An inexact Newton method is a generalization of Newton's method for solving F(x) = 0, F:/ -/, in which, at the kth iteration, the step sk from the current approximate solution xk is required to satisfy a condition ]lF(x) + F'(x)s]l _< /]lF(xk)]l for a "forcing term" / [0,1). In typical applications, the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well. Promising choices of the forcing terms arc given, their local convergence properties are analyzed, and their practical performance is shown on a representative set of test problems.

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Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.

The 3-term Lanczos process leads, for a symmetric matrix, to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of symmetric indefinite linear systems, by solving the reduced system in one way or another. This leads to well-known methods: MINRES, GMRES, and SYMMLQ. We will discuss in what way and to what extent these approaches differ in their sensitivity to rounding errors. In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the different methods, and we will not consider the errors in the Lanczos process itself. We will show that the method of solution may lead, under certain circumstances, to large additional errors, that are not corrected by continuing the iteration process. Our findings are supported and illustrated by numerical examples. 1 Introduction We will consider iterative methods for the construction of approximate solutions, starting with...

FPGAs are becoming more and more attractive for high precision scientific computations. One of the main problems in efficient resource utilization is the quadratically growing resource usage of multipliers depending on the operand size. Many research efforts have been devoted to the optimization of individual arithmetic and linear algebra operations. In this paper we take a higher level approach and seek to reduce the intermediate computational precision on the algorithmic level by optimizing the accuracy towards the final result of an algorithm. In our case this is the accurate solution of partial differential equations (PDEs). Using the Poisson Problem as a typical PDE example we show that most intermediate operations can be computed with floats or even smaller formats and only very few operations (e.g. 1%) must be performed in double precision to obtain the same accuracy as a full double precision solver. Thus the FPGA can be configured with many parallel float rather than few resource hungry double operations. To achieve this, we adapt the general concept of mixed precision iterative refinement methods to FPGAs and develop a fully pipelined version of the Conjugate Gradient solver. We combine this solver with different iterative refinement schemes and precision combinations to obtain resource efficient mappings of the pipelined algorithm core onto the FPGA. 1.

. It is well-known that Bi-CG can be adapted so that the operations with A T can be avoided, and hybrid methods can be constructed in which it is attempted to further improve the convergence behaviour. Examples of this are CGS, Bi-CGSTAB, and the more general BiCGstab(`) method. In this paper it is shown that BiCGstab(`) can be implemented in different ways. Each of the suggested approaches has its own advantages and disadvantages. Our implementations allow for combinations of Bi-CG with arbitrary polynomial methods. The choice for a specific implementation can also be made for reasons of numerical stability. This aspect receives much attention. Various effects have been illustrated by numerical examples. Key words. Bi-Conjugate gradients, non-symmetric linear systems, CGS, Bi-CGSTAB, iterative solvers, ORTHODIR, Krylov subspace. AMS subject classification. 65F10. 1. Introduction and background. The Bi-CG algorithm [2, 4] is an iterative solution method for linear systems Ax = b (...

By using a combination of 32-bit and 64-bit floating point arithmetic the performance of many sparse linear algebra algorithms can be significantly enhanced while maintaining the 64-bit accuracy of the resulting solution. These ideas can be applied to sparse multifrontal and supernodal direct techniques and sparse iterative techniques such as Krylov subspace methods. The approach presented here can apply not only to conventional processors but also to exotic technologies such as

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 fill-in 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...

By using a combination of 32-bit and 64-bit floating point arithmetic, the performance of many dense and sparse linear algebra algorithms can be significantly enhanced while maintaining the 64-bit accuracy of the resulting solution. The approach presented here can apply not only to conventional processors but also

The 3-term Lanczos process leads, for a symmetric matrix, to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of linear systems, by solving the reduced system in one way or another. This leads to well-known methods: MINRES (GMRES), CG, CR, and SYMMLQ. We will discuss in what way and to what extent the various approaches are sensitive to rounding errors. In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the different methods (except CR), and we will not consider the errors in the Lanczos process itself. These errors may lead to large perturbations with respect to the exact process, but convergence takes still place. Our attention is focussed to what happens in the solution phase. We will show that the way of solution may lead, under circumstances, to large additional errors, that are not corrected by continuing the iteration process. Our findings are...