Quadratic convergence estimate of scaled iterates by J-symmetric Jacobi method

This paper estimates the quadratic convergence reduction of scaled iterates by J-symmetric Jacobi method [Numer. Math. 64 (1993) 241]. Although, the method is well defined for a general definite pair (H,J), H = HT, J = diag(Im, (In−m), the paper considers the most important case when H is positive definite. In that case the method is an accurate floating point eigensolver for the pair (H,J). As such, it is used in a compound algorithm for accurate floating point computation of eigenvalues and eigenvectors of a non-singular indefinite symmetric matrix. The new result is proved for scaled diagonally dominant matrices in the general case of multiple eigenvalues. It uses Frobenius norm of the off-diagonal part of symmetrically scaled iteration matrix, and a relative gap in the spectrum of (H,J). It can be effectively used in connection with stopping criterion of the method, especially with its one-sided version.