2.4.3 Ordinary Interpolation Methodology
We now formalize the interpolation procedure illustrated in our two examples. The examples used polynomials as interpolation functions f. The following discussion extends the procedure to a broader range of interpolating functions.
Consider m points (x[k], y[k]) where x[k] n, y[k] , and the x[k] are distinct. We wish to interpolate a function f : n → of the form
in such a manner that f intercepts each of the points. Functions fj : n → can take any form. In our quadratic example, they were:
- f1(x) = 1
- f2(x) = x2
- f3(x) = x1
- f4(x) =
- f5(x) =
but exponentials, roots, logarithms, and other functions are permissible. Let’s express our problem with matrices. Define
This is unknown. It is what we want to solve for. Define f as the m × m matrix comprising values of each function fj evaluated at each point x[k].
Define the vector
Both the matrix f and vector y are constants. They are known. Our requirement that the function f intercept each point (x[k], y[k]) yields the equation
If the matrix f is invertible, this has the unique solution
Consider three points (x[k], y[k]) = (1, 2), (4, 2) and (5, 3). Interpolate a quadratic polynomial of the form
Interpolate a function f : 2 → of the form
such that f (1, 0) = 1 and f (1, 1) = 1.