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Complete stagnation of GMRES
, 2003
"... We study problems for which the iterative method GMRES for solving linear systems of equations makes no progress in its initial iterations. Our tool for analysis is a nonlinear system of equations, the stagnation system, that characterizes this behavior. We focus on complete stagnation, for which th ..."
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We study problems for which the iterative method GMRES for solving linear systems of equations makes no progress in its initial iterations. Our tool for analysis is a nonlinear system of equations, the stagnation system, that characterizes this behavior. We focus on complete stagnation, for which there is no progress until the last iteration. We give necessary and sufficient conditions for complete stagnation of systems involving unitary matrices, and show that if a normal matrix completely stagnates then so does an entire family of nonnormal matrices with the same eigenvalues. Finally, we show that there are real matrices for which complete stagnation occurs for certain complex righthand sides but not for real ones.
Preconditioning Strategies for Linear Systems Arising in Tire Design
, 1999
"... In this paper, we consider linear systems arising in static tire equilibrium computation. The heterogeneous material properties, nonlinear constraints, and a 3D finite element formulation make the linear systems arising in tire design difficult to solve by iterative methods. An analysis of matrix ..."
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In this paper, we consider linear systems arising in static tire equilibrium computation. The heterogeneous material properties, nonlinear constraints, and a 3D finite element formulation make the linear systems arising in tire design difficult to solve by iterative methods. An analysis of matrix characteristics attempts to explain this negative effect. This paper focuses on two preconditioning techniques  a variation of an incomplete LU factorization with threshold and a multilevel recursive solver  that are able to improve the convergence of a suitable iterative accelerator. In particular, we compare these techniques and assess their applicability when the linear system difficulty varies for the same class of problems. The effect of altering the values of parameters such as number of fillin elements, block size, and number of levels is considered. 1 Introduction Static equilibrium computation routinely takes place in the tire manufacturing process. Tire stability an...
Lucent Technologies
"... a collection of codes for finding zeros or fixed points of nonlinear systems using globally convergent probabilityone homotopy algorithms. Three qualitatively different algorithms— ordinary differential equation based, normal flow, quasiNewton augmented Jacobian matrix—are provided for tracking ho ..."
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a collection of codes for finding zeros or fixed points of nonlinear systems using globally convergent probabilityone homotopy algorithms. Three qualitatively different algorithms— ordinary differential equation based, normal flow, quasiNewton augmented Jacobian matrix—are provided for tracking homotopy zero curves, as well as separate routines for dense and sparse Jacobian matrices. A high level driver for the special case of polynomial systems is also provided. Changes to HOMPACK include numerous minor improvements, simpler and more elegant interfaces, use of modules, new end games, support for several sparse matrix data structures, and new iterative algorithms for large sparse Jacobian matrices.