Thread Subject:

Taylor vector expansion?

Hi all,

Is there a way to do a Taylor expansion for a function f(x) where x is a
p dimensional vector around some vector p0? (not just around one
component of x)

thanks, matt.

"Matt" <mmmabos@comcast.net> wrote in message
news:1124303215.77c7216acb8341860620356fb2e914ee@teranews...
> Hi all,
>
> Is there a way to do a Taylor expansion for a function f(x) where x is a
> p dimensional vector around some vector p0? (not just around one
> component of x)
>
> thanks, matt.

Yes. You will find this in almost any book on vector calculus. It get's
messy for more than 1 or 2 terms, though. Equation 37 on this page
(http://mathworld.wolfram.com/TaylorSeries.html) shows the expression

Ken Davis wrote:
> "Matt" <mmmabos@comcast.net> wrote in message
> news:1124303215.77c7216acb8341860620356fb2e914ee@teranews...
>
>>Hi all,
>>
>>Is there a way to do a Taylor expansion for a function f(x) where x is a
>>p dimensional vector around some vector p0? (not just around one
>>component of x)
>>
>>thanks, matt.
>
>
> Yes. You will find this in almost any book on vector calculus. It get's
> messy for more than 1 or 2 terms, though. Equation 37 on this page
> (http://mathworld.wolfram.com/TaylorSeries.html) shows the expression
>
>
Thanks. I was thinking more along the lines of how to use the Taylor
function in Matlab to the same effect without having to do the expansion
by hand... As far as I can see it only works for one variable.

In article <1124303215.77c7216acb8341860620356fb2e914ee@teranews>,
 Matt <mmmabos@comcast.net> wrote:

> Is there a way to do a Taylor expansion for a function f(x) where x is a
> p dimensional vector around some vector p0? (not just around one
> component of x)

Its known as a multivariate Taylor series. You can expand
a function of multiple variables around some point p0 in
a vector space. Assuming that x and x0 are row vectors,

  f(x) = f(p0) + grad(x0)*(x-x0)' + (x-x0)*H(x)*(x-x0)'/2 + ...

Here grad(x0) is the gradient of f at the point x0, and H(x0)
is the nxn Hessian matrix of second partial derivatives.

You can see that for scalar x, it reduces to the standard
boring old Taylor series.

HTH,
John D'Errico


--
The best material model of a cat is another, or
preferably the same, cat.
A. Rosenblueth, Philosophy of Science, 1945

John D'Errico wrote:
> In article <1124303215.77c7216acb8341860620356fb2e914ee@teranews>,
> Matt <mmmabos@comcast.net> wrote:
>
>
>>Is there a way to do a Taylor expansion for a function f(x) where x is a
>>p dimensional vector around some vector p0? (not just around one
>>component of x)
>
>
> Its known as a multivariate Taylor series. You can expand
> a function of multiple variables around some point p0 in
> a vector space. Assuming that x and x0 are row vectors,
>
> f(x) = f(p0) + grad(x0)*(x-x0)' + (x-x0)*H(x)*(x-x0)'/2 + ...
>
> Here grad(x0) is the gradient of f at the point x0, and H(x0)
> is the nxn Hessian matrix of second partial derivatives.
>
> You can see that for scalar x, it reduces to the standard
> boring old Taylor series.
>
> HTH,
> John D'Errico
>
>
Thanks. I understand this. What I am looking for is a way of
using/modifying the Taylor function in Matlab to do this for me. As far
as I can see the function is only designed to do expansions around
scalars and thus takes partials... Any ideas?

thanks, m.