Taylor series

From DDL Wiki

Jump to: navigation, search

In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It is common practice to use a finite number of terms of the series to approximate a function.

Unidimensional Taylor Series

The Taylor series of a function ƒ(x) that is infinitely differentiable in the neighbourhood of the point \mathbf{x}_0 is the power series

f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots

which in a more compact form can be written as

\sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n},


Multidimensional Taylor Series

A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as

T(\mathbf{x}) = f(\mathbf{x}_0) + \nabla f(\mathbf{x}_0)(\mathbf{x} - \mathbf{x}_0) + \frac{1}{2!} (\mathbf{x} - \mathbf{x}_0)^T \,\nabla^2 f(\mathbf{x}_0)\,(\mathbf{x} - \mathbf{x}_0) + \cdots\! \,,

where \nabla f(\mathbf{x}_0)\! is the gradient of \,f evaluated at \mathbf{x} = \mathbf{x}_0 and \nabla^2 f(\mathbf{x}_0)\! is the Hessian of \,f evaluated at \mathbf{x} = \mathbf{x}_0.

Links

Retrieved from "http://wiki.ece.cmu.edu/ddl/index.php/Taylor_series"
Personal tools