Scientific Computing

This module demonstrates how the same polynomial can be expressed in different ways as a linear combination of various sets of basis functions. It also gives insight into some of the differences among several common polynomial bases

The user first chooses a target polynomial
*f*(*x*)*Random Polynomial* to
have a polynomial generated randomly. To see the functions in a
particular polynomial basis combine to form the target polynomial, the
user selects a polynomial basis then repeatedly clicks the *Next*
button. At each step, the approximating polynomial of next higher
degree is added to the plot, and its coefficients are shown below in an
expansion in terms of the basis functions
*φ*_{i} of the selected basis.

Certain characteristics of the available polynomial bases are illustrated here. The monomial basis is built as a succession of Taylor polynomial approximations around zero. The Lagrange basis functions have the property of taking either the value 0 or 1 at the interpolation points. The Newton basis functions take the value 0 at all previously interpolated points. The orthogonal Chebyshev basis functions minimize the maximum error over the interval of interpolation, often getting a good approximation to the target polynomial with fewer basis functions than the other bases.

**Reference:** Michael T. Heath, *Scientific Computing,
An Introductory Survey*, 2nd edition, McGraw-Hill, New York,
2002. See Section 7.3.

**Developers:** Evan VanderZee and Michael Heath