[Maxima-bugs] [ maxima-Bugs-3220118 ] erf(1+5*%i) not accurate

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Bugs item #3220118, was opened at 2011-03-17 11:02
Message generated for change (Comment added) made by rtoy
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Category: None
Group: None
Status: Open
Resolution: None
Priority: 5
Private: No
Submitted By: Raymond Toy (rtoy)
Assigned to: Nobody/Anonymous (nobody)
Summary: erf(1+5*%i) not accurate

Initial Comment:
erf(1.0+5*%i) returns

 - 2.7837671212125964e+9 %i - 1.0786562260474622e+9

But the true answer (as given by Wolfram/Alpha) is

 - 2.7837702922089653e+9 %i - 1.0786931161985406e+9

Only about 4 digits are correct.

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>Comment By: Raymond Toy (rtoy)
Date: 2011-03-24 09:27

Message:
I think the inaccuracy comes from gamma-incomplete, which is used to
evaluate erf. For this particular argument, gamma-incomplete uses the
series.  

I think the test of when to use the series could be relaxed a bit.  When I
set *gamma-imag* to 0.1, the continued fraction is used, which gives

- 2.7837770292209086e+9 %i - 1.0786931161985383e+9

This is better.

I've done some experiments using the continued fraction
http://functions.wolfram.com/06.06.10.0005.01.  This seems to work a bit
better.

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Comment By: Barton Willis (willisbl)
Date: 2011-03-17 22:51

Message:
By the way: using a 1F1 representation for erf, the values agree with
Wolfram | Alpha. But the hypergeometric code has to switch to bigfloats to
do this: 

(%i11) my_erf(x) :=
nfloat(2*x*hypergeometric([1/2],[3/2],-x^2)/sqrt(%pi),[]);
(%o11)
my_erf(x):=nfloat((2*x*hypergeometric([1/2],[3/2],-x^2))/sqrt(%pi),[])

(%i12) load(hypergeometric)$

(%i14) my_erf(1.0+5*%i);
(%o14) -2.783777029220896b9*%i-1.078693116198541b9

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