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Finding multiple roots of nonlinear algebraic equations using S-system methodology. (English) Zbl 0771.65025

The paper concerns the determination of nonnegative real roots of generalized mass action equations of the form \[ \sum^ p_{k=1}\alpha_{ik}\prod^ n_{j=1}X^{g_{ijk}}_ j-\sum^ p_{k=1}\beta_{ik}\prod ^ n_{j=1}X^{h_{ijk}}_ j=0,\quad i=1,\dots,n \] where the coefficients \(\alpha_{ik}\), \(\beta_{ik}\) as well as the unknown \(X_ i\) are real, nonnegative, while the exponents \(g_{ijk}\), \(h_{ijk}\) are real. In terms of the logarithms of the unknown the positive real roots can be represented as solutions of an underdetermined linear system of equations coupled with a set of simple nonlinear equations. By approximating these nonlinear constraints by suitable combinations of monomials a square system of linear equations is obtained. This leads to an iterative process which is typically observed to have quadratic convergence. The overall methods is outlined using some simple examples and several numerical results are given.

MSC:

65H05 Numerical computation of solutions to single equations
26C10 Real polynomials: location of zeros
12Y05 Computational aspects of field theory and polynomials (MSC2010)

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