On the Global Convergence of the Complex HZ Method

Abstract

The paper considers a Jacobi method for solving the generalized eigenvalue problem $Ax=\lambda Bx$, where $A$ and $B$ are complex Hermitian matrices and $B$ is positive definite. The method is a proper generalization of the standard Jacobi method for the Hermitian matrix $A$ to the matrix pair $(A,B)$. The paper derives the method and proves its global convergence under the large class of generalized serial pivot strategies. If both matrices are positive definite, it can be implemented as a one-sided method. It then solves the initial problem as the generalized singular value problem. Its main application is to serve as a kernel algorithm in a block Jacobi method for the same problem with large matrices $A$ and $B$. The block Jacobi methods are methods of choice on contemporary CPU and GPU computing architectures. The proposed algorithm is very efficient on pairs of almost diagonal matrices, and diagonalization of such matrices is the main task of the kernel algorithm. The numerical tests indicate the high relative accuracy of the method on certain pairs of positive definite matrices.