Minimax Approximation and the Exchange Algorithm

In this video we'll discuss minimax approximation. This is a method of approximating functions by minimisation of the infinity (uniform) norm. The exchange algorithm is an iterative method of finding the approximation which minimises the infinity norm.

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[EdToml]

Years ago, in my first real job, I need to approximate some experimentally derived curves. These curves were used to figure out if an turbine was performing as designed. So the error in the approximations needed to be predicable. I first tried least squared, but quickly realized, that even though this algorithm minimized the total error it did not minimize the maximum error. Hence it would sometimes pass engines that should have been rejected. I then tried min max. It was 'fun' determining the coefficients. This was in the late 1970s and computer power was quite limited. Min max ended up working quite well. its a very good solution when errors mater.

[emrekisi]

Best explanations ever, it takes just 12 minutes and worth for every second. Thank you👍

[tuongnguyen]

this algorithm is used in filter design for electrical engineering, something call Remez algorithm

[The-Martian]

Dude can't imagine how this did help me a lot, big thanks ❤

[francescomazzoli]

Your videos are fantastic, thanks a lot for putting these out.

[yuankunzhu]

First of all, i like this videos.🙂 because, your video is almost the only one which talking about remez algo that i have found. And, because, this "only one that i found" is very detailed, professional and with good examples. However, as a foreigner, more references(like slids or pdf file) are welcomed, and that will help me a lot.

[kodirovsshik]

That is very useful, thank you for sharing. Lovely video!

[monsterplane-fi]

Your vedio is very claer, and the examples provided helps me to understand the whole algorithm and min max approximation. Excellent.Thanks for your sharing.

[Focke]

0:41 nom nom nomnom
😜Im sorry. The video was very informative👍

[hellomoto]

I got lost when at 3:05 he called it a 4th order polynomial P(x), but you can see from the visual that P(x) must be at least a 5th degree polynomial, due to P'(x) = 0 for 4 different x values in the given interval (or you can notice that the end behaviors are approaching opposite infinities, i.e. odd degree =/= 4th degree). Am i missing something? I'm assuming "order" and "degree" mean the same thing for this polynomial context

[Noissimsarm]

Is there a proof of the first theorem? Why should the minimal generate error terms like that?

[fork]

@DrWillWood 10:34 Should be 0.073, instead of 0.73.

[albertmagician]

Once you realise that there is a background music, I can no longer follow the mathematics. I try to follow the music. Why do they that?