Taylor Series
Taylor series are series that estimate a linearization.
Usage
To estimate an unknown value f(x), pick a value a near x. Building a series from the nth derivatives of f at a, estimate f(x) as:
The series (sometimes called the Taylor expansion) can be infinite, but if it converges then the omitted components represent an error term that approaches 0 as more components are added.
If a finite (n) number of terms are taken as an approximation, it is called the nth order Taylor polynomial.
If x is 0, then it is a Maclaurin series.
Multivariate Functions
For a multivariate function f(x,y), the first-order Taylor polynomial can be expressed as:
for a near x and b near y.
More conventionally however it is expressed using gradients and vectors, as:
The second-order Taylor polynomial can be expressed using Hessian matrices.
where
Vector-Valued Functions
For a vector-valued function F(X) = <f1(X), f2(X)> where X is a vector as X = <x1, x2, ... xn>, the first-order Taylor polynomial can be expressed using Jacobian matrices.
where A = <a1, a2, ... an> near X.