I am trying to solve non-linear systems and since I can't download matlab on this device, I was wondering if there is a way I can set it up in excel.
I know the formula for x$^{k+1}$=x$^k$-[Df(x)]$^-$f(x$^k$) but I realize to try and get a solution ∥x$_k$-x$_{k+1}$∥ $\le$ $10^{6}$ for each becomes very tedious and very prone to error if done manually.
x$^2$+x$^2$y-xz+6=0
e$^x$+e$^y$-z=0
x$^2$-2xz-4=0
use initial guess
$$ \begin{bmatrix} -1 \\ -2 \\ 1 \\ \end{bmatrix} $$
and I know the jacobian is $$ \begin{bmatrix} 2x_2xy-z & x^2 & -x \\ e^x & e^y & -1 \\ 2x-2z & 0 & -2x \\ \end{bmatrix} $$
This is not an answer but it is too long for a comment.
Solving nonlinear equations using Newton-Raphson can be quite a long iterative process depending on the quality of the initial guesses.
When you face systems like the one in your post, you can easily reduce the dimension of the problem eliminating as many variables as you can for linear or polynomila expressions. For example, in your case$$x^2 y+x^2-x z+6=0\tag 1$$ $$e^x+e^y-z=0 \tag 2$$ $$x^2-2 x z-4\tag 3$$ Use $(3)$ to get $$z=\frac{x^2-4}{2 x}\tag 4$$ Plug this in $(1)$ to get $$y=-\frac{x^2+16}{2 x^2}\tag 5$$ So, only remains equation $(2)$ which writes $$e^{-(\frac{8}{x^2}+\frac{1}{2})}+e^x-\frac{x}{2}+\frac{2}{x}=0$$ Assuming $x\neq 0$, multiply by $x$ and then look for the zero of $$g(x)=x\Big(e^{-(\frac{8}{x^2}+\frac{1}{2})}+e^x\Big)-\frac{x^2}2+2=0$$ which is much better conditionned.
Using Newton method with $x_0=-1$ as in your problem, the successive iterates will be $$x_1=-2.12801532799644$$ $$x_2=-1.82837114694732$$ $$x_3=-1.79536109066658$$ $$x_4=-1.79493929006255$$ $$x_5=-1.79493922117722$$ which is the solution for fifteen significant figures.
Now, reuse $(4)$ and $(5)$ to get $y=-2.9830787435267$ and $z=0.21677424591829$.