Refined Block Conjugate Gradient Method And Its Application

In many applications such as structural mechanics, electronic physics and biology solving eigenvalue problem with high accuracy and speed is very important. Recently LOBPCG was introduced to solve symmetric eigenvalue problems. This method could compute several eigenvalues simultaneously in an efficient way. However, the residual during the iterative steps is oscillating. In order to make the eigenvectors converge better, a refined method is presented. The new method preserves the advantages of LOBPCG and is able to make the residual monotonically decreasing as eigenvalues converge. And some numerical examples indicates that its performance is as good as that of LOBPCG.The Self Consistent Field (SCF) iteration method is mainly used for solving the non-linear eigenvalue problem and often requires more accurate approximate eigenvectors. The characteristic of the refined method could make the SCF iteration more efficient. And some experiments numerically confirm that.