# Raf Vandebril

## The Lanczos reduction to semiseparable matrices

The Lanczos tridiagonalization is the classical way to compute an approximation
of dominant singular subspaces. Recently, a similarity reduction of symmetric
matrices in semiseparable form has been introduced [2]. The latter reduction
has the same convergence properties as the Lanczos tridiagonalization for
the "extremal" eigenvalues and it is based on Householder and Givens transformations.

The interesting feature of the latter algorithm is that, if gaps are present in
the spectrum, after few steps the transformed matrix has almost a diagonal block
structure, and the largest eigenvalues of the lefttop block converge to the
largest eigenvalues of the original matrix.

In this talk we consider a new reduction of a matrix in semiseparable form [1].
Similarly to the Lanczos tridiagonalization, the main computation of each step
of the new algorithm is a product of the original matrix by a vector. This
can be done in an efficient way exploiting the structure of the original matrix
(i.e. sparsity, rank structure, low displacement rank structure, ...).
Moreover, during the reduction the already partially computed semiseparable
submatrix is almost block diagonal, and the largest eigenvalues of the lefttop
block converge to the largest eigenvalues of the original matrix.

### References

- Nicola Mastronardi, Mieke Schuermans, Marc Van Barel, Raf Vandebril, and Sabine Van Huffel. A Lanczos-like reduction of symmetric structured matrices into semiseparable ones. Calcolo, 2005. To appear.
- Marc Van Barel, Raf Vandebril, and Nicola Mastronardi. An orthogonal similarity reduction of a matrix into semiseparable form. SIAM Journal on Matrix Analysis and Applications, 27(1):176197, 2005.