Linear interpolation
A method for the approximate calculation of the value of a function , 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) |
Linear interpolation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_interpolation&oldid=27068