 # 2.4.3 Ordinary Interpolation Methodology

###### 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

[2.44] 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

[2.45] 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].

[2.46] Define the vector

[2.47] 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

[2.48] If the matrix f is invertible, this has the unique solution

[2.49] ###### Exercises
2.4

Consider three points (x[k], y[k]) = (1, 2), (4, 2) and (5, 3). Interpolate a quadratic polynomial of the form

[2.50] 2.5

Interpolate a function f : 2 of the form

[2.51] such that f (1, 0) = 1 and f (1, 1) = 1.