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A derivative-free method for solving large-scale nonlinear systems of equations. (English) Zbl 1187.65055

The paper is concerned with iteratively solving systems of nonlinear equations $g\left(x\right)=0$, where $g$ is a continuously differentiable mapping in $n$-dimensional real space. It is supposed that the systems are large-scale systems for which the Jacobian is not available or requires a prohibitive amount of storage. The author extends the conjugate gradient method to solve the system (a problem equivalent to an unconstrained optimization-minimization problem). For this, the known Polak-Ribiere-Polyak conjugate gradient direction , as a new line search direction, is used [see E. Polak and G. Ribière, Rev. Franç. Inform. Rech. Opér. 3, No. 16, 35–43 (1969; Zbl 0174.48001) and B. T. Polyak, U.S.S.R. Comput. Math. Math. Phys. 9(1969), No. 4, 94–112 (1971); translation from Zh. Vychisl. Mat. Mat. Fiz. 9, 807–821 (1969; Zbl 0229.49023)].

The author proposes the algorihm DFCGNE (Derivative Free Conjugate Gradient for Nonlinear Equations) for solving nonlinear systems and also, modification of this algorithm, called M-DFCGNE method, in the case of nonmonotone objective functions. Under some reasonable conditions, the global convergence of these algorithms is proved. Numerical experiments and comparisons with other methods are discussed.

##### MSC:
 65H10 Systems of nonlinear equations (numerical methods)