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[parent] linear interpolation (Definition)

Among the many interpolation techniques that are available, linear interpolation is one of the easiest to understand and implement, as the interpolating function is pieced together by a series of line segments connecting the breakpoints.

Suppose we have a finite set $ S$ of ordered pairs $ (x_1,y_1),\ldots,(x_n,y_n)$ of real numbers such that $ x_1<x_2<\cdots <x_n$. The linear interpolation function of $ S$ is a real-valued function $ f$ defined on $ [x_1,x_n]$ such that, for $ i=1,\ldots,n-1$,

$\displaystyle f(x)=y_i + m_i (x-x_i),$    where $\displaystyle m_i=\frac{y_{i+1}-y_i}{x_{i+1}-x_i}$ and $\displaystyle x\in [x_i,x_{i+1}].$
In other words, $ f$ is a piecewise linear function such that $ f$ is linear in each of the interval $ [x_i,x_{i+1}]$ for $ i=1,\ldots,n-1$. When the points (in $ S$) belong to the graph of a function $ g$ defined on a subset of $ [x_1,x_n]$, we say that $ f$ interpolates $ g$. We also say that $ f$ interpolates $ S$, as $ S$ can be viewed as the graph of the function $ g_S$ defined on $ \lbrace x_1,\ldots, x_n\rbrace$ such that $ g_S(x_i)=y_i$.

Visually, the interpolation function can be constructed by line segments whose end points are pairs of points $ (x_i,y_i)$ and $ (x_{i+1},y_{i+1})$ for each $ i=1,\ldots,n-1$. The follow graph shows the linear interpolation function $ f$ (in blue) of a set consisting of seven points (in dark green). Note that $ f$ interpolates any function $ g$ defined on a subset of $ [x_1,x_n]$ such that $ g(x_i)=y_i$.


\begin{pspicture} % latex2html id marker 98 (-7,-1.5)(7,3.5) \psset{unit=0.8cm} ... ...=darkgreen,dotsize=5pt](-6,1)(-4,3)(-3,3)(0,0.5)(1,1)(3,-1)(6,2) \end{pspicture}

Example. Interpolate $ \lbrace (4,7),(2,3),(6,1)\rbrace$ using linear interpolation.

Arrange the points so the $ x$-coordinates are in the ascending order. There are two line segments associated with these three points: $ \ell_1$ with end points $ (2,3),(4,7)$ and $ \ell_2$ with end points $ (4,7),(6,1)$. Next, calculate the slopes with respect to each line segments:

$\displaystyle m_1=\frac{7-3}{4-2}=2$    and $\displaystyle \qquad m_2=\frac{1-7}{6-4}=-3.$
Therefore, the linear interpolation function $ f$ is given by
\begin{displaymath} % latex2html id marker 394f(x) = \left\{ \begin{array}{ll}... ...(-3)(x-4)= -3x+19 & \textrm{if }x\in [4,6]. \end{array}\right. \end{displaymath}



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Cross-references: slopes, calculate, ascending order, end points, subset, graph, points, interval, piecewise, function, real numbers, ordered pairs, finite set, breakpoints, line segments, series, interpolating function, interpolation
There are 3 references to this entry.

This is version 10 of linear interpolation, born on 2007-08-13, modified 2007-09-27.
Object id is 9861, canonical name is LinearInterpolation.
Accessed 2693 times total.

Classification:
AMS MSC: 41A05 (Approximations and expansions :: Interpolation)
 65D05 (Numerical analysis :: Numerical approximation and computational geometry :: Interpolation)
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