The role eigenvalues play in forming GMRES residual norms with non-normal matrices

Abstract

In this paper we give explicit expressions for the norms of the residual vectors generated by the GMRES algorithm applied to a non-normal matrix. They involve the right-hand side of the linear system, the eigenvalues, the eigenvectors and, in the non-diagonalizable case, the principal vectors.

They give a complete description of how eigenvalues contribute in forming residual norms and offer insight in what quantities can prevent GMRES from being governed by eigenvalues.

Keywords

• GMRES convergence
• Non-normal matrix
• Eigenvalues
• Residual norms