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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 \cite{; ; ; ves-92}; ; ; . Although, the method is well defined for a general definite pair $(H, J)$, $H=H^T$, $J=\diag (I_m , I_{; ; ; n-m}; ; ; )$, 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 nonsingular 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.

Jacobi method; J-symmetric matrix; quadratic convergence; scaled iterates

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