Difference between revisions of "Taylor Series"
(Created page with '== Single Variable Taylor Series == If we know a function value and its derivatives at some point <var>x</var><sub>0</sub>, we can use a ''Taylor series'' to approximate its val...') |
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+ | |||
+ | == Multivariate Taylor Series == | ||
+ | |||
+ | Consider the case where we have several equations that are functions of multiple variables, | ||
+ | :<math> | ||
+ | \begin{array}{c} | ||
+ | f_{1}\left(x_{1},x_{2},\cdots x_{n}\right)\\ | ||
+ | f_{2}\left(x_{1},x_{2},\cdots x_{n}\right)\\ | ||
+ | \vdots \\ | ||
+ | f_{n}\left(x_{1},x_{2},\cdots x_{n}\right)\end{array} | ||
+ | </math> | ||
+ | |||
+ | We can write a multivariate Taylor series expansion for the | ||
+ | <var>i</var><sup>th</sup> function <math>f_i(x_j)</math> as | ||
+ | :<math> | ||
+ | f_{i}(\mathbf{x}) \approx f_{i}(\mathbf{x}_{0}) | ||
+ | + \sum_{j=1}^{n} | ||
+ | + (x_{j}-x_{j0})\frac{\partial f_{i}}{\partial x_{j}} | ||
+ | + \frac{1}{2}(x_{j}-x_{j0})^{2}\frac{\partial^{2}f}{\partial x^{2}} | ||
+ | + \frac{1}{3!}(x_{j}-x_{j0})^{3}\frac{\partial^{3}f}{\partial x^{3}} | ||
+ | + \cdots | ||
+ | </math> | ||
+ | |||
+ | where <math>\mathbf{x}_0</math> is the location where we evaluate the function and <math>n</math> is the number of independent variables. | ||
+ | |||
+ | |||
+ | The term <math>\frac{\partial f_{i}}{\partial x_{j}}</math> is a | ||
+ | matrix called the '''Jacobian''' matrix, <math>[\mathbf{J}]</math> It | ||
+ | is used in solving nonlinear systems of equations using | ||
+ | [[Nonlinear_equations#Newton.27s_Method_for_Systems_of_Equations|Newton's method]]. | ||
+ | |||
+ | |||
+ | === Example === | ||
+ | Consider the equations | ||
+ | :<math> | ||
+ | \begin{cases} | ||
+ | f(x,y) = x^2 + y^2 -1 \\ | ||
+ | g(x,y) = y - x^3 + x - 1 | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | |||
+ | Here <math>n=2</math>. Let's first look at the expansion for <math>f(x,y)</math>. Applying our general equation above we find | ||
+ | :<math> | ||
+ | \begin{align} | ||
+ | f(x,y) \approx f(x_{0},y_{0}) | ||
+ | &+ (x-x_{0})\frac{\partial f}{\partial x} | ||
+ | + \tfrac{1}{2}(x-x_{0})^{2}\frac{\partial^{2}f}{\partial x^{2}} | ||
+ | + \tfrac{1}{6}(x-x_{0})^{3}\frac{\partial^{3}f}{\partial x^{3}} | ||
+ | + \cdots \\ | ||
+ | &+ (y-y_{0})\frac{\partial f}{\partial y} | ||
+ | + \tfrac{1}{2}(y-y_{0})^{2}\frac{\partial^{2}f}{\partial x^{2}} | ||
+ | + \tfrac{1}{6}(y-y_{0})^{3}\frac{\partial^{3}f}{\partial x^{3}} | ||
+ | +\cdots | ||
+ | \end{align} | ||
+ | </math> | ||
+ | The partial derivatives of <math>f(x,y)</math> are: | ||
+ | :{| border=1 cellpadding=10 cellspacing=0 style="text-align:center" | ||
+ | !style="border-bottom:3px solid gray;" | <math>\alpha</math> | ||
+ | !style="border-bottom:3px solid gray;" | <math>\frac{\partial f}{\partial \alpha}</math> | ||
+ | !style="border-bottom:3px solid gray;" | <math>\frac{\partial^2 f}{\partial \alpha^2}</math> | ||
+ | !style="border-bottom:3px solid gray;" | <math>\frac{\partial^3 f}{\partial \alpha^3}</math> | ||
+ | !style="border-bottom:3px solid gray;" | <math>\frac{\partial^4 f}{\partial \alpha^4}</math> | ||
+ | |- | ||
+ | | <math>x</math> || <math>2x</math> || <math>2</math> || 0 || 0 | ||
+ | |- | ||
+ | | <math>y</math> || <math>2y</math> || <math>2</math> || 0 || 0 | ||
+ | |} | ||
+ | substituting this we find | ||
+ | :<math> | ||
+ | f = f(x_0,y_0) | ||
+ | + (x-x_0) 2x + (x-x_0)^2 + (y-y_0) 2y + (y-y_0)^2 | ||
+ | </math> | ||
+ | |||
+ | We can do the same thing for <math>g(x,y)</math> | ||
+ | :<math> | ||
+ | \begin{align} | ||
+ | g(x,y) \approx g(x_{0},y_{0}) | ||
+ | &+ (x-x_{0})\frac{\partial g}{\partial x} | ||
+ | + \tfrac{1}{2}(x-x_{0})^{2}\frac{\partial^{2}g}{\partial x^{2}} | ||
+ | + \tfrac{1}{6}(x-x_{0})^{3}\frac{\partial^{3}g}{\partial x^{3}} | ||
+ | + \cdots \\ | ||
+ | &+ (y-y_{0})\frac{\partial g}{\partial y} | ||
+ | + \tfrac{1}{2}(y-y_{0})^{2}\frac{\partial^{2}g}{\partial x^{2}} | ||
+ | + \tfrac{1}{6}(y-y_{0})^{3}\frac{\partial^{3}g}{\partial x^{3}} | ||
+ | +\cdots | ||
+ | \end{align} | ||
+ | </math> | ||
+ | The partial derivatives of <math>g(x,y)</math> are: | ||
+ | :{| border=1 cellpadding=10 cellspacing=0 style="text-align:center" | ||
+ | !style="border-bottom:3px solid gray;" | <math>\alpha</math> | ||
+ | !style="border-bottom:3px solid gray;" | <math>\frac{\partial g}{\partial \alpha}</math> | ||
+ | !style="border-bottom:3px solid gray;" | <math>\frac{\partial^2 g}{\partial \alpha^2}</math> | ||
+ | !style="border-bottom:3px solid gray;" | <math>\frac{\partial^3 g}{\partial \alpha^3}</math> | ||
+ | !style="border-bottom:3px solid gray;" | <math>\frac{\partial^4 g}{\partial \alpha^4}</math> | ||
+ | |- | ||
+ | | <math>x</math> || <math>-3x^2+1</math> || <math>-6x</math> || -6 || 0 | ||
+ | |- | ||
+ | | <math>y</math> || <math>1</math> || 0 || 0 || 0 | ||
+ | |} | ||
+ | substituting this we find | ||
+ | :<math> | ||
+ | g = g(x_0,y_0) + (x-x_0)(-3x^2+1) + (x-x_0)^2(-3x) + (x-x_0)^3(-x) | ||
+ | + (y-y_0) | ||
+ | </math> | ||
== Multivariate Taylor Series == | == Multivariate Taylor Series == |
Revision as of 18:55, 5 August 2009
Contents
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.
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
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 our function are:
0 0 0 0
substituting this we find