Gaussian Quadrature 3: The Explanation of the Technique

[pflintaryan]

this guy not only teaches but also inspires. The way he explains, it feels nothing in the world is more important. Massive respect!! :)

[samirkhan]

Thank you, Gauss, for revealing this incredible method to the world, and thank you, @MathTheBeautiful, for explaining it in such a beautifully amazing way.

[mohamedradwan]

I am a computer science student, but I really enjoy watching these lectures to understand terms that I hear all the time, such as Laplacian and Gaussian quadruture.

[ozzyfromspace]

This is easily one of the most beautiful methods of numerical analysis 😭❤️

[perivarfriborg]

This is beautiful, I'm lost for words.

[stratpap]

You are a very good teacher and you enjoy teaching with your heart!!Congratulations!!

[georgeorourke]

This lecture has a very suttle argument (especially from 4 min 30 on) that initially missed me completely. When you arrived at the point where you had ∫p(x) = ∫r(x) I told myself Ah! now we just need to go back to the previous lecture to solve the integral. I initially was confused why you worried about the zeros of the Legendre polynomials since, by the inner product argument, that integral was zero. I then realized that the whole point of choosing the zeros of the Legendre polynomial was A) because R(x) = P(x) at those points and B) the coefficients ωi of Ln have been calculated once and for all.Therefore the evaluation the integral of p(x) becomes simply the sum ωi * P(gi) where gi are the zeros of the Ln Legendre polynomial - I mention this should someone else have the same hesitation as I did.

Last question - at the end of te previous lecture (11:40) you stated that the problem with the method presented in that lecture was the very significant variation in the magnitude of the coefficients. Is the use of the zeros of Ln better because the coefficients have been determined and we do not need to rederive them or is there really less variation - in which case why?

As always thank you for a very interesting presentation.

[User-cvee]

Just How did these mathematicians had the vision to go that far?

[adarshkishore]

Thanks, great video! Btw, I think I can give an analogy to explain why the weights behave nice. In one of the previous videos (Why {1, X, x^2} is a terrible basis), you had explained the nice feature of orthogonal basis which makes it less error prone compared to an arbitrary basis. The Legendre polynomials are also orthogonal with respect to the inner product and thus they approximate better than arbitrary non-orthogonally polynomials. That may be the reason why the weights are nicer

[shifagoyal]

Logically buildinging concepts step by step.Upload more videos.

[User-cvee]

Great teaching, good video editing! Perfect

[coffeedotbean]

The amount of times I had to pause the video and go hold up hold up hOLD UP FOR A MOMENT

[jvdcaki]

Awesome lecture. I studied this at class but I did not understand anything. Your videos are much more interesting

[anhquocnguyen]

I think the logic is great but do you think that you made a mistake on the indexing of Ln?

[cantkeepitin]

Is there also an optimum rule if we only assume continuity, like |x|? How much we can gain over rectangular rule with equidistant samples of same weight? Or whay about integration from a to inf.

[iphykelvin]

I have a question. Why can we use this for any interval of integration [a, b] and not for only [-1, 1]. Thanks

[ozzyfromspace]

I’m gonna try this tonight! I’m so excited to code this marvelous idea from scratch 😊🔥🙏🏽🎊❤️💯🙌🏽😭👏🏽🥳

[bernhardriemann]

how do you get the roots of the legendre polynomials?

[moritzbecker]

Thank you very much for your wonderful lecture. I wish I was one of your students.

[MesbahSalekeen]

p(x) is the polynomial that we are trying to integrate, r(x) is the remainder. What about f(x)?