Computing Symmetric Positive Definite Solutions of Three Types of Nonlinear Matrix Equations

TL;DR

This paper introduces new iterative algorithms for efficiently computing symmetric positive definite solutions to three types of nonlinear matrix equations, with proven convergence and superior performance in numerical tests.

Contribution

The paper presents novel iterative algorithms that ensure solutions remain symmetric and positive definite, improving accuracy and convergence over existing methods.

Findings

Algorithms successfully compute positive definite solutions in all tested cases.

Proposed methods outperform existing approaches in accuracy and computational efficiency.

Numerical results demonstrate convergence and robustness of the algorithms.

Abstract

Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute a symmetric and positive definite solution. Here, we propose new iterative algorithms for solving three different types of nonlinear matrix equations. We have recently proposed a new algorithm for solving positive definite total least squares problems. Making use of an iterative process for inverse of a matrix, we convert the nonlinear matrix equation to an iterative linear one, and, in every iteration, we apply our algorithm for solving a positive definite total least squares problem to solve the linear subproblem and update the newly defined variables and the matrix inverse terms using appropriate formulas. Our proposed algorithms have a number of useful features. One is that the computed unknown matrix remains symmetric and positive definite in all iterations. As the second useful feature, numerical test results show that in most cases our proposed approach turns to compute solutions with smaller errors within lower computing times. Finally, we provide some test results showing that our proposed algorithm converges to a symmetric and positive definite solution in Matlab software environment on a PC, while other methods fail to do so.