Lagrange Interpolation

Lagrange Interpolation Formula

In this video, I present the extremely neat Lagrange Interpolation Formula, which gives a clean formula for a polynomial that goes through given points. And I do this purely using linear algebra techniques, which illustrates how powerful this subject actually is!

For instance, here I find the polynomial of degree 2 that goes through (1,2), (2,3), and (4,11).

It's not too good to be true - it's too good and it's true! Enjoy!

[bogdancorobean]

I find it hard to believe that there's anything in math that could take you 10 years to understand :) But I see your point, sometimes the simplest things are hardest to really grasp.


Also, I love how linear algebra just pops up everywhere, especially if you're trying to calculate stuff or do anything "practical".

[jiyoonpark]

Am I the only one who constantly looks for his videos after searching something??? Love his lectures!!!

[AakibKaushar]

Most satisfying video of Lagrange's Polynomial so far.

[Ensivion]

this is a neat way to construct any degree polynomial, super intuitive the way presented it but I could see how it's confusing if you write it out abstractly

[MrCigarro]

highly used in statistics for smoothing curves. Thank you Dr. Peyam.

[shaharyar]

I have no idea what's going on after the 9:00 mark, But the first 9 minutes were extremely helpful. Thanks!

[luishumbertoninoalvarez]

Thanks Dr. Peyam, great explanation!!!

[fernandobezerrademenezes]

Highly instructive. Congratulations for your great videos!!

[JPK]

It's worth noting that there isn't a polynomial for every set of pairs; the x coordinate needs to be different for each member of the set. In this case, this is equivalent to assuming the denominators are nonzero or assuming a function (not necessarily polynomial) of x exists which goes through those points

[mireksoja]

From the educational point of view it is a very good explanation.Thank you.

[tingweixia]

Very helpful and understandable for my seminar facharbeit( project). Many appreciation!

[namannarang]

I first saw this technique in a very basic polynomial question in my book i.e something like f(x) has some degree say 4 sat f(1)=3 f(2)=5 f(3)=7 find f(4)

[abdolkarimmehrparvar]

wow, that was a solid explanatin, thank you . could you make a video for newton interpolation as clear as this one?

[dattchan]

Thanks a lot, it took me so long to get it

[GhostyOcean]

I don't have time atm to watch the whole video and see why you're doing it this way, but the way I learned to solve for a unique polynomial of degree n given (n+1) points is to solve Xc=y where X is the matrix of exponents of x (x0^0, x0^1, ..., x0^n; x1^0, x1^1, ..., x1^n;...;xn^0, xn^1, ..., xn^n), c is the coefficient vector (c0, c1, ..., cn) and y is the vector of y's (y0, y1, ..., yn). You solve using RREF to find the coefficients.

[ethancheung]

It is a similar process to partial fraction, except the denominator and numerator is interchanged

[IFearlessINinja]

Any time I see things like this, I like to think of outlandish hypotheticals that might relate to the new method I've learned.
I'd imagine it would be possible to extend this to infinitely many points. If you were to trace e^x, for example, would you just arrive with the taylor series?
Assuming it does, what if you tried it with a polynomial that doesnt have a taylor series of finite convergence interval? What would it do?
Maybe it would it become a sinusoidal wave of infinite magnitude, and a period equal to the distance of your points? But then what if chose points that approached infinite density?

Probably some dumb questions in there, but I thought I'd share my thought process. I always have fun thinking about things like that

[armandoski-g]

Hi! Could you please do a lecture on Vandermonde matrix? I really enjoy your explanations, please don't stop making videos :))

[soumyachandrakar]

This is interesting. I just learned something new 🙂

[newtonnewtonnewton]

Very nice video thanks D peyam السلام عليكم