GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Topics in matrix analysis
An analysis of some element-by-element techniques
Computer Methods in Applied Mechanics and Engineering
Iterative methods for solving linear systems
Iterative methods for solving linear systems
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Iterative methods for large, sparse, nonsymmetric systems of linear equations
Iterative methods for large, sparse, nonsymmetric systems of linear equations
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In the early seventies, Fried formulated bounds on the spectrum of assembled Hermitian positive (semi-) definite finite element matrices using the extreme eigenvalues of the element matrices. In this paper we will generalise these results by presenting bounds on the field of values, the numerical radius and on the spectrum of general, possibly complex matrices, for both the standard and the generalised problem. The bounds are cheap to compute, involving operations with element matrices only. We illustrate our results with an example from acoustics involving a complex, non-Hermitian matrix. As an application, we show how our estimates can be used to derive an upper bound on the number of iterations needed to achieve a given residual reduction in the GMRES-algorithm for solving linear systems.