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I'm trying to use Newton's method to solve the following system of equations, where f and g are functions of x and y. (h,a,f,c,d,b and k are just constants).

$f(y,x)=\left[\begin{array}{c} y^{1}\\ y^{2}\\ y^{3}\\ y^{4} \end{array}\right]-\left[\begin{array}{c} y_{0}^{1}\\ y_{0}^{2}\\ y_{0}^{3}\\ y_{0}^{4} \end{array}\right]+h\left[\begin{array}{cccc} a_{1} & f_{1} & 0 & 0\\ c_{1} & a_{1} & f_{1} & 0\\ 0 & c_{1} & a_{1} & f_{1}\\ 0 & 0 & c_{1} & a_{1} \end{array}\right]\left[\begin{array}{c} y^{1}\\ y^{2}\\ y^{3}\\ y^{4} \end{array}\right]+\left[\begin{array}{cccc} b_{1} & 0 & 0 & 0\\ d_{1} & b_{1} & 0 & 0\\ 0 & d_{1} & b_{1} & 0\\ 0 & 0 & d_{1} & b_{1} \end{array}\right]\left[\begin{array}{c} x^{1}\\ x^{2}\\ x^{3}\\ x^{4} \end{array}\right]-\left[\begin{array}{c} k_{1}\\ 0\\ 0\\ k_{2} \end{array}\right]=0$

$g(y,x)=\left[\begin{array}{c} x^{1}\\ x^{2}\\ x^{3}\\ x^{4} \end{array}\right]-\left[\begin{array}{c} x_{0}^{1}\\ x_{0}^{2}\\ x_{0}^{3}\\ x_{0}^{4} \end{array}\right]+h\left[\begin{array}{cccc} a_{2} & f_{2} & 0 & 0\\ c_{2} & a_{2} & f_{2} & 0\\ 0 & c_{2} & a_{2} & f_{2}\\ 0 & 0 & c_{2} & a_{2} \end{array}\right]\left[\begin{array}{c} y^{1}\\ y^{2}\\ y^{3}\\ y^{4} \end{array}\right]+\left[\begin{array}{cccc} b_{2} & 0 & 0 & 0\\ d_{2} & b_{2} & 0 & 0\\ 0 & d_{2} & b_{2} & 0\\ 0 & 0 & d_{2} & b_{2} \end{array}\right]\left[\begin{array}{c} x^{1}\\ x^{2}\\ x^{3}\\ x^{4} \end{array}\right]-\left[\begin{array}{c} k_{1}\\ 0\\ 0\\ k_{2} \end{array}\right]=0$

Am I right in saying that the Newton Iteration equations be:

$y^{k+1}=y^{k}-\frac{f(y,x)}{\frac{\partial f(y,x)}{\partial y}}$ and $x^{k+1}=x^{k}-\frac{g(y,x)}{\frac{\partial g(x,y)}{\partial x}}$.

Or would the denominators be partial derivatives: $\left(x\frac{\partial f}{\partial y}+y\frac{\partial f}{\partial x}\right)$.

I know it was not necessary to write the entire functions but I have some more questions about the equations depending on the answer to this one.

Many Thanks!

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I've just been thinking a lot about this and realized that I'm not even sure how I can apply Newton Raphson method to these equations. If applying it to the first equations will give x or y? ... Please help if have any ideas. –  Hooman Jan 19 '12 at 13:29
    
Note: It's very possible I'm horribly wrong. With that said: I've never seen Newton's method applied to functions that aren't single variable. mathworld.wolfram.com/NewtonsMethod.html Mathworld does not seem to recognize this approach, either. Sorry if I am wrong, but I think Newton's method does not apply to functions that are more than one variable. –  000 Jan 21 '12 at 5:29
    
There's a method called the Multivariate Newton Raphson Method (MNRM) which is used for functions of more than one variable. But my equations are a bit more complicated. So I'm not quite sure how this could be applied in this case. –  Hooman Jan 22 '12 at 12:26
    
I think I might know the answer now. Will post as soon as I am sure. –  Hooman Jan 22 '12 at 14:31
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