2.4 Ordinary Interpolation
Interpolation is any procedure for fitting a function to a set of points in such a manner that the function intercepts each of the points. Consider m points (x[k], y[k]) where x[k] n, y[k] , and the x[k] are distinct. We wish to construct a function f : n → such that y[k] = f (x[k]) for all k. There are various solutions to this problem. We consider an approach called ordinary interpolation. This is applicable to a wide range of interpolation functions. Later, we will consider another approach that uses cubic splines as interpolation functions.
2.4.1 Example: Linear Interpolation
Between two points, (x, y) and (x, y), with x ≠ x, we may interpolate a linear polynomial of the form
Consider points (2, 1) and (5, 3). These and an interpolated linear polynomial are graphed in Exhibit 2.4. To find the formula for the interpolated polynomial, we must determine constants β1 and β2. If the linear polynomial is to intercept (2, 1) and (5, 3), it must satisfy
This is a system of two linear equations in two unknowns. We express it in matrix form:
Solving, we obtain β1 = –1/3 and β2 = 2/3, so our interpolated polynomial is
- Our name reflects the method’s similarity to the method of ordinary least squares. See Exercise 2.16.↵