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.
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.
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.
there are proofs for everything here in MJD Powell's "Approximation theory and methods" including the characterisation theorem, the inequality for the bound on |h|, proof of convergence of the exchange algorithm etc. The characterisation theorem actually applies not just to polynomials but a wider class of functions which satisfy a set of conditions called the "Haar conditions", but didn't want to complicate the video too much! As for why the error terms behave like that I don't have a great intuition beyond a straight line where it's easy to verify by playing around a bit. I feel like it has something to do with the number of minima and maxima of a polynomial increasing with its degree but can't say much beyond that i'm afraid!
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
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.
hey edtomi. good story lad. thanks for contributing to this. i like your story my bro. very good for expressing the real deal here. here, take a coin
this algorithm is used in filter design for electrical engineering, something call Remez algorithm
Beautifully and elegantly presented!
Best explanations ever, it takes just 12 minutes and worth for every second. Thank you👍
Thanks
thank you so much! Appreciate the support!
Dude can't imagine how this did help me a lot, big thanks ❤
Your videos are fantastic, thanks a lot for putting these out.
thanks , i appreciate that!
Beautiful video.
That is very useful, thank you for sharing. Lovely video!
Thanks!
Hi Dr. Wood!
This is nifty.
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.
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.
Is there a proof of the first theorem? Why should the minimal generate error terms like that?
there are proofs for everything here in MJD Powell's "Approximation theory and methods" including the characterisation theorem, the inequality for the bound on |h|, proof of convergence of the exchange algorithm etc. The characterisation theorem actually applies not just to polynomials but a wider class of functions which satisfy a set of conditions called the "Haar conditions", but didn't want to complicate the video too much! As for why the error terms behave like that I don't have a great intuition beyond a straight line where it's easy to verify by playing around a bit. I feel like it has something to do with the number of minima and maxima of a polynomial increasing with its degree but can't say much beyond that i'm afraid!
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
Awesome!
Lovely
0:41 nom nom nomnom
😜Im sorry. The video was very informative👍
@DrWillWood 10:34 Should be 0.073, instead of 0.73.
1:13 idk like y=1/2 or smth
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?
5:53 power of interpolation
manim?
regards from Chile!
Well spotted! Manim is used for the repeated iterations of the exchange algorithm. the rest is done in keynote!