Comments on: Interpolation errors

By: link dump #2

link dump #2 — Sun, 08 May 2011 14:26:07 +0000

[...] Interpolation errors [...]

By: Jan Galkowski

Jan Galkowski — Thu, 02 Dec 2010 20:45:44 +0000

Similar Gibbs effects afflict Fourier transforms when applied to non-stationary or non-periodic time series. This is one of the reason why some of the wavelet transforms are more attractive for analysis of time series.

By: wok

wok — Thu, 02 Dec 2010 19:14:31 +0000

Tchebychev nodes are really interesting.

By: Three surprises with the trapezoid rule — The Endeavour

Three surprises with the trapezoid rule — The Endeavour — Thu, 02 Dec 2010 14:50:50 +0000

[...] right problem. Conversely, a more sophisticated integration technique such as Gauss quadrature can fail miserably when naively [...]

By: Michael Croucher

Michael Croucher — Wed, 13 Jan 2010 09:18:14 +0000

Are the Chebyshev nodes the BEST choice or merely a good choice? How hard is the proof that they are good points?

Cheers,
Mike

By: Degrafa Natural Cubic Spline Demo 3 « The Algorithmist

Degrafa Natural Cubic Spline Demo 3 « The Algorithmist — Thu, 11 Jun 2009 13:08:19 +0000

[...] polynomials and Chebyshev nodes. I actually found a good blog post about this very topic at John Cook’s blog. John has a great blog on math and computation, [...]

By: Ana Lemos

Ana Lemos — Sat, 18 Apr 2009 18:20:09 +0000

With n+1 distinct interpolation nodes we can find a unique polynomial of degree LESS OR EQUAL THAN n interpolating the points (x_i,f(x_i)), i=0,1,…, n.

By: ekzept

ekzept — Thu, 02 Apr 2009 01:46:10 +0000

@Daniel Lemire,

It can be more than “overfitting”. For example, the satisfactory solution to designing a satisfactory low-pass filter in frequency space not only demands a change in polynomials, it requires a change in norm for measuring deviations. This is the FIR filter regimen, with its equiripple deviations and infinity norm, defeating the problem of “Gibbs towers” which result from using Fourier polynomials. There may be a correspondence between norms and different sampling points or polynomials, but that’s too deep for me to see personally if there is. I know Chebyshev are involved in equiripple.

By: Daniel Lemire

Daniel Lemire — Wed, 01 Apr 2009 21:43:34 +0000

In Machine Learning or statistics, this is known as overfitting.

What is the lesson here? Always try the simplest possible model that will do the job.

(This is a deep lesson for those doing actual work.)

By: EastwoodDC

EastwoodDC — Wed, 01 Apr 2009 19:13:04 +0000

I have encountered this sort of error using the normal CDF function, which is often implemented with a polynomial approximation. The error is not noticeable unless you look at the extreme left tail, where for practical purposes it should evaluate to (essentially) zero.
A quick check shows me that Excel 2007 does not have this problem, or at least the error is so small I can’t easily detect it by looking the PDF/CDF ratio.

By: sohail

sohail — Wed, 01 Apr 2009 18:07:49 +0000

My God man. I don’t know how you keep up this quality blogging but I sincerely hope that you are able to continue it for years.

This is by far my favourite blog on the Internet.