We shall present a new iterative method for finding some extreme eigenvalues of the symmetric definite generalized eigenvalue problem $Ax=\lambda B x$. The method takes a form of inner-outer iterations and is based on the Krylov subspace projection methods but differs from the known methods in that inversion (i.e. solution of shift-and-invert equations) is neither explicitly nor implicitly involved. We shall also present a convergence analysis that leads to some preconditioning transformations to accelerate the convergence. We shall finally describe a program called EIGIFP that has recently been developed to implement this new method.

This is a joint work with Gene H. Golub of Stanford University.