The Sherman–Morrison–Woodbury formula for generalized linear matrix equations and applications

We discuss the use of a matrix-oriented approach for numerically solving the dense matrix equation AX + XAT + M1XN1 + … + MℓXNℓ = F, with ℓ ≥ 1, and Mi, Ni, i = 1, …, ℓ of low rank. The approach relies on the Sherman–Morrison–Woodbury formula formally defined in the vectorized form of the problem, but applied in the matrix setting. This allows one to solve medium size dense problems with computational costs and memory requirements dramatically lower than with a Kronecker formulation. Application problems leading to medium size equations of this form are illustrated and the performance of the matrix-oriented method is reported. The application of the procedure as the core step in the solution of the large-scale problem is also shown. In addition, a new explicit method for linear tensor equations is proposed, that uses the discussed matrix equation procedure as a key building block.