Two Double Recursive Block Macaulay Matrix Algorithms to Solve Multiparameter Eigenvalue Problems

We present two double recursive block Macaulay matrix algorithms to solve multiparameter eigenvalue problems (MEPs). In earlier work, we have developed a non-recursive approach that finds the solutions of an MEP via a multidimensional realization problem in the null space of the block Macaulay matrix constructed from the coefficient matrices of that MEP. However, this approach requires an iterative increase of the degree of the block Macaulay matrix: in order to determine whether the null space contains all the (affine) solutions of the MEP, we need to compute a basis matrix of the null space for every degree and check its dimension or rank structure. In this letter, we employ a recursive/sparse technique to compute a basis matrix of the null space of the block Macaulay matrix and a recursive technique to perform the necessary rank checks. We provide two system identification examples to show our improvements in computation time and memory usage.