What Are the Error Approximations in Gaussian Integration?

[Mr.Brown]

Hi,

Im learning about gaussian integration right now but i can't really find anything about
the error approximation other than that it goes proportional to the 2nth derivative of the function at some point in the interval I am trying to integration the function over.
But that doesn't really help me since i don't know at which point the derivative is to be evaluated.

Also i don't learn anything about functions who don't have that derivative at any point in the interval like sqrt(x) in the interval (0,1) for example.

So if anyone knows any good resources about the subject id def appreciate if someone could help tell me :)

Thanks

 

[jasonRF]

Hi,

Im learning about gaussian integration right now but i can't really find anything about
the error approximation other than that it goes proportional to the 2nth derivative of the function at some point in the interval I am trying to integration the function over.
But that doesn't really help me since i don't know at which point the derivative is to be evaluated.

Yes, that is the way it goes. In order to exactly know the error you need to exactly know the value of the integral - if you knew that you wouldn't be doing it numerically!

EDIT: I was being a little flip about this. The error estimates can be useful for a couple of reasons. First, if you can bound your 2nth derivative, then the error formula allows you to bound the error. Second, such formulas allow you to compare different numerical approaches, although an integration formula with a better looking error term isn't necessarily better for a given function, but having the error terms are better than nothing.

Also i don't learn anything about functions who don't have that derivative at any point in the interval like sqrt(x) in the interval (0,1) for example.

So if anyone knows any good resources about the subject id def appreciate if someone could help tell me :)

Thanks

That is a great question. Unfortunately, I don't think there is a general way to deal with such "singularities." Sometimes a change of variable can get rid of it (try u = sqrt(x) ...). In a few special cases there may be nice formulas already - some of hte Gaussian integration formulas have singular kernels. Sometimes you may have to break up the integral into two pieces. For your example of an integral from 0 to 1 with a singularity at 0, make one integral from 0 to eps, and one from eps to 1. The integral from eps to 1 will be well behaved as long as eps doesn't get too close to zero, and your standard numerical approximations are happy. The integral from 0 to eps still has to be dealt with - perhaps over a small enough interval an adequate approximation to the integrand can be made that can be analytically integrated. I'm sure there are other approaches as well. The applicability of all of these depends on the specifics of the problem. So the case with singularities gives you a chance to be creative!

good luck,

jason