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A formula which is defined as half the sum of the Gauss formula (cf. [[Gauss interpolation formula|Gauss interpolation formula]]) for forward interpolation on the nodes
 
A formula which is defined as half the sum of the Gauss formula (cf. [[Gauss interpolation formula|Gauss interpolation formula]]) for forward interpolation on the nodes
  
<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/b/b015/b015860/b0158601.png" /></td> </tr></table>
+
$$
 +
x _ {0} ,\
 +
x _ {0} + h,\
 +
x _ {0} - h \dots x _ {0} + nh,\
 +
x _ {0} - nh,\
 +
x _ {0} + (n + 1) h ,
 +
$$
 +
 
 +
at the point  $  x = x _ {0} + th $:
  
at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015860/b0158602.png" />:
+
$$ \tag{1 }
 +
G _ {2n + 2 }  (x _ {0} + th)  = \
 +
f _ {0} + f _ {1/2}  ^ {1}
 +
t + f _ {0}  ^ {2}
  
<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/b/b015/b015860/b0158603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\frac{t (t - 1) }{2!}
 +
+ \dots +
 +
$$
  
<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/b/b015/b015860/b0158604.png" /></td> </tr></table>
+
$$
 +
+
 +
f _ {1/2} ^ {2n + 1 }
 +
\frac{t (t  ^ {2} - 1) \dots (t  ^ {2} - n  ^ {2} ) }{(2n + 1)! }
 +
,
 +
$$
  
and the Gauss formula of the same order for backward interpolation with respect to the node <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015860/b0158605.png" />, i.e. with respect to the population of nodes
+
and the Gauss formula of the same order for backward interpolation with respect to the node $  x _ {1} = x _ {0} + h $,  
 +
i.e. with respect to the population of nodes
  
<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/b/b015/b015860/b0158606.png" /></td> </tr></table>
+
$$
 +
x _ {0} + h, x _ {0} ,\
 +
x _ {0} + 2h,\
 +
x _ {0} - h \dots x _ {0} + (n + 1) h,\
 +
x _ {0} - nh:
 +
$$
  
<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/b/b015/b015860/b0158607.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
G _ {2n + 2 }  (x _ {0} + th)  = f _ {1} + f _ {1/2}  ^ {1} (t - 1) + f _ {1}  ^ {2}
 +
\frac{t (t - 1) }{2!}
 +
+ \dots +
 +
$$
  
<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/b/b015/b015860/b0158608.png" /></td> </tr></table>
+
$$
 +
+
 +
f _ {1/2} ^ {2n + 1 }
 +
\frac{t (t  ^ {2} - 1) \dots [t  ^ {2}
 +
- (n - 1)  ^ {2} ] (t - n) (t - n - 1) }{(2n + 1)! }
 +
.
 +
$$
  
 
Putting
 
Putting
  
<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/b/b015/b015860/b0158609.png" /></td> </tr></table>
+
$$
 +
f _ {1/2}  ^ {2k}  = \
 +
 
 +
\frac{(f _ {0}  ^ {2k} + f _ {1}  ^ {2k} ) }{2}
 +
,
 +
$$
  
 
Bessel's interpolation formula assumes the form ([[#References|[1]]], [[#References|[2]]]):
 
Bessel's interpolation formula assumes the form ([[#References|[1]]], [[#References|[2]]]):
  
<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/b/b015/b015860/b01586010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
B _ {2n + 2 }  (x _ {0} + th) =
 +
$$
 +
 
 +
$$
 +
= \
 +
f _ {1/2} + f _ {1/2}  ^ {1} \left ( t - {
 +
\frac{1}{2}
 +
}
 +
\right ) + f _ {1/2}  ^ {2}
 +
\frac{t (t - 1) }{2!}
 +
+ \dots +
 +
$$
  
<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/b/b015/b015860/b01586011.png" /></td> </tr></table>
+
$$
 +
+
 +
f _ {1/2}  ^ {2n}
 +
\frac{t (t  ^ {2} - 1) \dots [t  ^ {2} - (n - 1)  ^ {2} ] (t - n) }{(2n)!}
 +
+
 +
$$
  
<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/b/b015/b015860/b01586012.png" /></td> </tr></table>
+
$$
 +
+
 +
f _ {1/2} ^ {2n + 1 }
 +
\frac{t (t  ^ {2} - 1) \dots
 +
[t  ^ {2} - (n - 1)  ^ {2} ] (t - n) (t - 1/2) }{(2n + 1)! }
 +
.
 +
$$
  
<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/b/b015/b015860/b01586013.png" /></td> </tr></table>
+
Bessel's interpolation formula has certain advantages over Gauss' formulas (1), (2); in particular, if the interpolation is at the middle of the segment, i.e. at  $  t = 1/2 $,
 +
all coefficients at the differences of odd orders vanish. If the last term on the right-hand side of (3) is omitted, the polynomial  $  B _ {2n + 1 }  (x _ {0} + th) $,
 +
which is not a proper interpolation polynomial (it coincides with  $  f(x) $
 +
only in the  $  2n $
 +
nodes  $  x _ {0} - (n - 1)h \dots x _ {0} + nh $),
 +
represents a better estimate of the residual term (cf. [[Interpolation formula|Interpolation formula]]) than the interpolation polynomial of the same degree. Thus, for instance, if  $  x = x _ {0} + th \in (x _ {0} , x _ {1} ) $,
 +
the estimate of the last term using the polynomial which is most frequently employed
  
Bessel's interpolation formula has certain advantages over Gauss' formulas (1), (2); in particular, if the interpolation is at the middle of the segment, i.e. at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015860/b01586014.png" />, all coefficients at the differences of odd orders vanish. If the last term on the right-hand side of (3) is omitted, the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015860/b01586015.png" />, which is not a proper interpolation polynomial (it coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015860/b01586016.png" /> only in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015860/b01586017.png" /> nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015860/b01586018.png" />), represents a better estimate of the residual term (cf. [[Interpolation formula|Interpolation formula]]) than the interpolation polynomial of the same degree. Thus, for instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015860/b01586019.png" />, the estimate of the last term using the polynomial which is most frequently employed
+
$$
 +
B _ {3} (x _ {0} + th) = \
 +
f _ {1/2} + f _ {1/2}  ^ {1}
 +
\left ( t - {
 +
\frac{1}{2}
 +
} \right ) +
 +
f _ {1/2}  ^ {2}
  
<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/b/b015/b015860/b01586020.png" /></td> </tr></table>
+
\frac{t (t - 1) }{2}
 +
,
 +
$$
  
written with respect to the nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015860/b01586021.png" />, is almost 8 times better than that of the interpolation polynomial written with respect to the nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015860/b01586022.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015860/b01586023.png" /> ([[#References|[2]]]).
+
written with respect to the nodes $  x _ {0} - h, x _ {0} , x _ {0} + h, x _ {0} + 2h $,
 +
is almost 8 times better than that of the interpolation polynomial written with respect to the nodes $  x _ {0} - h, x _ {0} , x _ {0} + h $
 +
or $  x _ {0} , x _ {0} + h, x _ {0} + 2h $([[#References|[2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.S. Berezin,  N.P. Zhidkov,  "Computing methods" , '''1''' , Pergamon  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Bakhvalov,  "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.S. Berezin,  N.P. Zhidkov,  "Computing methods" , '''1''' , Pergamon  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Bakhvalov,  "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Abramowitz,  I.A. Stegun,  "Handbook of mathematical functions" , Dover, reprint  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F.B. Hildebrand,  "Introduction to numerical analysis" , Addison-Wesley  (1956)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Abramowitz,  I.A. Stegun,  "Handbook of mathematical functions" , Dover, reprint  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F.B. Hildebrand,  "Introduction to numerical analysis" , Addison-Wesley  (1956)</TD></TR></table>

Latest revision as of 10:58, 29 May 2020


A formula which is defined as half the sum of the Gauss formula (cf. Gauss interpolation formula) for forward interpolation on the nodes

$$ x _ {0} ,\ x _ {0} + h,\ x _ {0} - h \dots x _ {0} + nh,\ x _ {0} - nh,\ x _ {0} + (n + 1) h , $$

at the point $ x = x _ {0} + th $:

$$ \tag{1 } G _ {2n + 2 } (x _ {0} + th) = \ f _ {0} + f _ {1/2} ^ {1} t + f _ {0} ^ {2} \frac{t (t - 1) }{2!} + \dots + $$

$$ + f _ {1/2} ^ {2n + 1 } \frac{t (t ^ {2} - 1) \dots (t ^ {2} - n ^ {2} ) }{(2n + 1)! } , $$

and the Gauss formula of the same order for backward interpolation with respect to the node $ x _ {1} = x _ {0} + h $, i.e. with respect to the population of nodes

$$ x _ {0} + h, x _ {0} ,\ x _ {0} + 2h,\ x _ {0} - h \dots x _ {0} + (n + 1) h,\ x _ {0} - nh: $$

$$ \tag{2 } G _ {2n + 2 } (x _ {0} + th) = f _ {1} + f _ {1/2} ^ {1} (t - 1) + f _ {1} ^ {2} \frac{t (t - 1) }{2!} + \dots + $$

$$ + f _ {1/2} ^ {2n + 1 } \frac{t (t ^ {2} - 1) \dots [t ^ {2} - (n - 1) ^ {2} ] (t - n) (t - n - 1) }{(2n + 1)! } . $$

Putting

$$ f _ {1/2} ^ {2k} = \ \frac{(f _ {0} ^ {2k} + f _ {1} ^ {2k} ) }{2} , $$

Bessel's interpolation formula assumes the form ([1], [2]):

$$ \tag{3 } B _ {2n + 2 } (x _ {0} + th) = $$

$$ = \ f _ {1/2} + f _ {1/2} ^ {1} \left ( t - { \frac{1}{2} } \right ) + f _ {1/2} ^ {2} \frac{t (t - 1) }{2!} + \dots + $$

$$ + f _ {1/2} ^ {2n} \frac{t (t ^ {2} - 1) \dots [t ^ {2} - (n - 1) ^ {2} ] (t - n) }{(2n)!} + $$

$$ + f _ {1/2} ^ {2n + 1 } \frac{t (t ^ {2} - 1) \dots [t ^ {2} - (n - 1) ^ {2} ] (t - n) (t - 1/2) }{(2n + 1)! } . $$

Bessel's interpolation formula has certain advantages over Gauss' formulas (1), (2); in particular, if the interpolation is at the middle of the segment, i.e. at $ t = 1/2 $, all coefficients at the differences of odd orders vanish. If the last term on the right-hand side of (3) is omitted, the polynomial $ B _ {2n + 1 } (x _ {0} + th) $, which is not a proper interpolation polynomial (it coincides with $ f(x) $ only in the $ 2n $ nodes $ x _ {0} - (n - 1)h \dots x _ {0} + nh $), represents a better estimate of the residual term (cf. Interpolation formula) than the interpolation polynomial of the same degree. Thus, for instance, if $ x = x _ {0} + th \in (x _ {0} , x _ {1} ) $, the estimate of the last term using the polynomial which is most frequently employed

$$ B _ {3} (x _ {0} + th) = \ f _ {1/2} + f _ {1/2} ^ {1} \left ( t - { \frac{1}{2} } \right ) + f _ {1/2} ^ {2} \frac{t (t - 1) }{2} , $$

written with respect to the nodes $ x _ {0} - h, x _ {0} , x _ {0} + h, x _ {0} + 2h $, is almost 8 times better than that of the interpolation polynomial written with respect to the nodes $ x _ {0} - h, x _ {0} , x _ {0} + h $ or $ x _ {0} , x _ {0} + h, x _ {0} + 2h $([2]).

References

[1] I.S. Berezin, N.P. Zhidkov, "Computing methods" , 1 , Pergamon (1973) (Translated from Russian)
[2] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)

Comments

References

[a1] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1970)
[a2] F.B. Hildebrand, "Introduction to numerical analysis" , Addison-Wesley (1956)
How to Cite This Entry:
Bessel interpolation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bessel_interpolation_formula&oldid=46034
This article was adapted from an original article by M.K. Samarin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article