# Numerical method for nonlinear system of algebraic equations of special type

I have a nonlinear system of algebraic equations of special kind: $$\begin{array}{rcl} x_{i}+y_{i}+z_{0,1}+c_{i,1}z_{1,1} & = & d_{i,1}, \\ x_{i}^2 + y_{i}^2 + z_{0,2} + c_{i,1} z_{1,2} + c_{i,2} z_{2,2} & = & d_{i,2} \\ \ldots & = & \ldots \\ x_{i}^{m} + y_{i}^{m} +z_{0,m}+c_{i,1} z_{1,m} + \ldots + c_{i,m} z_{m,m} & = & d_{i,m} \end{array}$$ where $i = 1,\ldots.k$ and number of equations is not less than number of unknowns. I have to find $(x_{i},y_{i},z_{i,j})$; $c_{i,j}$ and $d_{i,j}$ are known. Is there some numerical method that take in account specificity of this system? If not, please tell me which numerical method will work better with this system?

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In general, for non-linear systems of equations, I would recommend newton's method. But I don't know of any specific method that can take advantage of your systems structure. Where do these equations come from? Perhaps others have researched this problem before and have come up with some recommendations. –  Paul Nov 10 '12 at 13:56
Well, it's a polynomial system of equations, so that structure could potentially be exploited. Also, the linear part is almost lower-triangular, which is helpful. You could try looking at Gröbner basis methods, perhaps? Nevertheless, Newton's method can be a good first hack at it. –  Geoff Oxberry Nov 10 '12 at 20:40
Is the number of equations equal to the number of variables? Or what will guarantee that there is a solution at all? –  Arnold Neumaier Nov 12 '12 at 19:22
Apart from Gröbner basis methods (which are mostly symbolic methods), there are also specific numerical methods for systems of polynomial equations such as homotopy continuation methods (implemented in, e.g., PHCpack). –  Christian Clason Nov 13 '12 at 20:00