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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(x)=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:
65H10Systems of nonlinear equations (numerical methods)