Preconditioned Conjugate Gradient Method for Large-scale Eigenvalue Problem of Quantum Problem:

Convergence Property of Adaptive-shift Preconditioner

Abstract

In order to improve the convergence property of the preconditioned conjugate gradient (PCG) method for solving eigenvalue problems of the Hamiltonian matrix, we propose a new preconditioning method. The preconditioner utilizes not only an approximate eigenvalue which is obtained during the CG iterations but also its residual error. We demonstrate that the PCG method with the new preconditioner can solve the eigenvalue problem for the Hamiltonian matrix several times faster than the PCG method with the conventional preconditioner.