# 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)