Suppose I have two nonlinear equations with two variables $\ell$ and $m$; the variables $\ell$ and $m$ are of degree eight in the first equation and of degree six in the second equation.

How it possible to solve these two nonlinear equations and find the possible values of the variables $\ell$ and $m$?

How many possible solutions for both variables $\ell$ and $m$?

Is there a Matlab code to solve these equations and find the all possible solutions?.

  • $\begingroup$ Can you post an example of the equations? $\endgroup$ – Amzoti Jan 10 '14 at 17:29
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    $\begingroup$ Hi @CS2013! $$\color{red}{\Large\text{Welcome to Math.SE!}}$$ Don't worry about it now (because you're new) but you might like to know that we prefer to use MathJax here :) $\endgroup$ – Shaun Jan 10 '14 at 18:31

You can try Two Variable Bisection.

$\quad\mathbf2$D Bisection

Bisection in $2$ dimensions amount to finding zeros of two equations in two unknowns, say $F(x,y)=0$ and $G(x,y)=0$ on a rectangular region in the plane. The algorithm is effectively a quadtree decomposition of the region into subregions, each subregion (presumably) having $F\neq0$ or $G\neq0$. I say presumably since the algorithm check the signs of $F$ and $G$ at the vertices of the subregions. If $F$ has the same sign at the four vertices, then the algorithm stops processing that subregion (maybe missing roots).

related post: https://stackoverflow.com/q/3513660/380384

  • $\begingroup$ Multivariable Newton's method sounds more reliable to me. $\endgroup$ – busman Jan 16 '14 at 12:43
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    $\begingroup$ Depends if you know the jacobian (derivatives) or you have to numerically calculate them. The latter, being unstable for sure. Also if the problem is bound bisection is generally better. $\endgroup$ – ja72 Jan 16 '14 at 13:09

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