Working with tensor-structured matrices and vectors

A new publication on tensor-structured matrices, with a tentative title “Linear algebra for tensor problems” is added. It is submitted to the special issue of Computing.

This paper is about  structured iterations with matrices of low tensor rank in three dimensions. We show that in three dimensions we can use Tucker decomposition instead of the canonical decomposition without any substantial increase in the computational cost. To achieve this goal we have to compute quite complex six-fold sum, but it is shown that they can be computed with calls to BLAS/LAPACK. Therefore Tucker format is highly recommended for 3-dimensional problems. With a current MATLAB code (will be posted here soon)  it is possible to handle dense matrices (i.e., approximate inversion) on a grid of size 256^3.  In a future research by using additional approximation techniques we will be able to increase this number to at least 1024^3.