HP71B Integral Questions

[Albert Chan]

Amazingly, Aitken's extrapolation trick work even for the troublesome 1/sqrt(1-x^2)

lua> I=require'integ'
lua> function f(x) return 1/sqrt(1-x*x) end
lua> u = I.u(f,-1,1)
lua> a = I.integ(I.simpson(u,-1,1,0),2^10) -- with bad assumption, T1 = 0

4     2.6032971925091086   3.131182235954578
8     2.876452993708263    3.141072676119944
16    3.009535987413591    3.1415792702305954
32    3.075628788308323    3.1415925017013078
64    3.1086187748017564   3.141592652903347
128   3.1251067205395437   3.1415926535886425
256   3.1333498128442754   3.141592653589793
512   3.1374712489390655   3.141592653589794
1024  3.139531953229666    3.141592653589784

Code:

Aitken Extrapolation on Trapezoid numbers
64    3.10861877480
128   3.12510672054
256   3.13334981284  3.14159114440
512   3.13747124894  3.14159246493
1024  3.13953195323  3.14159263001  3.14159265359

As expected, with correct T1 value, trapezoid numbers converge better.

lua> I.integ(I.simpson(u,-1,1,2*sqrt(3)),2^10)

4     3.1421574437527586   3.131182235954578
8     3.141606450669424    3.141072676119944
16    3.1415928071550314   3.1415792702305954
32    3.141592654278877    3.1415925017013078
64    3.141592653590944    3.141592652903347
128   3.1415926535897936   3.1415926535886425
256   3.1415926535897936   3.141592653589793
512   3.1415926535897944   3.141592653589794
1024  3.1415926535897887   3.141592653589784