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''Christoffel coefficients''
 
''Christoffel coefficients''
  
The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c0222001.png" /> of a quadrature formula
+
The coefficients $  \lambda _ {k} $
 +
of a quadrature formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c0222002.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { a } ^ { b }
 +
f ( x)  d \alpha ( x)  \approx \
 +
\sum _ {k = 1 } ^ { n }
 +
\lambda _ {k} f ( x _ {k} ),
 +
$$
  
which is exact for algebraic polynomials of degrees <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c0222003.png" />. The interpolation nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c0222004.png" /> of such a formula are the zeros of a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c0222005.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c0222006.png" /> which is orthogonal on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c0222007.png" /> relative to the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c0222008.png" /> to all polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c0222009.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220010.png" />, the Christoffel numbers are uniquely determined. One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220012.png" /> and
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220013.png" /></td> </tr></table>
+
$$
 +
\lambda _ {k}  = \
 +
\int\limits _ { a } ^ { b }
 +
\left [
  
If the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220014.png" /> are orthonormal, the Christoffel numbers may be expressed as
+
\frac{p _ {n} ( x) }{p _ {n}  ^  \prime  ( x) ( x - x _ {k} ) }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220015.png" /></td> </tr></table>
+
\right ]  ^ {2}  d \alpha ( x),\ \
 +
k = 1 \dots n.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220016.png" /></td> </tr></table>
+
If the polynomials  $  p _ {n} ( x) $
 +
are orthonormal, the Christoffel numbers may be expressed as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220017.png" /></td> </tr></table>
+
$$
 +
\lambda _ {k}  ^ {-} 1  = \
 +
p _ {0} ( x _ {k} )
 +
+ \dots + p _ {n} ( x _ {k} ),\ \
 +
k = 1 \dots n,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220018.png" /> is the leading coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220019.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220022.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220023.png" /> are the [[Legendre polynomials|Legendre polynomials]], and
+
$$
 +
\lambda _ {k}  = -
 +
\frac{K _ {n + 1 }  }{K _ {n} }
 +
 +
\frac{1}{p _ {n + 1 }  ( x _ {k} ) p _ {n}  ^  \prime  ( x _ {k} ) }
 +
=
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220024.png" /></td> </tr></table>
+
$$
 +
= \
  
These expressions are due to E.B. Christoffel [[#References|[1]]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022200/c02220025.png" /> they were evaluated by C.F. Gauss. See also [[Gauss quadrature formula|Gauss quadrature formula]].
+
\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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.B. Hildebrand,  "Introduction to numerical analysis" , McGraw-Hill  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.B. Hildebrand,  "Introduction to numerical analysis" , McGraw-Hill  (1974)</TD></TR></table>

Revision as of 16:44, 4 June 2020


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)

Comments

References

[a1] F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)
How to Cite This Entry:
Christoffel numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Christoffel_numbers&oldid=46344
This article was adapted from an original article by N.P. KorneichukV.P. Motornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article