[GiNaC-list] numerical quadrature versus arbitrary-precision calculation (was: Airy functions)

[Alexei Sheplyakov]

On Mon, Dec 31, 2007 at 09:14:27AM +0000, alexander baker wrote:
> Alexei,
> 
> Could you provide an example to highlight the following statement?
> 


> >However, for arbitrary precision calculation the accuracy can be
> > arbitrary high,
> >so the interval could be arbitrary big.

Consider \int_1^\intfty dx/x^2. Of course, this integral is elementary
(it's equal to 1), but for sake of example let's compute it with some
numeric quadrature method using IEEE double precision floating point (FP)
numbers.  There are two kinds of errors: the errors of the numerical
quadrature algorithm itself (cutoffs to replace improper integral with
Riemann one, which in turn gets replaced by a finite sum, etc) and
the roundoff errors. The accuracy of calculation can not be better
than roundoff error. In this example that means the error is always
bigger (or equal) than some \epsilon, being the smallest FP number
such that 1 + \epsilon \ne 1. Thus, it's fine to cut off the upper
limit of integration at 1/\epsilon (\int_{1/\epsilon}^\infty = \epsilon),
since its contribution will be lost due to roundoff errors anyway.
And any request to calculate the integral with higher accuracy is just
bogus.

Now, that \epsilon is basically a hardware- (platform-) dependent constant
for IEEE doubles (and floats). Thus, the maximal cutoff does not change
at the runtime. This is not the case for arbitrary-precision calculation,
since the size of the mantissa (and hence, \epsilon) of the FP numbers is
(almost) arbitrary user-specified quantity. Now, the arbitrary high target
accuracy *does* make sense, so, the interval of integration can be
arbitrary big.

Hope that helps,
	Alexei

-- 
All science is either physics or stamp collecting.

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