Mehler quadrature formula

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

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

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 [1].

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

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

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

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.

References

[1]	F.G. Mehler, "Bemerkungen zur Theorie der mechanischen Quadraturen" J. Reine Angew. Math. , 63 (1864) pp. 152–157
[2]	N.M. Krylov, "Approximate calculation of integrals" , Macmillan (1962) (Translated from Russian)

Comments

The quadrature formula above is more commonly referred to as Gauss–Chebyshev quadrature (see Gauss quadrature formula). It may be viewed as being based on Hermite (oscillatory) interpolation given the weight function $ 1 / \sqrt {1 - x ^ {2} } $. Hermite's quadrature formula is a Gauss-type quadrature formula with weight $ e ^ {- x ^ {2} } $ and integration interval $ ( - \infty , \infty ) $.

References

[a1]	P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1975)
[a2]	F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)