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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...
Algorithm 777: HOMPACK90: A Suite of . . .
, 1997
"... ... 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 trackin ..."
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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.
Scalability analysis of parallel GMRES implementations
"... Applications involving large sparse nonsymmetric linear systems encourage parallel implementations of robust iterative solution methods, such as GMRES(k).Two parallel versions of GMRES(k) based on different data distributions and using Householder reflections in the orthogonalization phase, and vari ..."
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Applications involving large sparse nonsymmetric linear systems encourage parallel implementations of robust iterative solution methods, such as GMRES(k).Two parallel versions of GMRES(k) based on different data distributions and using Householder reflections in the orthogonalization phase, and variations of these which adapt the restart value k, are analyzed with respect to scalability (their ability to maintain fixed efficiency with an increase in problem size and number of processors).A theoretical algorithmmachine model for scalability is derived and validated by experiments on three parallel computers, each with different machine characteristics. 1.
Transposefree Matrix Padé Via Lanczos Method
, 1998
"... A transposefree twosided nonsymmetric Lanczos method is developed for multiple starting vectors on both the left and right. The method is applied to the computation of the matrix Pad'e approximation to a linear dynamical system. The result is a method which can be labeled TransposeFree Matri ..."
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A transposefree twosided nonsymmetric Lanczos method is developed for multiple starting vectors on both the left and right. The method is applied to the computation of the matrix Pad'e approximation to a linear dynamical system. The result is a method which can be labeled TransposeFree Matrix Pad'e Via Lanczos. The method is mathematically equivalent to the twosided methods, but avoids the use of the transpose of the system matrix, and under certain circumstances will actually reduce the total number of matrixvector products needed. The method is illustrated with some numerical examples. Key words: model reduction, Pad'e approximation, MPVL method, transposefree Lanczos method. AMS subject classifications: Primary 65F15; Secondary 65G05. 1 Introduction We consider the task of model reduction via Pad'e approximation on a multiinput multioutput (MIMO) linear dynamical system dx dt = Ax(t) + V w(t); y(t) = U T x(t): where A 2 C N \ThetaN , V 2 C N \Thetam , U 2 C ...