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)
integ1samp.m
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