  We discuss the close connection between eigenvalue computation and
optimization using the Newton method and subspace methods. From the connection
we derive a new class of Newton updates. The new update formulation is similar
to the well-known Jacobi-Davidson method. This similarity leads to simplified
versions of the Jacobi-Davidson method and the inverse iteration generalized
Davidson (IIGD) method. We prove that the projection subspace augmented by the
updating direction from each of these methods is able to include the Rayleigh
quotient iteration (RQI) direction. Hence, the locally quadratic (cubic for
normal matrices) convergence rate of the RQI method is retained and
strengthened by the subspace methods. The theory is supported by extensive
numerical results. Preconditioned formulations are also briefly discussed for
large scale eigenvalue problems.
