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The quadrature formula for the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m0633501.png" /> and the weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m0633502.png" /> which gives the highest algebraic degree of accuracy. It has the form
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<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/m/m063/m063350/m0633503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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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
 
The nodes are the roots of the Chebyshev polynomial
  
<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/m/m063/m063350/m0633504.png" /></td> </tr></table>
+
$$
 +
T _ {N} ( x)  = \cos  N  \mathop{\rm arc}  \cos  x;
 +
$$
  
the coefficients are identical and equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m0633505.png" />. The algebraic degree of accuracy equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m0633506.png" />. Formula (1) was established by F.G. Mehler [[#References|[1]]].
+
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 [[#References|[1]]].
  
The quadrature formula of highest algebraic degree of accuracy for the weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m0633507.png" /> and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m0633508.png" /> nodes for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m0633509.png" /> fixed nodes coincide with the nodes of the quadrature formula (1), has the form
+
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
  
<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/m/m063/m063350/m06335010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\int\limits _ { - } 1 ^ { 1 } 
 +
\frac{1}{\sqrt {1- x  ^ {2} } }
 +
f( x) dx
 +
\approx \
 +
$$
  
<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/m/m063/m063350/m06335011.png" /></td> </tr></table>
+
$$
 +
\approx 
 +
\frac \pi {2N}
 +
\left [ f(- 1)+ f(
 +
\frac{1)}{2}
 +
+ \sum _ { k= } 1 ^ { 2N- }  1 f \left ( \cos 
 +
\frac{k \pi }{2N}
 +
\right ) \right ] .
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m06335012.png" /> supplementary nodes only. Formula (2) represents also the quadrature formula of highest algebraic degree of accuracy with the weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m06335013.png" /> for which the fixed nodes are the end points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m06335014.png" />, and hence the other nodes of which are the roots of the orthogonal polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m06335015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m06335016.png" /> with weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m06335017.png" />, i.e. of the Chebyshev polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m06335018.png" /> of the second kind. The algebraic degree of accuracy of the quadrature formula (2) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m06335019.png" />.
+
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.
 
Formula (1) is sometimes referred to as Hermite's quadrature formula.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F.G. Mehler,  "Bemerkungen zur Theorie der mechanischen Quadraturen"  ''J. Reine Angew. Math.'' , '''63'''  (1864)  pp. 152–157</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.M. Krylov,  "Approximate calculation of integrals" , Macmillan  (1962)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F.G. Mehler,  "Bemerkungen zur Theorie der mechanischen Quadraturen"  ''J. Reine Angew. Math.'' , '''63'''  (1864)  pp. 152–157</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.M. Krylov,  "Approximate calculation of integrals" , Macmillan  (1962)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The quadrature formula above is more commonly referred to as Gauss–Chebyshev quadrature (see [[Gauss quadrature formula|Gauss quadrature formula]]). It may be viewed as being based on Hermite (oscillatory) interpolation given the weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m06335020.png" />. Hermite's quadrature formula is a Gauss-type quadrature formula with weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m06335021.png" /> and integration interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063350/m06335022.png" />.
+
The quadrature formula above is more commonly referred to as Gauss–Chebyshev quadrature (see [[Gauss quadrature formula|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====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.J. Davis,  P. Rabinowitz,  "Methods of numerical integration" , Acad. Press  (1975)</TD></TR><TR><TD valign="top">[a2]</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">  P.J. Davis,  P. Rabinowitz,  "Methods of numerical integration" , Acad. Press  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F.B. Hildebrand,  "Introduction to numerical analysis" , McGraw-Hill  (1974)</TD></TR></table>

Revision as of 08:00, 6 June 2020


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)
How to Cite This Entry:
Mehler quadrature formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mehler_quadrature_formula&oldid=47818
This article was adapted from an original article by I.P. Mysovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article