Generalisations

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(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:

 

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

 

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.