A $μ$-mode BLAS approach for multidimensional tensor-structured problems

In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension $d$ by means of tensor product formulas. This is useful, for example, in the context of multivariate interpolation, multidimensional function approximation using pseudospectral expansions and solution of stiff differential equations on tensor product domains. The key point to obtain an efficient-to-implement BLAS formulation consists in the suitable usage of the $\mu$-mode product (also known as tensor-matrix product or mode-$n$ product) and related operations, whose MATLAB/GNU Octave implementations are discussed in the paper as well. We present numerical results on three- and four-dimensional problems from different fields of numerical analysis, which show the effectiveness of the approach.