Gaussian Quadrature 2: How to Determine the Weights

[HarpreetSingh-kezk]

This is pretty much exactly what I look for in presenters.

The simple explanation of the terms/variables/parameters used in the formula comes first, followed by the simple and basic math.

Thanks Professor.

[TheCsePower]

This is much easier to understand than these crazy maths textbooks where everything is either trivial or left as an exercise for the reader.

[econguy]

As a postgrad, I've been using gaussian quadrature quite a lot in stochastic dynamic programming in economics, and it has never been clear to me how everything is tied together (since Python takes care of the difficult parts). Glad to have found this series of lectures. It really helps me understand how the engine works. Thank you for all the wonderful uploads!

[balajisriram]

Sir, I am using gauss quadrature for FEM. Your explanation was great. Now I understand how to choose the weights for a polynomial of degree "n"

[daviddavini]

Your videos are so helpful and inspiring!

[debendragurung]

At 10:00 Isnt Gauss-Legendre quadrature approximated at N points, get exact integration for upto 2N-1 degree polynomials...

[shifagoyal]

What do we mean by 4 degrees of freedom?

[gerardoperez]

absolutely beautiful. Thank you very much!

[sosoyo]

Wow who knew David Wallace would be such a good teacher?

[김승환-gc]

Oh my god, i finally understand basis of function space from your vedio.
Thank you very much!! Much than the below comment

[patipateeke]

5:50: okay, but why are the weights wi the same when you integrate f(x)=1 and f(x)=x, and f(x)=x^2?

[vasantk]

Is there a reason you cannot use an arbitrary power (like you mentioned x^7) when filling the matrix? Also, what makes the approximation only as good as the cubic power if you were to have the last polynomial be a power of 7 instead of cubic?

[ronaldssebadduka]

where do u get the values 2, 0, 2/3, and 0? Also, why did u use 1111 in the first matrix

[christossofianos]

Professor if i may ask another question!
You say that the problem with this method is that the weights are all over the place. And i understand the magic with Gauss quadrature!
But why is this a problem? Even if weights are all over the place arent they the exact solution to the integral?

[raquelpicado]

How can we predict the error when using Gaussian Quadrature?

[NicolasSchmidMusic]

"linearity [...] it means something so simple that talking about it makes it more complicated"

[subramaniannk]

Prof, I have a doubt and I wish you helped.

I am trying to understand the link between orthogonality and linearly independent.

Is it valid to show Wronskian of Legendre polynomials is non-zero instead of proving orthogonality between Legendre polynomials using inner product?

Or, are orthogonality and linear independence two very different things?

[christossofianos]

Professor, thank you for your videos they are very well presented and informative.
I have a question, is it possible to calculate the error when computing the integral in some other points other than gauss points for an arbitrary polynomial function?
For example we know that for a third degree polynomial when using 2 gauss points the evaluation of the integral is going to be exact.
But if these 2 points are not in the optimal positions of Gauss, but in arbitrary positions is there a way to calculate the error?
I understand that if the polynomial is given then we can of course compute the error by evaluating the polynomial. But if the polynomial is arbitrary and we only know its degree, would it be possible to calculate the error?
Thank you in advance!

[ghandricheahcene]

How we can scale the weights to an interval [a, b] ?

[snnwstt]

8:34 I feel that we have 8 degree of freedom, four weights and ... four x[sub]i[/sub].