WP34s 3924 Calculation Error

[Albert Chan]

(09-02-2024 06:44 PM)Nigel (UK) Wrote:  For many integrals the new method is quicker and more accurate. It is particularly good when there is an integrable singularity at either end of the interval of integration. However it is less good when integrating functions that vary periodically many times over the interval, which is the case for your integral.

You are right. DE algorithm does not like oscillatory integrand.
It assumed transformed function is bell-shaped, then simply sum the rectangles.

For this case, DE algorithm work too. It is just that software had placed a hard limit on levels deep.
Note: my code changed meaning of levels, start from 1 instead of 0 --> levels = 12 get 12 rows output.

lua> f = fn'x: exp(sin(x)*cos(x))'
lua> ; Q.quad(f, 0, 1000, 1, 12, 1e-16, true)

Code:

1103.4366650221054      0.370273314848271
1168.093244328251       0.055352241458539
996.2599753713897       0.17247834220460842
1024.9010969740907      0.027945254119896016
1017.6160848617153      0.0071589003168768695
1059.4116213556815      0.0394516500021799
1055.105792170186       0.004080945453478147
1038.9227093485367      0.01557679187876928
1062.9411055724377      0.02259616840292012
1063.8404586620848      0.0008453834241068643
1063.8468942997813      6.049402156368027e-06
1063.8468944036285      9.76149629141775e-11

Problem is this take about 2^(levels + 2) = 2^14 samples.

Still, it beat Romberg by a factor of 2

lua> ; Q.integ(Q.simpson(Q.u(f, 0, 1000)), 2^15)

Code:

4       865.1840001860048       299.21141073191774
8       778.0020184493249       693.544473641916
16      896.6700594243979       1014.4110063293531
32      1107.3997862356239      1317.7179315491724
64      1035.9832383865798      964.6015618987897
128     1049.6893911940772      1063.3938708872174
256     1084.9625972289223      1120.234726810947
512     1057.8616940259467      1030.7609975864543
1024    1064.5204424618387      1071.1791781971756
2048    1063.8045493118993      1063.0886565033247
4096    1063.8480952794305      1063.8916412417711
8192    1063.8468965924299      1063.8456979054642
16384   1063.846894123814       1063.846891655198
32768   1063.8468944049175      1063.8468946860226

col 1 = subintervals
col 2 = extrapolated trapezoid areas
col 3 = extrapolated rectangle areas, height = midpoint
col 3 are less good, but it cost nothing to generate, and useful to cross check col 2