Jeroen Demeyer on Wed, 13 Nov 2013 01:06:51 +0100

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Re: Large errors for eint()

I did some tests and it still looks like the absolute error on
eint1(x, n)[15] is unbounded as x goes to 0 and n to infinity. Interestingly, it is always the 15th component which has the worst absolute error.

gp> \p38
   realprecision = 38 significant digits
gp> X=eint1(1.>>20, 10^5)[15];
gp> \p96
   realprecision = 96 significant digits
gp> Y=eint1(1.>>20, 10^5)[15];
gp> abs(X-Y)<<128
%7 = 208720.00000000000000

gp> \p134
   realprecision = 134 significant digits
gp> Z=eint1(1.>>20, 10^5)[15];
gp> abs(Y-Z)<<320
%18 = 208672.00000000000000

Note also that the absolute error divided by 2^bitprecision is almost independent of the precision, which seems to indicate some algorithmic error rather than a problem with floating-point cancellation.