Taylor Polynomial Interpolation
This module illustrates Taylor polynomial interpolation. The Taylor
polynomial interpolant of degree n for a smooth function
f(x) about a point a is given by the
truncated Taylor series
pn(x) =
f(a) +
f ′(a)(x − a) +
(f ″(a) ⁄ 2) (x −
a)2 + ⋅ ⋅ ⋅ +
(f (n)(a) ⁄ n!)
(x − a)n. It is the
unique polynomial pn(x) of
degree n whose value and those of its first n derivatives
at a agree with those of f, i.e.,
pn(k)(a)
= f (k)(a) for
k = 1,…,n, where superscripts
indicate derivatives. The Taylor polynomial interpolants approximate
f well in some neighborhood of a whose size depends on
the location of any singularities of f in the complex plane.
The user begins by selecting a function
f(x) from the list of available functions
and using the slider to choose the point a at which the function
is to be interpolated. The graph of the function is drawn, and the
chosen point a is indicated by a black dot. Clicking
Next causes Taylor polynomial interpolants of increasing degree
to be added to the graph. The color of the interpolants 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
interpolant shown.
Reference: Michael T. Heath, Scientific Computing,
An Introductory Survey, 2nd edition, McGraw-Hill, New York,
2002. See Section 7.3.5.
Developers: Evan VanderZee and Michael Heath
Department of Computer Science