The Coefficients of the FOM and GMRES Residual Polynomials

Abstract

In this paper we derive closed-form expressions for the coefficients of the residual and characteristic polynomials in the full orthogonalization method and GMRES iterative Krylov method for solving linear systems with diagonalizable matrices. The coefficients are given as functions of the eigenvalues and eigenvectors of the matrix $A$ and of the right-hand side $b$. These results yield the residual vectors and the explicit solution of the optimization problem $\min_{p\in\pi_k} \Vert p(A)b\Vert$, where $\pi_k$ is the set of polynomials of degree $k$ with a value 1 at the origin. In addition, the Ritz values and harmonic Ritz values can be written explicitly for the first four iterations of the Arnoldi algorithm. Moreover, from the coefficients of the characteristic polynomials, we obtain lower bounds for the distances of the eigenvalues of $A$ to the Ritz and harmonic Ritz values.