I had never known about Padé approximations, and you did such a good job motivating and explaining them. Also, the way you mentioned how it seems almost unreasonably effective, like getting something for nothing, definitely felt like it was speaking directly to the thought passing through my mind at that moment.

Padé approximations shine in analog filter design where you have poles and zeros. They are particularly effective in analog delay lines.

In general, expressions of the form at 0:23 are interesting since they have a few nice properties: 1.) Much like how polynomials are capable of expressing any function composed from addition, subtraction, and multiplication (for a finite number of operations, at the very least), expressions of the form at 0:23 do the same thing, but for addition, subtraction, multiplication, *and division*. Another way of putting it is that they are capable of describing any function defined on a field. This might help towards explaining why Padé approximates can be so much more effective than Taylor series. 2.) Approximates of that form are used all the time in highly realistic graphics shaders. This is because they can be used to create fast approximates of functions whose real values could not be calculated in the time it takes to render a frame. Unlike polynomials, they can behave very well over their entire domain, and they avoid large exponents that could introduce floating point precision issues, both of which are important when you need to guarantee that a shader will not create graphical artifacts in a limited environment where all you have to work with is 32 bit floating point precision. They also avoid calls to advanced functions like sin() or exp(), which again makes their execution especially fast. 3.) You don't always need the derivatives of a function to find such an approximate. For instance, if you know that a function has an asymptote, or that it assumes a certain value at 0, or that it's symmetric, or that it tends towards a number at ±∞, then that automatically tells you something about the coefficients within the approximate. It then becomes much easier for you to run an optimization algorithm on a dataset to find good values for the remaining coefficients. Christophe Schlick gives an excellent example of this approach in "An Inexpensive BRDF Model for Physically-based Rendering" (1994). 4.) Multivariate versions of the approximate are a thing, too. To see how such a thing can be done, simply start from the proof for the statement in 1.) but now instead of working with real values and a variable "x" as elements, you'll also be working with another variable "y". As an example, for 3rd order bivariate approximates you'll wind up with polynomials in your numerators and denominators that have the form p₁xxx +p₂xxy + p₃xyy + p₄yyy + p₅xx + p₆xy + p₇yy + p₈x + p₉y + p₁₀

The algorithm recommended this video to me, I'm so thankful because this is beatiful and very useful.

Oh nice. e^-x is a very common function to Padé approximate in linear control theory because it's the Laplace transform of a uniform time delay. Notably, x in this context is a complex number, yet it still works. I've never understood how it was computed until now. I think the aha moment is realizing we are discarding all higher order terms when we perform the equation balance. This is the key reason why the Padé approximation isn't just equal to the Taylor approximation.

I've never heard of this before, and after judging so many bad entries, this is a breath of fresh air

This is an excellent explanation of something I didn't know existed. Yet it's so simple and elegant. I'm working on a Machine Learning playlist on linear regression and kernel methods and I wish I had seen this video earlier! I'll play around with Padé approximants for a while and see where this leads me. Thank you for this interesting new perspective!

Finally! I encountered these so often in physics papers. Finally I get it!

Mech Eng grad student here. This is my "did you know?!" flex for the next couple weeks. Amazing video, thanks!!

I really like how fast you managed to explain it! Only few math videos get a topic like this explained in under 7 minutes

Having spent a great deal of time reading up on Pade approximants and struggling to find easy to understand introductory examples it is extremely exciting to see content such as this being put out there for people to learn. Fantastic job motivating the need and demonstrating the utility for these rational approximations. In my personal explorations, I have found multipoint Pade approximations to be very cool, being able to capture asymptotic behaviors for both large and small x, or around poles / points of interest is very cool. Keep up the awesome work!

It seems natural to choose M = N. What are the situations where there is an advantage in choosing M > N or N > M, where I have a "budget" of M+N coefficients that I want to work with?

I really didn't like calculus in University but I find this very interesting. I can appreciate the beauty much more now that I'm not suffering through it

The best and yet the simplest explanation for Padé approximation I have seen! We use it a lot in finite element simulation software in engineering, but I was always in search for a more intuitive explanation for its merits over default Taylor series! I am happy today.

The Padé Approximant might be closer to the actual function in the long run, but it actually has a larger relative error compared to the Taylor series around x=0. Since we only care about approximating sin(x) from x=0 to x=pi/4, because we can then use reflection and other properties to get the value for other angles, the benefits are overcome by the disadvantages (i.e. you have to do more arithmetic operations, including a division).

A hidden gem of a channel! Never really considered other approximations because the Taylor ones are so commonly used in computation. I remember reading about polynomial approximations of trigonometric functions for low-end hardware but maybe those were less general than the Padé approximation.

Pretty cool video! The general idea for the approximation reminds me a lot of the process one uses to expand a fraction into a p-adic sum, to get its p-adic representation. After years of doing math, this whole idea of actually using rational functions, a thing that we commonly ignore due to them being "ugly" is shinning a new light. Keep up the good work!

This is a great video. Well made, simple, clearly explained, genuinely interesting. Awesome.

Definitively interesting, but if I get it right: if you decide you need a higher order you cannot re-use the low order coefficients that you already have. That I'd consider a disadvantage.

Superb introduction. I was browsing thru Hall's book on continued fractions and happened upon a section on Padé approximants, which peaked my curiosity and led me to this video. I can't wait to study these further. Thank you.

I found this video a year ago, before I knew calculus, so I wasn't able to follow through the part about "Taylor series(s)". Having just finished calc 1 and 2, I'm so glad I found this video again! Very insightful, thank you for sharing.

Thank you very much! All the explanations I found on the Internet were quite difficult for me to understand. You've done a really cool work!

I once needed to look into a dataset that seemed to have an asymptotic line. I remembered of rational equations from high school algebra, did a regression analysis with that instead of polynomials, and it worked wonders. I never expected this to be also a thing for approximating existing functions and I am so happy to learn about it here.

This video is way better than the book I read which skipped a lot of the information relating to the Pade approximation. Thank you! This is brilliant!

Excellent. Hadn't known about that method. I like (typically) that the estimate tends to zero in the long term (no mention of NM Effects)

I appreciate that you switched to stressing the correct syllable midway through the video. Not only is the man French, but there's even an explicit diacritic in the last syllable of his name to make stress extra clear.

I have read about Padé schemes to discretize a partial differential operator in Computational fluid dynamics a while ago but thanks for making the advantage over Taylor series more visible. This could be of great use in computational engineering simulations to avoid divergence.

Immediately subbed. Definitely a great content, keep up the good work

After seeing this video I remembered that I actually did learn about this in an advanced mathematics class during my Master's, although then with with much higher order terms. However, they never explained very well how usefull it actually is, for simple approximations as well, so I quickly forgot about this. Thank you for the refesher, very well explained, and I will definitely keep PAdé approximation in mind as a useful tool in the future! EDIT: I remember now that the assignment was to find a singularity in a polynomial of order 130. By using Mathematica and gradually decreasing the orders of N and M in the Padé polynomial allowed the finding of a function with the same symmetry as the original giant polynomial, but with much reduced terms. This derivative of the approximated polynomial could then be used to find the singularity, which did not change location during approximation. Just a cool example of a more complicated application of the Padé approximation method for those who are interested.

This was fun to watch! Definitely good points on the typical Taylor series divergence too. How cool!

Such a great video! I'm cramming for exams in my uni right now, and this was super useful and pleasant to listen to! Way more understandable than our professor's notes lol

This stuff is like the authors you never heard of, but were notable for their time :-)

I'm stunned I've never heard of this before, given how important Taylor series approximations are.

Very interesting explanation, I'm currently studying electrical engineering and I've just been slightly introduced to a Padé approximant during a Control Systems practice, but never learned the algebra behind it in previous calculus classes.

You have such a nice style, in the communicating (both voice and visual) but also / maybe even more so, a nice "pace" of thinking and a swell difficulty level.

Physics often use power series expansion because its so easy to just cut off terms higher than some order we want for small values of x, saying that those are "negligible". I imagine picking a polynomial order "just high enough" would be tougher if its in a ratio like the Padé approximant

Very nice and very simple, makes perfect sense. I feel that these Padé approximants can greatly improve approximate series solutions to difference and differential equations in general.

Short, clear, and instructive. Congratulations for the great work 👍🏾👍🏾

As a calc 2 TA, I absolutely loved this

I´ve met Padé-Approximation at university in 5th semester. The name of the course was - as you can guess - "Approximation". :D There are another very interesting methods as well. Nice video from you. :)

This was really interesting. A professor at my college did a lot of research with approximants known as “Chromatic Derivatives”, and these share a similar motivation.

So clear and concise! Thank you.

This is fantastic, genuinely might use these at work 👍🏻

Very cool. I'm somewhat surprised I was never introduced to this in numerical analysis

this is a really neat method of approximation explained very clearly and practically! nice vid

Great explanation. These are the kinds of topics you want to share with everyone, as a scientist, but want to keep quiet, as a company, in order to have an edge. Thank you much.

I have friends who live in a house in rue Paul Padé near Paris. I'm going to ring them and explain what the video has done so well. I'm sure it will make their day.

Very cool channel, thanks for making these videos!

Oh thank you for the explanation, I will try to apply the padé approximants

Excellent Presentation. Concise, clear, and thoroughly enjoyable.

Very well explained. Thank you!

Rewatching because the thing has a lot of potential and i might apply it in my projects.

This is incredible. I never knew about this, thank you!

Very interesting and very well explained ! Thank you

Such an informative video. Thank you so much!

Thank you very much!! I didn't know about that. Very interesting 😀🤔

This is absolutely brilliant.

I didnt know and i liked, so simple and useful.thanks

Thanks a lot, you deserve more than million subscribers ❤

I always regretted not taking a splines class when i was a grad student because I didn't understand NURBS (Non-uniform rational B-splines) and how to compute their coefficients. In very few minutes, you made it apparent.

Great video with a concise and effective explanation!

Remarkable that in my university Maths degree, I didn't meet Pade Approximants. You explain the topic very clearly. I believe the B_0 = 1 assumption is fine unless the denominator polynomial vanishes at x = 0, which would imply a vertical asymptote in the overall function.

Cool! Thanks for sharing!

Awesome video! Thank you!

Thanks, never heard of this approximation before.

The first time I saw Padé approximants was in a paper proving the transcendence of e. Thanks for the useful discussion!

Never heard of it, may enhance the results i am currently having with Taylor series alone. Tankyou.

I really like this video. Very well explained. The concept could obviously (so im sure oneone did and that it has fancy name) be generalized to the idea of also thinking about how we want the approximation to behave for lim x -> inf by modelling things as a sum of functions F(X) (fractions of polynomials here) lim x -> F(x) where F defines how you want the approximation to behave for extrapolation. That way we could also think about the derivatives of the approximations at infinity.

Very interesting. I heard a few times Pade approximaton, but i did not know why we use that.

I'm missing the point. It's cool and all, but there are two buts: 1) yes, taylor series do not extrapolate well, but it's not the point of taylor series, they are specifically used to approximate the function in some small area near to some point. 2) [N/0] padé approximant is the same as taylor series, and then you have the other N versions for padé approximants - [N-1/1], [N-2/2], etc. It seems unfair to say that padé approximants work better than taylor series, since padé approximants are a direct extension of taylor series, plus you can cheat by freely choosing how to split N into [N-M, M].

I knew already the Pade approximants (see e.g. Bender-Orszag book) but I never understood why they give an advantage. Thanks for explaining.

I'm fairly sure this has givne me an idea that'll be useful, however indirectly, in a paper I'm working on. Thanks for the video!

Thanks for sharing! I learned something new today 🙂

Incredibly useful for curve fitting with functions that you know should tend to zero as x-> inf

It give you a lot of imporvement for only the extra step of rewritting the coefficients, quite impressive, indeed

Excellent presentation. vow !!

Simply amazing!

A great explanation for Padé approximation. It is clear and simple. Allow me only to indicate a small error in Recap (at 6:27). In the last line the equation shows {0 = C_{2n} + B_1*C_{2n+1} + ...} and I think it should be written as {0 = C_{n+m} + B_1*C_{n+m+1} + ..} because the total number of equations is n+m; unless you were implying that n = m.

Another great thing is that rational functions are often procedurally integratable. We can always factor the denominator into linears and quadratics (because of conjugate pairs) then we apply partial fractions. Terms of the form linear/quadratic can be dealt with using u-sub and arctan. If the quadratic is a perfect square then we only need u-sub. If there is an (ax+b)^2 - c form with positive c, we can further factor using difference of squares.

Nice video, interesting stuff.

Nice video, @DrWillWood!

Never heard of Pade Approx. Thank you.

Very cool. This looks super useful in implementing DSP and FPGA based algorithms (my wheelhouse).

Very amazing video. You really explained the concepts very nicely and concisely. A few things: 1.) When we say that B_0 can be made 1 WLOG, are we not missing the cases when B_0=0? As such, we may be losing some generality 2.) Do we always know that the M+N dimensional linear system is non-singular ie has a solution? 3.) Since we are using an M+N dimensional Taylor expansion of f to get the [N/M] Padé approximation, would it not make more sense to compare against an M+N term Taylor expansion? For sin(x) the issue of diverging to infinity would still remain but atleast the expansion will agree with the curve for a bit longer.

Such a greta video! So simple, elegant and effective. I would like to know however why the examples were given for the case where m=n, as in, when n=0, we just have a taylor polynomial. But you just explained how those are less efficient. I'm left wondering the question what is the best possible m, n pair for a given function, or perhaps more generally, a given degree.

Also used in the scipy matrix exponential algorithm AND, it was used to solve the ising model at some point

I remember there was a Fortran course where we had to compute all sorts of fits using matrix algebra. Padé was one of them (as was Fourier, etc..). I had all but forgotten about it, thanks for sharing!

This is so cool??? How is this the first time I'm hearing of this after a decade of knowing about Taylor Series?????

Would you mind doing a followup about these? You could talk a lot more about end behavior, applications, and (the thing I wondered about the most) how to decide the best N and M.

Great video thanks !

Saw this in complex analysis lessons. When a question asks for a gap between numbers and the tylor series gets infinite there, we had to use that so it goes to 0 instead. Didn't know it had a name and thought it was a part of the tylor series.

Helpful! Thanks.

Nicely done!

Wow thanks! So interesting

Thanks! Will put this to good use! I'll have to try a few examples. The sine function worked, but both approximations seemed "bad" at similar points. Since we know sine is cyclic, maybe I need to compare the two approximations over 1, 1/2 and 1/4 cycle to see which might perform better (or even if there's a clear winner). Also, I would want to look more into e^-x, and I need to check the approximations for x

Very nice, thanks!

This was very well made. Great job!

Very nice approximation to know

Amazing job

It's really interesting that you choose the degree of the numerator and denominator, which sort of means you're deciding the asymptotic behavior. Then again, if a coefficient turns out to be 0 then the degree might not be what you wanted. On that note, you set this up as a 0th degree approximation but ended up with a -1st degree approximation. That's crazy to me. It's like the approximation knew that the function oscillates around 0 even though the first few terms of the Taylor series don't suggest this at all.