Let us consider the problem of computing the smallest eigenvalue $\lambda$ and the corresponding eigenvector $x$ of a real symmetric matrix $A.$ We restrict ourselves to the class of polynomial iterative methods, i. e. methods that generate approximations to $x$ within the Krylov subspaces, and we use the Rayleigh quotient to measure the approximation quality. Under these assumptions, the Lanczos method gives the best possible approximation as it provides the global minimization of the Rayleigh quotient on Krylov subspaces. Is there a possible competitor?

As such, we analyze the local minimization of the Rayleigh quotient on the subspace spanned by the current approximation, the current residual and the previous approximation. This is, in fact, just a trivial particular case of the locally optimal preconditioned conjugate gradient method suggested by the author earlier. The costs per iteration and the memory use here are competitive with those of the Lanczos method. According to preliminary numerical tests, this locally optimal method is capable of capturing the same linear (but not super-linear) convergence speed rate as that of the Lanczos method. It computes an approximation to the eigenvector directly on every iteration and has no numerical stability issues similar to those of the Lanczos method.