The Taylor Polynomial

Most mathematical functions (exp, sin, cos, etc) cannot be in general evaluated exactly. By hand, and on a computer as well, we usually calculate some approximations to these functions. The approximate functions need to be easy to evaluate, and have to provide values which are close enough to the values of the original function. One of the most convenient ways to compute functions is to approximate them by polynomials.

A Taylor polynomial of degree is constructed to mimic the behavior of near a point . If is a degree Taylor polynomial,

it has coefficients through .

We have to choose the coefficients such that for . For this, it is natural to require that the value of the polynomial (and the values of its first derivatives) coincides with the value of the function (and the values of function's first derivatives, respectively) for . This means that , and since we have determined . This one condition is clearly not sufficient to determine all coefficients of the polynomial. An extra condition, that forces more resemblance between and when is . Since , we have determined the second coefficient also. If this is enough; if not, we continue imposing conditions on higher derivates of , until we have conditions for the coefficients:

To fix the ideas, let , and . Then

has two unknown coefficients, and . We therefore use the conditions

Hence, .

For , we add the third condition , which reads , or . Thus the quadratic approximation of near is

figure=Eps/taylor.eps,width=6.0in,height=3.0in

In the general case, if is easy to see that

(to check this take the first several derivatives of and evaluate them at ).

In order to be able to write this formula, we need of course to assume that is at least times continuously differentiable; we will actually assume for the purpose of writing an error estimate also.

For ( has derivatives of any order) we can increase indefinitely; the approximations get better and better, and, in the limit, they will coincide with . In the limit the summation becomes infinite, and we obtain the Taylor series expansion

Question: to define we have to evaluate , together with several of its derivatives. Aren't we better off just evaluating , and forgetting about this whole approximation stuff? The answer is that, indeed, we have to evaluate at once. Then , once constructed, will be a cheap approximation for thousands of future calls. Moreover, and its derivatives are evaluated at one point , while is a good approximation of for an entire interval .