On the Semidefinite B-Arnoldi Method

The B-Arnoldi method is a variant of the ordinary Arnoldi method in which orthogonalization is done with respect to the inner product generated by a positive definite matrix $B$. It arises in connection with the generalized eigenvalue problem $Ax = \lambda Bx$. When $B$ is semidefinite, the algorithm can proceed formally, with “orthogonalization” taking place in the semi-inner product generated by $B$. However, it has been observed that components of the Arnoldi vectors lying in the null space of $B$ can grow rapidly. In this paper we examine the source and consequences of this growth.