First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 3. INTERPOLATION 91
for [a 0 , ..., a n ] T where [·] T stands for the transpose of the vector. The coefficient matrix A is
known as the van der Monde matrix. This is usually an ill-conditioned matrix, which means
solving the system of equations could result in large error in the coefficients a i . An intuitive
way to understand the ill-conditioning is to plot several basis monomials, and note how less
distinguishable they are as the degree increases, making the columns of the matrix nearly
linearly dependent.
Figure 3.1: Monomial basis functions
Solving the matrix equation Aa = b could also be expensive. Using Gaussian elimination
to solve the matrix equation for a general matrix A requires O(n 3 ) operations. This means
the number of operations grows like Cn 3 , where C is a positive constant. 1 However, there
are some advantages to the monomial form: evaluating the polynomial is very efficient using
Horner’s method, which is the nested form discussed in Exercises 1.3-4, 1.3-5 of Chapter 1,
requiring O(n) operations. Differentiation and integration are also relatively efficient.
Lagrange form of polynomial interpolation
The ill-conditioning of the van der Monde matrix, as well as the high complexity of solving
the resulting matrix equation in the monomial form of polynomial interpolation, motivate us
to explore other basis functions for polynomials. As before, we start with data (x i ,y i ),i =
0, 1, ..., n, and call our interpolating polynomial of degree at most n, p n (x). The Lagrange
1 The formal definition of the big O notation is as follows: We write f(n) =O(g(n)) as n →∞if and
only if there exists a positive constant M and a positive integer n ∗ such that |f(n)| ≤Mg(n) for all n ≥ n ∗ .