An Adaptive Solver for Systems of Linear Equations

TL;DR

This paper introduces an adaptive solver within the Armadillo C++ library that automatically detects matrix properties and selects optimal solving methods, improving speed and usability over traditional LAPACK-based approaches.

Contribution

The paper presents a novel adaptive solver that automatically identifies matrix properties and chooses the most efficient solving method, enhancing performance and user-friendliness.

Findings

Notable speedups achieved with the adaptive solver.

Automatic detection of matrix properties improves solving efficiency.

Fallback to SVD for poorly conditioned systems enhances robustness.

Abstract

Computational implementations for solving systems of linear equations often rely on a one-size-fits-all approach based on LU decomposition of dense matrices stored in column-major format. Such solvers are typically implemented with the aid of the xGESV set of functions available in the low-level LAPACK software, with the aim of reducing development time by taking advantage of well-tested routines. However, this straightforward approach does not take into account various matrix properties which can be exploited to reduce the computational effort and/or to increase numerical stability. Furthermore, direct use of LAPACK functions can be error-prone for non-expert users and results in source code that has little resemblance to originating mathematical expressions. We describe an adaptive solver that we have implemented inside recent versions of the high-level Armadillo C++ library for linear algebra. The solver automatically detects several common properties of a given system (banded, triangular, symmetric positive definite), followed by solving the system via mapping to a set of suitable LAPACK functions best matched to each property. The solver also detects poorly conditioned systems and automatically seeks a solution via singular value decomposition as a fallback. We show that the adaptive solver leads to notable speedups, while also freeing the user from using direct calls to cumbersome LAPACK functions.