On the global convergence of the block Jacobi method for the positive definite generalized eigenvalue problem

Abstract

The paper proves the global convergence of a general block Jacobi method for the generalized eigenvalue problem $$\mathbf {A}x=\lambda \mathbf {B}x$$ with symmetric matrices $$\mathbf {A}$$ , $$\mathbf {B}$$ such that $$\mathbf {B}$$ is positive definite. The proof is made for a large class of generalized serial strategies that includes important serial and parallel strategies. The sequence of matrix pairs generated by the block method converges to $$(\varvec{\varLambda } , \mathbf {I})$$ where $$\varvec{\varLambda }$$ is a diagonal matrix of the eigenvalues of the initial matrix pair $$(\mathbf {A},\mathbf {B})$$ and $$\mathbf {I}$$ is the identity matrix. First, the convergence to diagonal form is proved. After that several conditions are imposed to ensure the global convergence of the block method.