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

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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].

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Define the vector

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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

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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]

Solution

2.5

Interpolate a function f : 2 → of the form

[2.51]

such that f (1, 0) = 1 and f (1, 1) = 1.

Solution