# The use of lower semiseparable matrices to solve generalized eigenvalue problems

## Yvette Vanberghen

Lower semiseparable matrices have already been used as an alternative to Hessenberg matrices for computing the eigenvalues [1].

In this talk we will
discuss the use of lower semiseparable matrices when solving generalized
eigenvalue problems. Given a matrix pair *(A, B)*, the matrix A is classically
reduced to Hessenberg form by unitary transformations, while the matrix B is
reduced to upper triangular form. As an alternative, we will reduce the matrix A
into lower semiseparable form. Furthermore we will design a QZ-step that
preserves the (lower semiseparable, upper triangular) structure. The algorithms
will be illustrated with some numerical examples.

### References

- R. Vandebril. Semiseparable matrices and the symmetric eigenvalue problem. PhD thesis, Dept. of Computer Science, K.U.Leuven, Celestijnenlaan 200A, 3000 Leuven, May 2004.