A generalized multivariable Newton method

• Published in: Fixed point theory and algorithms for sciences and engineering Vol. 2021; no. 1
• Main Authors: Burachik, Regina S., Caldwell, Bethany I., Kaya, C. Yalçın
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 20-09-2021; Springer Nature B.V
• Subjects: Algorithms; Analysis; Applications of Mathematics; Asymptotic methods; Convergence; Differential Geometry; Experiments; Mathematical analysis; Mathematical and Computational Biology; Mathematical programming; Mathematics; Mathematics and Statistics; Methods; Newton methods; Optimization and Real World Applications; Polynomials; Signal processing; Topology; Fixed-point theory; Fixed-point algorithms; Nonlinear systems of equations; Newton’s method; 65H10; 49M15; Polynomial equations; 65H04
• ISSN: 2730-5422; 2730-5422
• DOI: 10.1186/s13663-021-00700-9

It is well known that the Newton method may not converge when the initial guess does not belong to a specific quadratic convergence region. We propose a family of new variants of the Newton method with the potential advantage of having a larger convergence region as well as more desirable properties near a solution. We prove quadratic convergence of the new family, and provide specific bounds for the asymptotic error constant. We illustrate the advantages of the new methods by means of test problems, including two and six variable polynomial systems, as well as a challenging signal processing example. We present a numerical experimental methodology which uses a large number of randomized initial guesses for a number of methods from the new family, in turn providing advice as to which of the methods employed is preferable to use in a particular search domain.