Christoffel numbers

Christoffel coefficients

The coefficients $ \lambda _ {k} $ of a quadrature formula

$$ \int\limits _ { a } ^ { b } f ( x) d \alpha ( x) \approx \ \sum _ {k = 1 } ^ { n } \lambda _ {k} f ( x _ {k} ), $$

which is exact for algebraic polynomials of degrees $ \leq 2n - 1 $. The interpolation nodes $ x _ {k} $ of such a formula are the zeros of a polynomial $ p _ {n} ( x) $ of degree $ n $ which is orthogonal on $ [ a, b] $ relative to the distribution $ d \alpha ( x) $ to all polynomials of degree $ n - 1 $; if $ x _ {1} < \dots < x _ {n} $, the Christoffel numbers are uniquely determined. One has $ \lambda _ {k} > 0 $, $ \sum _ {k = 1 } ^ {n} \lambda _ {k} = \alpha ( b) - \alpha ( a) $ and

$$ \lambda _ {k} = \ \int\limits _ { a } ^ { b } \left [ \frac{p _ {n} ( x) }{p _ {n} ^ \prime ( x) ( x - x _ {k} ) } \right ] ^ {2} d \alpha ( x),\ \ k = 1 \dots n. $$

If the polynomials $ p _ {n} ( x) $ are orthonormal, the Christoffel numbers may be expressed as

$$ \lambda _ {k} ^ {-1} = \ p _ {0} ( x _ {k} ) + \dots + p _ {n} ( x _ {k} ),\ \ k = 1 \dots n, $$

$$ \lambda _ {k} = - \frac{K _ {n + 1 } }{K _ {n} } \frac{1}{p _ {n + 1 } ( x _ {k} ) p _ {n} ^ \prime ( x _ {k} ) } = $$

$$ = \ \frac{K _ {n} }{K _ {n - 1 } } \frac{1}{p _ {n - 1 } ( x _ {k} ) p _ {n} ^ \prime ( x _ {k} ) } ,\ k = 1 \dots n, $$

where $ K _ {n} $ is the leading coefficient of $ p _ {n} ( x) $. In the case $ a = - 1 $, $ b = 1 $ and $ d \alpha ( x) = dx $, the $ p _ {n} ( x) $ are the Legendre polynomials, and

$$ \lambda _ {k} = \ \frac{2}{( 1 - x _ {k} ^ {2} ) [ p _ {n} ^ \prime ( x _ {k} )] ^ {2} } . $$

These expressions are due to E.B. Christoffel [1]. For $ n = 1 \dots 7 $ they were evaluated by C.F. Gauss. See also Gauss quadrature formula.

References

[1]	E.B. Christoffel, "Ueber die Gaussche Quadratur und eine Verallgemeinerung derselben" J. Reine Angew. Math. , 55 (1858) pp. 61–82
[2]	G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)
[3]	I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian)

References

[a1]	F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)