User talk:Maddog Battie

Generalisations
(temp for Newtons Method which is now believed to be incorrect)

I've been looking into the generalisation issue and have come across the following problem: For a system with two variables then the Jacobian matrix looks like this:

$$J_F(x) = \begin{pmatrix} {\delta f_1(x_1,x_2) \over \delta x_1} & {\delta f_1(x_1,x_2) \over \delta x_2} \\ \\ {\delta f_2(x_1,x_2) \over \delta x_1} & {\delta f_2(x_1,x_2) \over \delta x_2} \end{pmatrix}$$

Now to use this in Newton's method, you need to calculate the inverse which ends up as:

$${J_F(x)}^{-1} = {1 \over {\delta f_1(x_1,x_2) \over \delta x_1}{\delta f_2(x_1,x_2) \over \delta x_2} - {\delta f_1(x_1,x_2) \over \delta x_2}{\delta f_2(x_1,x_2) \over \delta x_1}}\begin{pmatrix} {\delta f_2(x_1,x_2) \over \delta x_2} & -{\delta f_1(x_1,x_2) \over \delta x_2} \\ \\ -{\delta f_2(x_1,x_2) \over \delta x_1} & {\delta f_1(x_1,x_2) \over \delta x_1} \end{pmatrix}$$

Unfortunately, the first section ends up as a divide by zero. I assume I must be doing something wrong here. I'd love to know what as I'm keen to use Newton's method for a 12th order problem.