Gaussian quadrature

[hobz]

I have used this way of approximating the integral of a function. It is pretty straight forward as it incorporates the sum of a product of the function to be integrated and a weight function.

 

The weight function is a Legendre polynomial which is a solution to Legendre's differential equation.

 

My problem is I do no understand how these solutions are connected in approximating the integral of a function?

Anyone can use Gaussian quadrature once the solutions to the L. polynomials have been calculated, but understanding how Gauss saw a connection from a differential equation to a (very nice way) of approximating an integral is not apparent.

 

I took an introduction course in numerical analysis, and asked my teacher about this, to which he replied: "It is not worth the time to examine how the connection is established".

 

Can anyone here shed some light on this matter?

 

Thanks!

[Dr. Jekyll]

I took an introduction course in numerical analysis, and asked my teacher about this, to which he replied: "It is not worth the time to examine how the connection is established".

 

It probably was me!

 

More seriously. I dont have this fresh in mind, but Gauss quadrature is merely the integral of a polynomial that is fitted to the given function points (the function you wish to integrate). I.e., if you have two points you do a linear approximation. Three points you do a quadratic approximation (interpolation), etc.

 

The weights, and basis functions, falls out in that way. You could probably derive the same formula by yourself with the above knowledge, and a load of stamina.

 

More rigorous, they are given by setting up a linear system of equations, but not actually very interesting (since I don't have the details in my mind heheh), and I dont have the interest to refresh them either . More important is to know what the actual quadrature do, i.e., approximates the given integrand with a polynomial and computes the integral value.

 

Maybe I just stated what you already knew, cheers anyway!

Edited July 25, 2008 by Dr. Jekyll

[hobz]

Cheers mate!

 

I dont have this fresh in mind, but Gauss quadrature is merely the integral of a polynomial that is fitted to the given function points

Me neither, but doens't it strike you as amazingly simple and easy to use the G-Q?

 

[...] but not actually very interesting (since I don't have the details in my mind heheh), and I dont have the interest to refresh them either .

Sounds like it is really NOT worth the effort!

 

As a certain greek states (as well as people who know me): All I know is that I do not know anything!

No arrogance intended. The insight required to understand G-Q is sheer genius!

[Bignose]

You might want to get your hands on a copy of J.P. Boyd's book Chebyshev and Fourier Spectral Methods -- I found the second edition from Dover an exceptionally good book with plenty of theory (and my memory may not be 100% correct, but I am pretty sure that a pretty detailed derivation of the quadrature points is presented) and plenty of applications. The best thing is that the author's style made the book very readable to me. Very highly recommended -- and since it is a Dover book, the price really cannot be beat should you choose to own your own copy of it.

[hobz]

Thanks for the tip!

I can see the price is $27.70 which hardly scares anyone.

[Dr. Jekyll]

Here is a link to a describing .doc-file: http://numericalmethods.eng.usf.edu/nbm/gen/07int/nbm_gen_int_txt_gaussquadrature.doc

 

They only deal with lower orders, but the main procedure is there. I figure a proof of a general n-order Gauss quadrature becomes quite messy with lotsa algebra.

 

I don't think it was Gauss that invented this formula, it is only named after him. Anyone know who in that case (or have the stamina to Google )?