5.4: Multidimensional Interpolation

Suppose we are interpolating the value of a function of two variables,

$$z=f(x, y) \nonumber$$

The known values are given by

$$z_{i j}=f\left(x_{i}, y_{j}\right) \nonumber$$

with $i=0,1, \ldots, n$ and $j=0,1, \ldots, n$. Note that the $(x, y)$ points at which $f(x, y)$ are known lie on a grid in the $x-y$ plane.

Let $z=g(x, y)$ be the interpolating function, satisfying $z_{i j}=g\left(x_{i}, y_{j}\right) .$ A twodimensional interpolation to find the value of $g$ at the point $(x, y)$ may be done by first performing $n+1$ one-dimensional interpolations in $y$ to find the value of $g$ at the $n+1$ points $\left(x_{0}, y\right),\left(x_{1}, y\right), \ldots,\left(x_{n}, y\right)$, followed by a single one-dimensional interpolation in $x$ to find the value of $g$ at $(x, y)$.

In other words, two-dimensional interpolation on a grid of dimension $(n+1) \times$ $(n+1)$ is done by first performing $n+1$ one-dimensional interpolations to the value $y$ followed by a single one-dimensional interpolation to the value $x$. Twodimensional interpolation can be generalized to higher dimensions. The MATLAB functions to perform two-and three-dimensional interpolation are interp2.m and interp3.m.