On the complex HZ method for PGEP

The paper considers a Jacobi-type method for solving the generalized eigenvalue problem Ax = λBx, 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 Hermitian matrices since it reduces to it when B is diagonal. Originally, it is a two-sided method, but it can be implemented as one-sided method and then it solves the generalized singular value problem. To further enhance its efficiency on contemporary CPU and GPU architectures it can be implemented as a block Jacobi-type method. The one-sided block method has proved to be very efficient and compares favorably to the LAPACK DTGSJA algorithm. There are several open problems related to the original method and more to its one-sided and block versions. The problems refer to the global and asymptotic convergence, high relative accuracy and speed. The aim of this short communication is to briefly describe the element-wise method and to report how well it is understood.