##### Actions

The quadrature formula for the segment $[- 1, 1]$ and the weight $1/ \sqrt {1- x ^ {2} }$ which gives the highest algebraic degree of accuracy. It has the form

$$\tag{1 } \int\limits_{-1}^{1} \frac{1}{\sqrt {1- x ^ {2} } } f( x) dx \approx \frac \pi {N} \sum_{ k=1 } ^ { N } f \left ( \cos \frac{2k-1}{2N} \pi \right ) .$$

The nodes are the roots of the Chebyshev polynomial

$$T _ {N} ( x) = \cos N \mathop{\rm arc} \cos x;$$

the coefficients are identical and equal to $\pi / N$. The algebraic degree of accuracy equals $2N- 1$. Formula (1) was established by F.G. Mehler .

The quadrature formula of highest algebraic degree of accuracy for the weight $1/ \sqrt {1- x ^ {2} }$ and with $2N+ 1$ nodes for which $N$ fixed nodes coincide with the nodes of the quadrature formula (1), has the form

$$\tag{2 } \int\limits_{-1}^{1} \frac{1}{\sqrt {1- x ^ {2} } } f( x) dx \approx \$$

$$\approx \frac \pi {2N} \left [{\frac{f(-1)+ f(1)}{2} + \sum _ { k=1}^{ 2N-1 } f \left ({\cos\frac{k \pi }{2N} }\right ) }\right ] .$$

Formula (2) is used to improve the accuracy of the approximate value of the integral obtained by means of formula (1); since the values of the integrand at the nodes of formula (1) have already been computed, it is necessary to compute its values at $N+ 1$ supplementary nodes only. Formula (2) represents also the quadrature formula of highest algebraic degree of accuracy with the weight $1/ \sqrt {1- x ^ {2} }$ for which the fixed nodes are the end points of $[- 1, 1]$, and hence the other nodes of which are the roots of the orthogonal polynomial of degree $2N- 1$ on $[- 1, 1]$ with weight $\sqrt {1- x ^ {2} }$, i.e. of the Chebyshev polynomial $U _ {2N-1}( x)$ of the second kind. The algebraic degree of accuracy of the quadrature formula (2) is $4N- 1$.

Formula (1) is sometimes referred to as Hermite's quadrature formula.

How to Cite This Entry: