View source for Linear interpolation

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

<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>

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" />:

<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>

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>

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

<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>

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====
<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>



====Comments====


====References====
<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>