Gaussian Quadrature 2: How to Determine the Weights

[@MathTheBeautiful]

Go to LEM.MA/LA for videos, exercises, and to ask us questions directly.

[@HarpreetSingh-ke2zk]

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.

[@econguy1437]

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!

[@MathTheBeautiful]

Hi, thanks for letting me know! -Pavel

[@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.

[@luck4393]

this!

[@Qbabxtra]

Its a joy watching you teach Math!

[@balajisriram6363]

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"

[@sosoyo180]

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

[@MathTheBeautiful]

Yesh

[@MeteOguc]

But I think the voice actor is Matt Damon. Production cost must have been really high but as a result the lecture is high quality in the end too. So thanks 😄

[@gerardoperez4067]

absolutely beautiful. Thank you very much!

[@shifagoyal8221]

What do we mean by 4 degrees of freedom?

[@김승환-g3c]

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

[@daviddavini847]

Your videos are so helpful and inspiring!

[@NicolasSchmidMusic]

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

[@christossofianos8540]

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?

[@MathTheBeautiful]

Yes, but in numerical calculations, it becomes a problem. Subtracting two large similar numbers is inaccurate. For example, try this (in most numerical environments): (10^17 + 1) - 10^17. E.g.: www.google.com/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=(10^17+%2B+1)+-+10^17

[@christossofianos8540]

Ok i understand now. Thank you professor! Greetings from Greece!

[@debendragurung3033]

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

[@jiashuotong6922]

You are right. I think there should be more x^n rows, up to n=7.

[@ronaldssebadduka6837]

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

[@ZMKmagic]

you get them when you integrate the functions from -1 to 1 ... for example if you integrate f(x)=1 from -1 to 1 you get =2..if you do the same with f(x)=x you get 0, f(x)=x^2 you get 2/3 and so on. Also, the 1111 is from when f(x)=1, so when you evaluate f(x1) or f(x2) or f(x3) or f(x4) you always get 1. Hope this helped

[@vasantk6663]

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?

[@ohboyd]

You can use an arbitrary power such as x^7 but this now means that the family of functions for which the method is exact is f(x) = ax^7 + bx^2 + cx + d. Using powers of x other than 0, 1, 2 or 7 would result in poor approximations, or results which are no longer exact.

[@raquelpicado7878]

How can we predict the error when using Gaussian Quadrature?

[@MathTheBeautiful]

Estimate the difference between the integrand and the (2n-1)-degree interpolating polynomial.

[@raquelpicado7878]

thank you for your help

[@subramaniannk3650]

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?

[@JJC007]

think of 2 vectors, if they form an angle which is not 0 or 180 deg, then they are linearly independent. Orthogonality dictates that the angle they form cannot be arbitrary, it has to be 90 deg exactly.

[@Fsogge]

Great video! Thank you so much.

[@christossofianos8540]

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!

[@MathTheBeautiful]

Yes, the theory of upper bounds on errors is very well developed. You should be able to find it in most books on numerical integration!

[@zahirjan9935]

Thanks from your great service

[@MathTheBeautiful]

Thanks, that means a lot!

[@snnwstt]

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

[@snnwstt]

Oh, 11:50 answers exactly to that question.

[@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?

[@rookiecookie8258]

Because every polynomial can be written as linear combination of 1,x,x^2,x^3 and so on. If each of these terms integrates exactly then f that is a polynomial integrates exactly

[@丁涵-k3q]

super fabulous!

[@atiqueahmed8201]

Simply amazing ❤❤

[@MathTheBeautiful]

So glad you liked it!

[@ghandricheahcene9645]

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

[@Merthalophor]

You use a so called pullback function: If your polynomial p is defined on [a, b], then define a linear function g [-1, 1] -> [a, b] and pop that into your polynomial, so you get p' = p(g(x)), which is now defined on [-1, 1] (it's "squeezed" together from [a, b] to [-1, 1]). Since g is linear, p' has the same degree as p, and thus you can apply your quadrature formula on p' to optain an approximation of p on [a, b]. p doesn't even have to be a polynomial, but that's a bit harder to show .

[@AnkitKumar-ev9nf]

can someone help me with this like why he wrote 1 1 1 1in the first row?

[@MathTheBeautiful]

Hi Ankit, it is explained in the following video: kzbin.info/www/bejne/eGPaoo2Bj9tmhZI