TABLE OF CONTENTS Interpolation is the process of defining a function that takes on specified values at specified points. This chapter concentrates on two closely related interpolants: the piecewise cubic spline and the shape-preserving piecewise cubic named "pchip."
We all know that two points determine a straight line. More precisely, any two points in the plane, ( x 1, y 1) and ( x 2, y 2), with x 1 ? x 2, determine a unique first-degree polynomial in x whose graph passes through the two points. There are many different formulas for the polynomial, but they all lead to the same straight line graph.
This generalizes to more than two points. Given n points in the plane, ( x k, y k), k = 1, , n, with distinct x k's, there is a unique polynomial in x of degree less than n whose graph passes through the points. It is easiest to remember that n, the number of data points, is also the number of coefficients, although some of the leading coefficients might be zero, so the degree might actually be less than n ? 1. Again, there are many different formulas for the polynomial, but they all define the same function.
This polynomial is called the interpolating polynomial because it exactly reproduces the given data:
Later, we examine other polynomials, of lower degree, that only approximate the data.
