[GiNaC-list] Re: Airy functions

Jan Bos jb68 at xs4all.nl
Sun Dec 23 14:53:55 CET 2007

Dear Alexei,

Thank you for your interest and careful examination of the code. I made
an effort to fit the functions in the GiNaC framework and got series
expansions etc. more or less for free. I'm looking forward to seeing the
result when it meets the GiNaC standards. 

To answer your questions a fundamental property of the algorithm is

Depending on the argument a so called "steepest descent" contour is
determined along which the integrand is monotonically decreasing. 

This ensures that a) roundoff does not occur, b) the range over which
the integrand has to be evaluated to obtain a certain accuracy is
minimized and c) the dominant factor (possibly very larger or small) can
be placed outside the integral. Furthermore all derivatives at +/-
infinity are zero. For functions with this property the trapezoidal rule
has exponential convergence, that is, halving the stepsize more than
doubles the number of significant digits of the result. For an
integrator with non-constant step size this property would be lost. 

>     AiryContour* pcntr=createAiryContour(zn);
> Pointers are evil. What is the reason for explicit

Again depending on the argument (zn) but more involved, 3 different
types of contours are needed of which one is the truly steepest descent
contour and the other 2 are approximations thereof. AiryContour is the
base class of them.

I hope this answers your questions

Best regards,


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