SIR—An efficient solver for systems of equations

Jan Scheffel,Kristoffer Lindvall

doi:10.1016/j.softx.2018.01.003

Abstract

The Semi-Implicit Root solver (SIR) is an iterative method for globally convergent solution of systems of nonlinear equations. 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 Newton’s method, being a special case of the method. Furthermore the algorithm cannot land on local minima, as may be the case for Newton’s method with line search.

Highlights

Systems of algebraic equations generally need be solved computationally, whether it be with direct methods or iterative methods
The Semi-Implicit Root solver (SIR [5]), reported on here, was developed in order to improve on the global convergence characteristics of the widely used Newton method
Linesearch [6] is often combined with Newton’s method to improve convergence but, unlike SIR, it may lead to local extrema rather than to the roots of the equations

Summary

Motivation and significance

Systems of algebraic equations generally need be solved computationally, whether it be with direct methods or iterative methods. The Semi-Implicit Root solver (SIR [5]), reported on here, was developed in order to improve on the global convergence characteristics of the widely used Newton method. Linesearch [6] is often combined with Newton’s method to improve convergence but, unlike SIR, it may lead to local extrema rather than to the roots of the equations. SIR development was initially inspired by semi-implicit PDE algorithms ([1],[2],[3]) and evolved as a robust equation solver for the time-spectral method GWRM [4] for systems of PDEs; it is generally applicable to systems of equations. Its coding in MATLAB and MAPLE, has been substantially improved with respect to efficiency. In Appendix, MATLAB and MAPLE codes are provided

Software description

Illustrative example

MATLAB code for the example

Pseudocode

Impact

Conclusion