Taylor Series
Single Variable Taylor Series
If we know a function value and its derivatives at some point x0, we can use a Taylor series to approximate its value at a nearb point x as
If x-x0 is small then we can truncate the series to find
Note that for an nth order polynomial functions, the Taylor series converges in exactly n+1 terms.
Taylor series approximation of various functions showing the effect of retaining up to 4 terms in the expansion. The expansions are: Note that the third expansion,
recovers the function exactly at all points since all higher derivatives (and therefore all higher terms in the expansion) are zero.
The expansions are: Note that as we retain more terms we become increasingly more accurate. However, this is an infinite series for the exponential function since all of its derivatives are nonzero.
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Multivariate Taylor Series
Consider the case where we have several equations that are functions of multiple variables,
We can write a multivariate Taylor series expansion for the
ith function as
where is the location where we evaluate the function and
is the number of independent variables.
The term is a
matrix called the Jacobian matrix,
It
is used in solving nonlinear systems of equations using
Newton's method.
Example
Consider the equations
Here . Let's first look at the expansion for
. Applying our general equation above we find
The partial derivatives of are:
0 0 0 0
substituting this we find
We can do the same thing for
The partial derivatives of are:
-6 0 0 0 0
substituting this we find