Why Is My Gaussian Quadrature Implementation Inaccurate?

[valis]

I'm trying to make a generalized quadrature method and I seem to be running into some bizarre errors. For n=2 my answer is twice what it should be and for n greater the innaccuracy increases (answer/n is close but worse than answer/2 with n=2). My general algorithm is:
p = nth legendre polynomial
$g = x \rightarrow f\left(\frac{b-a}{2}x+\frac{b+a}{2}\right)$ where f is the function to integrate
xs = n roots of p
ws = weights based on value of n (2 => +/-0.57735026918963)
$\frac{b-a}{2}\sum^n_0{w_ig(x_i)}$

I've checked my generation of legendre functions and they're consistently correct. I also generated a list of roots and verified some against wikipedia's list. Have I stupidly mangled the actual gaussian quadrature algorithm?

I'm writing this in haskell so my code is 1:1 with the pseduo-code above.

Thanks ahead for the help.

Sorry if this is in the wrong forum, it's not coursework, but it is academic.

 

[mathman]

I suggest you try it with f(x)=1 and see what the weights are.

 

[tacman]

It sounds like you may have made a mistake in your implementation of the Gaussian quadrature algorithm. It's always a good idea to double check your code and make sure it matches the pseudo-code you are following. In particular, you may want to check the calculation of the weights, as those seem to be the source of your error.

Another possibility is that your function f is not well-behaved enough for the Gaussian quadrature method to work accurately. This method is best suited for integrating smooth, continuous functions. If your function has discontinuities or rapidly changing behavior, it may not give accurate results.

I would suggest trying to debug your code and also experimenting with different functions to see if the issue persists. Good luck!