# Recursive blocked algorithms for linear systems with Kronecker product structure

## TL;DR

This paper extends recursive blocked algorithms to higher-dimensional Sylvester-like equations, enabling efficient solutions for PDE discretizations and economic models, outperforming existing methods.

## Contribution

It introduces a novel recursive algorithm that handles higher-dimensional Kronecker-structured equations more efficiently than previous approaches.

## Key findings

- Algorithm outperforms existing Sylvester solvers
- Efficiently solves PDE discretizations with separable coefficients
- Applicable to macroeconomic model approximations

## Abstract

Recursive blocked algorithms have proven to be highly efficient at the numerical solution of the Sylvester matrix equation and its generalizations. In this work, we show that these algorithms extend in a seamless fashion to higher-dimensional variants of generalized Sylvester matrix equations, as they arise from the discretization of PDEs with separable coefficients or the approximation of certain models in macroeconomics. By combining recursions with a mechanism for merging dimensions, an efficient algorithm is derived that outperforms existing approaches based on Sylvester solvers.