# SIR - an Efficient Solver for Systems of Equations

## TL;DR

The paper introduces SIR, an efficient, robust iterative solver for nonlinear systems, with implementations in MATLAB and MAPLE that leverage sparse algorithms and numerical differentiation for improved convergence.

## Contribution

It presents a globally convergent solver for nonlinear equations, with implementations that outperform Newton's method in robustness and avoid local minima.

## Key findings

- SIR demonstrates superior global convergence compared to Newton's method.
- The solver approaches second-order accuracy near roots.
- Implementation uses efficient sparse matrix algorithms and numerical differentiation.

## Abstract

The Semi-Implicit Root solver (SIR) is an iterative method for globally convergent solution of systems of nonlinear equations. Since publication, SIR has proven robustness for a great variety of problems. We here present MATLAB and MAPLE codes for SIR, that can be easily implemented in any application where linear or nonlinear systems of equations need be solved efficiently. The codes employ recently developed efficient sparse matrix algorithms and improved numerical differentiation. SIR convergence is quasi-monotonous and approaches second order in the proximity of the real roots. Global convergence is usually superior to that of Newtons method, being a special case of the method. Furthermore the algorithm cannot land on local minima, as may be the case for Newtons method with linesearch.