Newton’s Iteration Method for Solving the Nonlinear Matrix Equation X + ∑ i = 1 m A i * X − 1 A i = Q

Chang-Zhou Li, Chao Yuan () and An-Gang Cui
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Chang-Zhou Li: School of Mathematics, Jilin University, Changchun 130012, China
Chao Yuan: School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
An-Gang Cui: School of Mathematics and Statistics, Yulin University, Yulin 719000, China

Mathematics, 2023, vol. 11, issue 7, 1-11

Abstract: In this paper, we study the nonlinear matrix equation (NME) X + ∑ i = 1 m A i * X − 1 A i = Q . We transform this equation into an equivalent zero-point equation, then we use Newton’s iteration method to solve the equivalent equation. Under some mild conditions, we obtain the domain of approximation solutions and prove that the sequence of approximation solutions generated by Newton’s iteration method converges to the unique solution of this equation. In addition, the error estimation of the approximation solution is given. Finally, the comparison of two well-known approaches with Newton’s iteration method by some numerical examples demonstrates the superiority of Newton’s iteration method in the convergence speed.

Keywords: nonlinear matrix equation; Newton’s iteration method; Fr é chet derivative; local convergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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