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A method for the approximate calculation of the value of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059330/l0593301.png" />, based on the replacement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059330/l0593302.png" /> by a linear function
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A method for the approximate calculation of the value of a function $f(x)$, based on the replacement of $f(x)$ by a linear function
  
<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/l/l059/l059330/l0593303.png" /></td> </tr></table>
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\[ L(x)=a(x-x_1)+b,\]
  
the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059330/l0593304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059330/l0593305.png" /> being chosen in such a way that the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059330/l0593306.png" /> coincide with the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059330/l0593307.png" /> at given points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059330/l0593308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059330/l0593309.png" />:
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the parameters $a$ and $b$ being chosen in such a way that the values of $L(x)$ coincide with the values of $f(x)$ at given points $x_1$ and $x_2$:
  
<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/l/l059/l059330/l05933010.png" /></td> </tr></table>
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\[L(x_1)=f(x_1),\quad L(x_2)=f(x_2).\]
  
 
These conditions are satisfied by the unique function
 
These conditions are satisfied by the unique function
  
<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/l/l059/l059330/l05933011.png" /></td> </tr></table>
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\[L(x)=\frac{ f(x_2)-f(x_1)}{x_2-x_1}(x-x_1)+f(x_1),\]
  
which approximates the given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059330/l05933012.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059330/l05933013.png" /> with error
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which approximates the given function $f(x)$ on the interval $[x_1,x_2]$ with error
  
<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/l/l059/l059330/l05933014.png" /></td> </tr></table>
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\[ f(x)-L(x)=\frac{f''(\xi)}{2}(x-x_1)(x-x_2),\quad \xi\in [x_1,x_2].\]
  
 
The calculations necessary for linear interpolation are easily realized by hand; for this reason this method is widely used for the interpolation of tabular data.
 
The calculations necessary for linear interpolation are easily realized by hand; for this reason this method is widely used for the interpolation of tabular data.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"N.S. Bakhvalov,  "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"I.S. Berezin,  N.P. Zhidkov,  "Computing methods" , Pergamon  (1973)  (Translated from Russian)</TD></TR></table>
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|-
 
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|valign="top"|{{Ref|Ba}}||valign="top"| N.S. Bakhvalov,  "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)
 
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|-
====Comments====
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|valign="top"|{{Ref|Be}}||valign="top"| I.S. Berezin,  N.P. Zhidkov,  "Computing methods" , Pergamon  (1973)  (Translated from Russian)
 
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|-
 
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|valign="top"|{{Ref|Da}}||valign="top"| P.J. Davis,  "Interpolation and approximation" , Dover, reprint  (1975)  pp. 108–126
====References====
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|-
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"P.J. Davis,  "Interpolation and approximation" , Dover, reprint  (1975)  pp. 108–126</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.F. Steffensen,  "Interpolation" , Chelsea, reprint  (1950)</TD></TR></table>
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|valign="top"|{{Ref|De}}||valign="top"| B.N. Delone, "The Peterburg school of number theory", Moscow-Leningrad (1947) (In Russian)
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|valign="top"|{{Ref|St}}||valign="top"| J.F. Steffensen,  "Interpolation" , Chelsea, reprint  (1950)
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Latest revision as of 21:21, 15 July 2012

A method for the approximate calculation of the value of a function $f(x)$, based on the replacement of $f(x)$ by a linear function

\[ L(x)=a(x-x_1)+b,\]

the parameters $a$ and $b$ being chosen in such a way that the values of $L(x)$ coincide with the values of $f(x)$ at given points $x_1$ and $x_2$:

\[L(x_1)=f(x_1),\quad L(x_2)=f(x_2).\]

These conditions are satisfied by the unique function

\[L(x)=\frac{ f(x_2)-f(x_1)}{x_2-x_1}(x-x_1)+f(x_1),\]

which approximates the given function $f(x)$ on the interval $[x_1,x_2]$ with error

\[ f(x)-L(x)=\frac{f''(\xi)}{2}(x-x_1)(x-x_2),\quad \xi\in [x_1,x_2].\]

The calculations necessary for linear interpolation are easily realized by hand; for this reason this method is widely used for the interpolation of tabular data.

References

[Ba] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)
[Be] I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)
[Da] P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 108–126
[De] B.N. Delone, "The Peterburg school of number theory", Moscow-Leningrad (1947) (In Russian)
[St] J.F. Steffensen, "Interpolation" , Chelsea, reprint (1950)
How to Cite This Entry:
Linear interpolation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_interpolation&oldid=16431
This article was adapted from an original article by M.K. Samarin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article