Gaussian Quadrature 1: Summary of Legendre Polynomials

[mr.ontological]

So I usually don't comment on these videos when I'm burning the midnight oil to finish my assignments, but MY GOD is your delivery captivating! To the point where I am now positively motivated to learn more about the theories that go into Gaussian Quadrature. HAVE A LIKE GOOD SIR!

[SamSarwat]

I don't think it's possible to teach this better. You are a talent man. Keep it up and thanks for sharing this.

[ozzyfromspace]

A few years ago, I found your videos irritating, and just wanted you to get to the point. Now, with more maturity and Gilbert Strang’s linear algebra lectures, I have come to appreciate your talks and teachings. My apologies for not giving you a chance earlier. I know I’m one of hundreds of thousands of viewers (so, insignificant), but I hope you realize that the way you convey information to patient math enthusiasts is really something special. Best wishes, and thanks for the series of videos you did on Gaussian Quadrature. They really came in handy for my computational electromagnetics class (self-taught)

[tristotech]

I finished Special function quiz and I really thank you for your video. I never learn Gram-Schmidt in classroom so I try to study it and your explanation is the best

[ayanaxhye]

Came here to understand Gaussian quadrature but stayed for your delivery lol, cool concept

[holyshit]

Orthogonal polynomials appear also
from solving linear recurrence relations second order, linear second order ordinary differential equations etc
In fact the orthogonalization is the slowest way to compute them (in my opinion)

[holyshit]

4:34 Laguerre polynomials are also orthogonal but break this pattern

[imrem]

I m sure there was a nr 4 video with a worked example. I don't get how you find the weights? You solve the wenderman matrix for the legendre polynomials? The x are clear and straight forward but w's less clear here...

[holyshit]

At first look I wanted to calculate coefficients of Legendre polynomials
but for numerical methods we don't need them
We need nodes (roots) and weights
which can be calculated from eigen problem with symmetric tridiagonal matrix
b_{k, k+1} =b_{k+1, k}= k/sqrt((2k-1)(2k+1))
For nodes we need eigenvalues and
for weights we need first entry of eigenvector corresponding with eigenvalue
There is other method based on Newton's method and asymptotic approximations but i dont know the details
Maybe video with code written from scratch

[OtterMorrisDance]

What is the motivates the standard inner product for polynomials to be the the integral between -1 and 1?

[holyshit]

In fact the quickest way for getting Legendre polynomials
is recursion from solving Legendre ODE

c_{k+2} =
P_{n}(-x) = (-1)^nP_{n}(x)
P_{n}(1) = 1

Recursion for Chebyshov polynomials is slightly different

c_{k+2} = (k-n)*(k+n)/((k+2)(k+1))c_{k}
T_{n}(-x) = (-1)^nT_{n}(x)
T_{n}(1) = 1


As you can see these two polynomials can be got in linear time
but fastest polynomial multiplication algorithm is a little bit slower than nlog(n)
and in method presented on video polynomial multiplication is in loop

but method of solving ode is not in algebra

[stipepavic]

you are one of the best teachers in the worldddd!!!, you alot :DD

[debendragurung]

2:50 where to find other sources for this list. The actual Gram-Schmidt orthogialization. Every where I see is Legendre's

[darrenpeck]

Would not keeping the abitrary nature of inner products, well there variety and choice, hidden in a standard course be like reducing all metric spaces to just one. That way seems like maths is being delivered because people are engineers and physicists. This course treats maths on its own terms! A metric or inner product is then just one choice that fits the axioms not an engineering edifice that is inscrutable. By ignoring the options, I feel that the engineers actually make math harder in the longterm. So it is not the simplification they seek.

[kepler_b]

Here studying for a computer course exam... I don't understand anything. Newton-cotes methods are direct, but when I search for Gaussian quadrature, everyone speaks alien. Okay... Keep your secrets. I now have to seek another path to the room, for here the doors are closed...

[thebullybuffalo]

why is it that if I shift two orthogonal functions (I'm thinking of the expressions for different n states of quantum harmonic oscillators that are shifted), they suddenly cease to be orthogonal?