## Krylov methods for tensors

Using Wedderburn rank-reduction formulae (for reference what is that, please read a nice paper by Gene Golub and Moody Chu) , we managed to create several robust Krylov methods for the approximation of three-dimensional tensors. It is a preprint 2010-1 of Institute of Numerical Mathematics. A new general interpretation is given for rank-revealing methods, which use only matrix-by-vector and tensor-by-vector-by-vector products. This seems to be also a new (and simple) derivation of the Lanczos algorithm.

These methods are very crucial for sparse structured tensors (as pioneered by the work of Elden and Savas), but they become critical for the approximation of structured tensor operatiions, for example for the compression of matrix-by-vector product, where both matrix and vector are in the Tucker format. In the sequel that is under preparation, these methods will be extended to TT case, and included in future versions of TT-Toolbox. This will greatly speed up almost all approximate linear algebra in it (by a factor of 10 at least).