Hello,

I'm attempting to modify a standard Gauss-Legendre quadrature set to include fixed nodes that I specify.

A quick description of my problem: Construct an $\displaystyle n$-point quadrature ($\displaystyle n$ is even) with 2 symmetric fixed nodes so that the degree of exactness $\displaystyle d$ is as high as possible. The quadrature should look like

$\displaystyle \int_{-1}^1 f(x) dx = w_f f(x_f) + w_f f(-x_f) + \sum_{i=1}^{n-2} w_i f(x_i) + E_n(f)

$

$\displaystyle f(x)$ is the function to integrate, $\displaystyle E_n(f)$ is the error, $\displaystyle x_f$ is the fixed node with weight $\displaystyle w_f$, and $\displaystyle \{x_i, w_i\}_{i=1}^{n-2}$ is the set of free nodes. The degree of exactness is defined as

$\displaystyle E_n(f) = 0 \quad \forall f \in \mathbb{P}_d$

where $\displaystyle \mathbb{P}_d$ is the space of polynomials of degree less than or equal to $\displaystyle d$.

Initially, I thought with 2 constraints and $\displaystyle 2n-2$ unknowns, the highest degree of exactness is $\displaystyle d_{max} = 2n-3$ (since there are $\displaystyle d+1$ equations).

Unfortunately, it seems that for an arbitrary positioning of the fixed nodes, the solution of those equations can give nodes outside the interval of integration (or even imaginary solutions). I'm guessing that if I'm willing to give up another one or two (or more) degrees of exactness, I could find a quadrature set which includes the constrained nodes, but I haven't figured out how to do this.

I've read a number of papers on related problems, but none of them address arbitrary interior fixed nodes.

Thanks for any help you can offer.