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A form of writing the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h0470001.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h0470002.png" /> that solves the problem of interpolating a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h0470003.png" /> and its derivatives at points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h0470004.png" />, that is, satisfying the conditions
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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/h/h047/h047000/h0470005.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|done}}
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A form of writing the polynomial  $  H _ {m} $
 +
of degree  $  m $
 +
that solves the problem of interpolating a function  $  f $
 +
and its derivatives at points  $  x _ {0} \dots x _ {n} $,
 +
that is, satisfying the conditions
 +
 
 +
$$ \tag{1 }
 +
\left .
 +
\begin{array}{c}
 +
 
 +
{H _ {m} ( x _ {0} )  = f( x _ {0} ) \dots H _ {m} ^ {( \alpha _ {0} - 1) } ( x _ {0} )  = f ^ { ( \alpha _ {0} - 1) } ( x _ {0} ) , }
 +
\\
 +
 
 +
{\dots \dots \dots \dots \dots }
 +
\\
 +
 
 +
{H _ {m} ( x _ {n} )  = f ( x _ {n} ) \dots H _ {m} ^ {( \alpha _ {n} - 1 ) } ( x _ {n} )  = f ^ { ( \alpha _ {n} - 1 ) } ( x _ {n} ),
 +
}
 +
\\
 +
{m  = \sum _ { i= 0} ^ { n }  \alpha _ {i} - 1 . }
 +
\end{array}
 +
\right \}
 +
$$
  
 
The Hermite interpolation formula can be written in the form
 
The Hermite interpolation formula can be written in 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/h/h047/h047000/h0470006.png" /></td> </tr></table>
+
$$
 +
H _ {m} ( x)  = \sum _ { i=0} ^ { n }  \sum _ { j=0 }^ { {\alpha _ i} - 1 } \
 +
\sum _ { k=0} ^ { {\alpha _ i} - j - 1 } f ^ { ( j) } ( x _ {i} )
 +
 
 +
\frac{1}{k!}
 +
 +
\frac{1}{j!}
 +
\left [
  
<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/h/h047/h047000/h0470007.png" /></td> </tr></table>
+
\frac{( x - x _ {i} ) ^ {\alpha _ {i} } }{\Omega ( x) }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h0470008.png" />.
+
\right ] _ {x = x _ {i}  }  ^ {( k)} \times
  
====References====
+
\frac{\Omega ( x) }{( x - x _ {i} ) ^ {\alpha _ {i} - j - k } }
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.S. Berezin,  N.P. Zhidkov,  "Computing methods" , Pergamon  (1973) (Translated from Russian)</TD></TR></table>
+
,
 +
$$
  
 +
where  $  \Omega ( x) = ( x - x _ {0} ) ^ {\alpha _ {0} } \dots ( x - x _ {n} ) ^ {\alpha _ {n} } $.
  
 +
====References====
 +
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  I.S. Berezin,  N.P. Zhidkov,  "Computing methods" , Pergamon  (1973)  (Translated from Russian)</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====
Hermite interpolation can be regarded as a special case of Birkhoff interpolation (also called lacunary interpolation). In the latter, not all values of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h0470009.png" /> and its derivatives are known at given points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700010.png" /> (whereas there is complete information in the case of Hermite interpolation). Data such as (1) naturally give rise to a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700011.png" />, a so-called interpolation matrix, constructed as follows. Write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700012.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700014.png" />. Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700015.png" /> if the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700016.png" /> is known (given) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700017.png" /> if it is not (for Hermite interpolation all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700018.png" />). Now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700019.png" />.
+
Hermite interpolation can be regarded as a special case of Birkhoff interpolation (also called lacunary interpolation). In the latter, not all values of a function $  f $
 +
and its derivatives are known at given points $  x _ {0} < \dots < x _ {n} $(
 +
whereas there is complete information in the case of Hermite interpolation). Data such as (1) naturally give rise to a matrix $  E $,  
 +
a so-called interpolation matrix, constructed as follows. Write $  f ^ { ( k) } ( x _ {i} ) = c _ {i,k} $
 +
for $  k = k ( i) = 0 \dots \alpha _ {i} - 1 $
 +
and $  i = 0 \dots n $.  
 +
Put $  e _ {i,k} = 1 $
 +
if the constant $  c _ {i,k} $
 +
is known (given) and $  e _ {i,k} = 0 $
 +
if it is not (for Hermite interpolation all $  e _ {i,k} = 1 $).  
 +
Now $  E = ( e _ {i,k} ) _ {i,k} $.
  
Such a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700020.png" /> is called order regular if it is associated to a solvable problem (i.e. (1) is solvable for all choices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700021.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700022.png" />). (Similarly, if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700023.png" /> of interpolation points may vary over a given class, a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700024.png" /> is called regular if (1) is solvable for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700025.png" /> in this class and all choices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700026.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700027.png" />.) A basic theme in Birkhoff interpolation is to find the regular pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047000/h04700028.png" />. More information can be found in [[#References|[a1]]].
+
Such a matrix $  E $
 +
is called order regular if it is associated to a solvable problem (i.e. (1) is solvable for all choices of $  c _ {i,k} $
 +
for which $  e _ {i,k} = 1 $).  
 +
(Similarly, if the set $  X $
 +
of interpolation points may vary over a given class, a pair $  E , X $
 +
is called regular if (1) is solvable for all $  X $
 +
in this class and all choices of $  c _ {i,k} $
 +
for which $  e _ {i,k} = 1 $.)  
 +
A basic theme in Birkhoff interpolation is to find the regular pairs $  E , X $.  
 +
More information can be found in [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.G. Lorentz,  K. Jetter,  S.D. Riemenschneider,  "Birkhoff interpolation" , Addison-Wesley  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.P. Mysovskih,  "Lectures on numerical methods" , Wolters-Noordhoff  (1969)  pp. Chapt. 2, Sect. 10</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Wendroff,  "Theoretical numerical analysis" , Acad. Press  (1966)  pp. Chapt. 1</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.G. Lorentz,  K. Jetter,  S.D. Riemenschneider,  "Birkhoff interpolation" , Addison-Wesley  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.P. Mysovskih,  "Lectures on numerical methods" , Wolters-Noordhoff  (1969)  pp. Chapt. 2, Sect. 10</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Wendroff,  "Theoretical numerical analysis" , Acad. Press  (1966)  pp. Chapt. 1</TD></TR></table>

Latest revision as of 20:12, 10 January 2024


A form of writing the polynomial $ H _ {m} $ of degree $ m $ that solves the problem of interpolating a function $ f $ and its derivatives at points $ x _ {0} \dots x _ {n} $, that is, satisfying the conditions

$$ \tag{1 } \left . \begin{array}{c} {H _ {m} ( x _ {0} ) = f( x _ {0} ) \dots H _ {m} ^ {( \alpha _ {0} - 1) } ( x _ {0} ) = f ^ { ( \alpha _ {0} - 1) } ( x _ {0} ) , } \\ {\dots \dots \dots \dots \dots } \\ {H _ {m} ( x _ {n} ) = f ( x _ {n} ) \dots H _ {m} ^ {( \alpha _ {n} - 1 ) } ( x _ {n} ) = f ^ { ( \alpha _ {n} - 1 ) } ( x _ {n} ), } \\ {m = \sum _ { i= 0} ^ { n } \alpha _ {i} - 1 . } \end{array} \right \} $$

The Hermite interpolation formula can be written in the form

$$ H _ {m} ( x) = \sum _ { i=0} ^ { n } \sum _ { j=0 }^ { {\alpha _ i} - 1 } \ \sum _ { k=0} ^ { {\alpha _ i} - j - 1 } f ^ { ( j) } ( x _ {i} ) \frac{1}{k!} \frac{1}{j!} \left [ \frac{( x - x _ {i} ) ^ {\alpha _ {i} } }{\Omega ( x) } \right ] _ {x = x _ {i} } ^ {( k)} \times \frac{\Omega ( x) }{( x - x _ {i} ) ^ {\alpha _ {i} - j - k } } , $$

where $ \Omega ( x) = ( x - x _ {0} ) ^ {\alpha _ {0} } \dots ( x - x _ {n} ) ^ {\alpha _ {n} } $.

References

[1] I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)

Comments

Hermite interpolation can be regarded as a special case of Birkhoff interpolation (also called lacunary interpolation). In the latter, not all values of a function $ f $ and its derivatives are known at given points $ x _ {0} < \dots < x _ {n} $( whereas there is complete information in the case of Hermite interpolation). Data such as (1) naturally give rise to a matrix $ E $, a so-called interpolation matrix, constructed as follows. Write $ f ^ { ( k) } ( x _ {i} ) = c _ {i,k} $ for $ k = k ( i) = 0 \dots \alpha _ {i} - 1 $ and $ i = 0 \dots n $. Put $ e _ {i,k} = 1 $ if the constant $ c _ {i,k} $ is known (given) and $ e _ {i,k} = 0 $ if it is not (for Hermite interpolation all $ e _ {i,k} = 1 $). Now $ E = ( e _ {i,k} ) _ {i,k} $.

Such a matrix $ E $ is called order regular if it is associated to a solvable problem (i.e. (1) is solvable for all choices of $ c _ {i,k} $ for which $ e _ {i,k} = 1 $). (Similarly, if the set $ X $ of interpolation points may vary over a given class, a pair $ E , X $ is called regular if (1) is solvable for all $ X $ in this class and all choices of $ c _ {i,k} $ for which $ e _ {i,k} = 1 $.) A basic theme in Birkhoff interpolation is to find the regular pairs $ E , X $. More information can be found in [a1].

References

[a1] G.G. Lorentz, K. Jetter, S.D. Riemenschneider, "Birkhoff interpolation" , Addison-Wesley (1983)
[a2] I.P. Mysovskih, "Lectures on numerical methods" , Wolters-Noordhoff (1969) pp. Chapt. 2, Sect. 10
[a3] B. Wendroff, "Theoretical numerical analysis" , Acad. Press (1966) pp. Chapt. 1
How to Cite This Entry:
Hermite interpolation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_interpolation_formula&oldid=13280
This article was adapted from an original article by M.K. Samarin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article