Scientific Computing

This module illustrates Bernstein polynomial approximation. The
Weierstrass approximation theorem states that for any continuous
function *f* on a closed interval, say *ε* > 0, there is a polynomial *p* such that
for any *x* in *f* (*x*)
− *p*(*x*) | < *ε*.

that converge uniformly to *f* as *n* increases,
where the binomial coefficient

is the number of distinct ways a set of *k* objects can be chosen
from a set of *n* objects (“*n* choose
*k*”). By shifting and scaling, the Bernstein polynomials
*B*_{n}(*x*) can be used to approximate a
function *f* on any closed interval *a*,
*b*]

The user begins by selecting a function *f*(*x*)*Next* causes Bernstein polynomial approximations of increasing
degree to be added to the graph. The color of the approximating
polynomials changes from blue to red as the degree increases, and the
polynomial written below the graph is updated to reflect the
coefficients of the highest degree Bernstein polynomial shown.

**Reference:** George M. Phillips, *Interpolation and
Approximation by Polynomials*, Springer, New York, 2003. See Chapter
7, pages 247-290.

**Developers:** Evan VanderZee and Michael Heath