0 0 1 2
24 0 0 0 0 0 0 4 49 51
25 0 0 0 0 0 0 0 2 0
26 0 0 0 0 0 0 0 4 3
27 0 0 0 0 0 0 0 1 0
28 0 0 0 0 0 0 0 6 11
29 0 0 0 0 0 0 0 0 0
30 0 0 0 0 0 0 0 0 0
31 0 0 0 0 0 0 0 0 0
32 0 0 0 0 0 0 0 1 4
33 0 0 0 0 0 0 0 0 0
34 0 0 0 0 0 0 0 0 0
35 0 0 0 0 0 0 0 0 0
36 0 0 0 0 0 0 0 0 0
37 0 0 0 0 0 0 0 0 0
38 0 0 0 0 0 0 0 0 0
39 0 0 0 0 0 0 0 0 0
40 0 0 0 0 0 0 0 0 0
41 0 0 0 0 0 0 0 0 0
42 0 0 0 0 0 0 0 0 0
43 0 0 0 0 0 0 0 0 0
44 0 0 0 0 0 0 0 0 0
45 0 0 0 0 0 0 0 0 0
46 0 0 0 0 0 0 0 0 0
47 0 0 0 0 0 0 0 0 0
48 0 0 0 0 0 0 0 0 0
49 0 0 0 0 0 0 0 0 0
50 0 0 0 0 0 0 0 0 0
51 0 0 0 0 0 0 0 0 0
52 0 0 0 0 0 0 0 0 0
53 0 0 0 0 0 0 0 0 0
54 0 0 0 0 0 0 0 0 0
55 0 0 0 0 0 0 0 0 0
56 0 0 0 0 0 0 0 0 0
57 0 0 0 0 0 0 0 0 0
58 0 0 0 0 0 0 0 0 0
59 0 0 0 0 0 0 0 0 0
60 0 0 0 0 0 0 0 0 0
61 0 0 0 0 0 0 0 0 0
62 0 0 0 0 0 0 0 0 0
63 0 0 0 0 0 0 0 0 0
64 0 0 0 0 0 0 0 0 0
I snipped iterations 22-43 because they are very similar to 21 and 44.
There are several interesting trends visible here. Although the overall trend
is approximately Gaussian, there is clearly a strong preference for even-
numbered bins, and an even stronger preference for power-of-2-numbered bins.
Definitely worthy of further study, but I've got to get some dinner now.
Cheers,
-Ernst
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- Mersenne: Re: Roundoff errors EWMAYER
- Re: Mersenne: Re: Roundoff errors Brian J. Beesley
- Mersenne: Re: Roundoff errors EWMAYER
- Mersenne: Re: Roundoff errors EWMAYER
- Mersenne: Re: Roundoff errors George Woltman
