Spectral residual method without gradient information for solving large-scale nonlinear systems of equations

• Published in: Mathematics of computation Vol. 75; no. 255; pp. 1429 - 1448
• Main Authors: LA CRUZ, William, MARTINEZ, José Mario, RAYDAN, Marcos
• Format: Journal Article
• Language: English
• Published: Providence, RI American Mathematical Society 01-07-2006
• Subjects: Applied sciences; Approximation; Calculus of variations and optimal control; Computer software; Directional derivatives; Exact sciences and technology; Globalization; Gradient method; Jacobians; Mathematical analysis; Mathematical programming; Mathematics; Newtons method; Nonlinear equations; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in mathematical programming, optimization and calculus of variations; Numerical methods in optimization and calculus of variations; Operational research and scientific management; Operational research. Management science; Perceptron convergence procedure; Sciences and techniques of general use; Spectral methods; nonmonotone line search; Nonlinear systems; Newton-Krylov methods; spectral gradient method; Line search; Spectral method; Numerical analysis; Scientific computation; Newton Krylov method; Large system; Large scale; Non linear system; Gradient method; Convergence; Large scale system
• ISSN: 0025-5718; 1088-6842
• DOI: 10.1090/S0025-5718-06-01840-0

A fully derivative-free spectral residual method for solving large-scale nonlinear systems of equations is presented. It uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for this nonmonotone behavior. The global convergence analysis of the combined scheme is presented. An extensive set of numerical experiments that indicate that the new combination is competitive and frequently better than well-known Newton-Krylov methods for large-scale problems is also presented.