Solving a system of nonlinear algebraic equations of a special type (Q2746876)

The author proposes a new fast and effective numerical algorithm for solving the following system of nonlinear algebraic equations: NEWLINE\[NEWLINE \begin{aligned} &a_{11}x_1+\dots +a_{1n}x_n=0, \\ &\dots \\ &a_{k1}x_1+\dots +a_{kn}x_n=0, \\ &a_{(k+1)1}|x_1|x_1+\dots +a_{(k+1)n}|x_n|x_n=H_1, \\ &\dots \\ &a_{n1}|x_1|x_1+\dots +a_{nn}|x_n|x_n=H_{n-k}. \end{aligned} NEWLINE\]NEWLINE Here \(a_{ij}\) and \(H_i\) are real-valued coefficients. Some uniqueness and convergence theorems are proven. Such systems of equations appear, in particular, in hydraulics.