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''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|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 [[#References|[1]]]. For  $  n = 1 \dots 7 $
they were evaluated by C.F. Gauss. See also [[Gauss quadrature formula|Gauss quadrature formula]].

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

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.B. Hildebrand,   "Introduction to numerical analysis" , McGraw-Hill  (1974)</TD></TR></table>