Dear All,
Yes, all our texts have this example.
However, the stricter criteria of the radius of convergence from any
point
on the real axis, x, for a power series representation of 1/(a^2+x^2), a
real, being R=(x^2+a^2)^0.5 should be observed instead, since this is
where it stops working. The theorem deals with where the series becomes
divergent, whereas the radius of convergence indicates when the series will
not be sufficiently convergent.
I am having trouble understanding the statement "poor performance",
since this does quite well, both on my Pentium 2 and the Core 2, although
the answers are not identical. Apparently the math handling in the Core 2
is better.
This does go to show that you never want to take data for an arctan
process. Instead you want it to simply tell you what its functionality is,
so
you can better integrate it.
dmelliott
________________________________________________________________
----- Original Message -----
From: Carlo de Falco
To: dmelliott
Cc: Søren Hauberg ; [email protected]
Sent: Sunday, June 07, 2009 7:47 AM
Subject: Re: [OctDev] Developer Registration Request
On 7 Jun 2009, at 00:25, dmelliott wrote:
>
> Dear Mr. de Falco,
>
> While it does not have a singularity on the path of
> integration, it does at x = +/- i/5. Thus the radius of
> convergence, from zero, interferes with the construction
> of a polynomial approximation therefrom. This is also
> early calculus, and not subtle.
>
> In the real world data comes to you as it comes to you,
> usually on an equal step domain and often without
> tolerance (necessary for any coherent data processing).
> If you can convince the all manufacturers of measuring
> equipment to modify all their machines, I am sure we
> would all be better off. The Octave routines that give
> this type of output might be a place to start.
>
> If you have a better method for equal step integration,
> it would indeed be of great value.
>
>
> dmelliott
On 7 Jun 2009, at 00:25, dmelliott wrote:
>
> Dear Mr. de Falco,
>
> While it does not have a singularity on the path of
> integration, it does at x = +/- i/5. Thus the radius of
> convergence, from zero, interferes with the construction
> of a polynomial approximation therefrom. This is also
> early calculus, and not subtle.
The counterexample I proposed is indeed a very well-known one
(usually referred to as "Runge's counterexample") you can find
in any elementary text book on numerical analysis,
see for example [1], page 390 example 9.3.
But complex roots in the denominator do not have anything to do with
the poor
performance of high order Newton-Cotes formulae.
The reason why Runge's function exposes the limitations of Lagrange
interpolation on equispaced nodes (and therefore of quadrature rules
based on this kind of interpolation)
has rather to do, to put it simply, with the fact that it has large
areas near the
ends of the integration interval where it is nearly flat so that high
order, highly oscillating polynomials are not able to approximate it
correctly.
You should see the same effect by substituting a gaussian curve for
Runge's function in the example.
> In the real world data comes to you as it comes to you,
> usually on an equal step domain and often without
> tolerance (necessary for any coherent data processing).
> If you can convince the all manufacturers of measuring
> equipment to modify all their machines, I am sure we
> would all be better off. The Octave routines that give
> this type of output might be a place to start.
I did not object your idea to provide a function for doing quadrature on
(equispaced) data samples.
I just wanted to put out a warning that increasing the order of the
quadrature
formula does not always improve the accuracy, in the extreme case of
Runge's function it's even the other way around.
That being said, I have nothing to object about your approach
as long as the docs explain exactly what your function is doing:
it is then left to the user to apply it wisely.
> If you have a better method for equal step integration,
> it would indeed be of great value.
My suggestion was to simply use composit formulae of a fixed order and
let the users
choose the order depending on whatever a-priori knowledge they might
have about
how the data they are analyzing were generated.
An example of how to do this is in the attached function, you can see
a test by typing
"demo integ1samp"
I'm not proposing this as a substitute for your approach, it is just a
quick hack to explain my idea, do whatever you wish with it as I'm not
willing to pursue this subject any further.
c.
>
> dmelliott
[1] A. Quarteroni, R. Sacco, F. Saleri,
Numerical Mathematics, secon edition.
Springer Verlag (2007)
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