Linear Interpolation

Lerp is a quasi-acronym for ''linear interpolation'', which can also be used as a verb .

LINEAR INTERPOLATION BETWEEN TWO KNOWN POINTS

If the two known points are given by the coordinates

\scriptstyle(x_0,y_0)

and

\scriptstyle(x_1,y_1)

, the linear interpolant is the straight line between these points. For a value ''x'' in the interval

\scriptstyle(x_0, x_1)

, the value ''y'' along the straight line is given from the equation

:

rac{y - y_0}{y_1 - y_0} = rac{x - x_0}{x_1 - x_0}

which can be derived geometrically from the figure on the right.

Solving this equation for ''y'', which is the unknown value at ''x'', gives

:

y = y_0 + (x-x_0)rac{y_1 - y_0}{x_1-x_0}

which is the formula for linear interpolation in the interval

\scriptstyle(x_0,x_1)

. Outside this interval, the formula is identical to Linear Extrapolation .

INTERPOLATION OF A DATA SET

Linear interpolation on a set of data points

\scriptstyle(x_0, y_0),\, (x_1, y_1),\,\dots,\,(x_n, y_n)

is defined as the concatenation of linear interpolants between each pair of data points. This results in a continuous curve, with a discontinuous derivative.

LINEAR INTERPOLATION AS APPROXIMATION

Linear interpolation is often used to approximate a value of some function ''f'' using two known values of that function at other points. The ''error'' of this approximation is defined as

:

R_T = f(x) - p(x) \,\!

where ''p'' denotes the linear interpolation Polynomial defined above

:

p(x) = f(x_0) + rac{f(x_1)-f(x_0)}{x_1-x_0}(x-x_0). \,\!

It can be proven using Rolle's Theorem that if ''f'' has two continuous derivatives, the error is bounded by

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Information About

Linear Interpolation