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Difference between revisions of "Interpolation spline"

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S _ {m} ( \Delta _ {n} ;  x )  = \  
 
S _ {m} ( \Delta _ {n} ;  x )  = \  
 
a _ {0} + a _ {1} x + \dots
 
a _ {0} + a _ {1} x + \dots
+ a _ {m-} 1 x  ^ {m-} 1 +
+
+ a _ {m-1} x  ^ {m-1} +
\sum _ { k= } 0 ^ { n- } 1
+
\sum _ { k= 0} ^ { n-1 }
 
C _ {k} ( x - x _ {k} ) _ {+}  ^ {m} ,
 
C _ {k} ( x - x _ {k} ) _ {+}  ^ {m} ,
 
$$
 
$$
Line 38: Line 38:
 
one usually takes  $  \overline{x}\; _ {i} = x _ {i} $,  
 
one usually takes  $  \overline{x}\; _ {i} = x _ {i} $,  
 
$  i = 0 \dots n $,  
 
$  i = 0 \dots n $,  
and since for  $  S _ {2k+} 1 ( \Delta _ {n} ;  x ) $
+
and since for  $  S _ {2k+1} ( \Delta _ {n} ;  x ) $
 
there still are  $  2 k $
 
there still are  $  2 k $
 
free parameters, one prescribes  $  k $
 
free parameters, one prescribes  $  k $
 
additional conditions at  $  x _ {0} $
 
additional conditions at  $  x _ {0} $
 
and  $  x _ {n} $,  
 
and  $  x _ {n} $,  
e.g.  $  S _ {2k+} 1 ^ {(} j) ( \Delta _ {n} ;  z ) = y _ {z}  ^ {(} j) $,  
+
e.g.  $  S _ {2k+1}  ^ {( j)} ( \Delta _ {n} ;  z ) = y _ {z}  ^ {( j)} $,  
 
$  j = 1 \dots k $,  
 
$  j = 1 \dots k $,  
 
$  z = x _ {0} , x _ {n} $,  
 
$  z = x _ {0} , x _ {n} $,  
where  $  y _ {z}  ^ {(} j) $
+
where  $  y _ {z}  ^ {( j)} $
are given numbers. If the  $  y _ {z}  ^ {(} j) $
+
are given numbers. If the  $  y _ {z}  ^ {( j)} $
 
linearly depend on the given function, then the corresponding spline linearly depends on this function. For  $  m = 2 k $
 
linearly depend on the given function, then the corresponding spline linearly depends on this function. For  $  m = 2 k $
 
one usually takes  $  \overline{x}\; _ {0} = x _ {0} $,  
 
one usually takes  $  \overline{x}\; _ {0} = x _ {0} $,  
 
$  \overline{x}\; _ {n} = x _ {n} $
 
$  \overline{x}\; _ {n} = x _ {n} $
and  $  x _ {i} = ( \overline{x}\; _ {i-} 1 + \overline{x}\; _ {i} ) / 2 $,  
+
and  $  x _ {i} = ( \overline{x}\; _ {i-1} + \overline{x}\; _ {i} ) / 2 $,  
 
$  i = 1 \dots n - 1 $,  
 
$  i = 1 \dots n - 1 $,  
 
and  $  k $
 
and  $  k $
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has an  $  ( m - s ) $-
 
has an  $  ( m - s ) $-
 
th continuous and an  $  ( m - s + 1 ) $-
 
th continuous and an  $  ( m - s + 1 ) $-
st discontinuous derivative at  $  x _ {1} \dots x _ {n-} 1 $,  
+
st discontinuous derivative at  $  x _ {1} \dots x _ {n-1} $,  
 
then for  $  s \geq  2 $
 
then for  $  s \geq  2 $
 
the first  $  ( s - 1 ) $-
 
the first  $  ( s - 1 ) $-
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. de Boor,  "Splines as linear combinations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198033.png" />-splines, a survey"  G.G. Lorentz (ed.)  C.K. Chri (ed.)  L.L. Schumaker (ed.) , ''Approximation theory'' , '''2''' , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.J. Schoenberg,  "Cardinal spline interpolation" , SIAM  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.L. Schumaker,  "Spline functions, basic theory" , Wiley  (1981)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. de Boor,  "Splines as linear combinations of $B$-splines, a survey"  G.G. Lorentz (ed.)  C.K. Chri (ed.)  L.L. Schumaker (ed.) , ''Approximation theory'' , '''2''' , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.J. Schoenberg,  "Cardinal spline interpolation" , SIAM  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.L. Schumaker,  "Spline functions, basic theory" , Wiley  (1981)</TD></TR></table>

Latest revision as of 12:11, 16 December 2020


A spline

$$ S _ {m} ( \Delta _ {n} ; x ) = \ a _ {0} + a _ {1} x + \dots + a _ {m-1} x ^ {m-1} + \sum _ { k= 0} ^ { n-1 } C _ {k} ( x - x _ {k} ) _ {+} ^ {m} , $$

where

$$ t _ {+} = \ \left \{ \begin{array}{ll} t & t \geq 0 , \\ 0 & t < 0 , \\ \end{array} \ \ x _ {0} < \dots < x _ {n} , \right .$$

that coincides with a given function at given distinct points $ \{ \overline{x}\; _ {i} \} $. For $ m = 2 k + 1 $ one usually takes $ \overline{x}\; _ {i} = x _ {i} $, $ i = 0 \dots n $, and since for $ S _ {2k+1} ( \Delta _ {n} ; x ) $ there still are $ 2 k $ free parameters, one prescribes $ k $ additional conditions at $ x _ {0} $ and $ x _ {n} $, e.g. $ S _ {2k+1} ^ {( j)} ( \Delta _ {n} ; z ) = y _ {z} ^ {( j)} $, $ j = 1 \dots k $, $ z = x _ {0} , x _ {n} $, where $ y _ {z} ^ {( j)} $ are given numbers. If the $ y _ {z} ^ {( j)} $ linearly depend on the given function, then the corresponding spline linearly depends on this function. For $ m = 2 k $ one usually takes $ \overline{x}\; _ {0} = x _ {0} $, $ \overline{x}\; _ {n} = x _ {n} $ and $ x _ {i} = ( \overline{x}\; _ {i-1} + \overline{x}\; _ {i} ) / 2 $, $ i = 1 \dots n - 1 $, and $ k $ additional conditions are prescribed at $ x _ {0} $ and $ x _ {n} $. If the spline $ S _ {m} ( \Delta _ {n} ; x ) $ has an $ ( m - s ) $- th continuous and an $ ( m - s + 1 ) $- st discontinuous derivative at $ x _ {1} \dots x _ {n-1} $, then for $ s \geq 2 $ the first $ ( s - 1 ) $- st derivatives of the spline are prescribed at these points, requiring them to coincide with the corresponding derivatives of the function to be interpolated. Interpolation $ L $- and $ L _ {q} $- splines, as well as interpolation splines in several variables, have also been considered. Interpolation splines are used to approximate a function using its values on a grid. In contrast to interpolation polynomials, there exist matrices of nodes such that the interpolation splines converge to an arbitrary given continuous integrable function.

References

[1] S.B. Stechkin, Yu.N. Subbotin, "Splines in numerical mathematics" , Moscow (1976) (In Russian)
[2] J.H. Ahlberg, E.N. Nilson, J.F. Walsh, "Theory of splines and their applications" , Acad. Press (1967)

Comments

Cf. also Spline; Spline approximation; Spline interpolation.

References

[a1] C. de Boor, "Splines as linear combinations of $B$-splines, a survey" G.G. Lorentz (ed.) C.K. Chri (ed.) L.L. Schumaker (ed.) , Approximation theory , 2 , Acad. Press (1976)
[a2] I.J. Schoenberg, "Cardinal spline interpolation" , SIAM (1973)
[a3] L.L. Schumaker, "Spline functions, basic theory" , Wiley (1981)
How to Cite This Entry:
Interpolation spline. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interpolation_spline&oldid=50985
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article