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For a large indefinite linear system, there exists the option to directly precondition for the normal equations. Matrix nearness problems are formulated to assess the attractiveness of this alternative. Polynomial preconditioning leads to polynomial approximation problems involving lemniscate-like sets, both in the plane and in $$\,\mathbb {C}^{n \times n}$$ . A natural matrix analytic extension for lemniscates is introduced. Operator theoretically one is concerned with polynomial unitarity and associated factorizations for the inverse. For the speed of convergence and lemniscate asymptotics, the notion of quasilemniscate arises. In the $$L^2$$ -norm algorithms for solving the problem are devised. [ABSTRACT FROM AUTHOR]

SUMMARY We prove that ℝ-linear GMRES for solving a class of ℝ-linear systems is faster than GMRES applied to the related ℂ-linear systems in terms of matrix-vector products. Numerical examples are given to demonstrate the theoretical result. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We discuss upper bounds for the resolvent of an $${\mathbb{R}}$$-linear operator in $${\mathbb{C}^d}$$. [ABSTRACT FROM AUTHOR]

Summary. We study how singular values and approximation numbers by algebraic operators of a bounded linear operator A on a separable Hilbert space H can be related to the speed of convergence of iterative methods. In infinite dimensional H these two quantities lead us to consider those A that can be split as $A=N+K$ with N normal and K compact. Then the spectrum of N, via a classical polynomial approximation problem, bounds below the speed of convergence of GMRES. In finite dimensional H we combine these two approximation numbers to approximation numbers by normal operators of a matrix A that lead us to consider splittings of A as $A=N+F$ with N normal and F of smallest possible rank. We show, analogously to infinite dimensions, that the eigenvalues of N can be used, via a new polynomial approximation problem, in assessing the speed of convergence of GMRES. For upper bounds we obtain estimates that can be combined with the results of [28]. [ABSTRACT FROM AUTHOR]

Standard analysis of multistep methods for ODE's assumes the application of an initialization routine that generates the starting points. Here a k-step method is considered directly as a mapping R→ R. It is shown to approximate a mapping which is expressible directly in terms of the flow of the vector field. Some useful properties of that mapping are shown and for strictly stable methods these are applied to the question of invariant circles near a hyperbolic periodic solution. [ABSTRACT FROM AUTHOR]

We continue our research concerning relations between accuracy and stability of time discretizations. Here we concentrate, rather than on the size and shape of the stability region, on the obtainable accuracy when the number of parameters in the formulas is large. [ABSTRACT FROM AUTHOR]

This paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems. Here the treatment is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general 'method-free' statements are again obtained. More specialized results are additionally given for linear multistep methods and for the Taylor series method. [ABSTRACT FROM AUTHOR]

Stability regions of explicit 'linear' time discretization methods for solving initial value problems are treated. If an integration method needs m function evaluations per time step, then we scale the stability region by dividing by m. We show that the scaled stability region of a method, satisfying some reasonable conditions, cannot be properly contained in the scaled stability region of another method. Bounds for the size of the stability regions for three different purposes are then given: for 'general' nonlinear ordinary differential systems, for systems obtained from parabolic problems and for systems obtained from hyperbolic problems. We also show how these bounds can be approached by high order methods. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of the nonlinear initial value problem by some linear multistep methods. We show that if g is monotone and x(t) is smooth enough at infinity, then the global error tends to a trigonometric polynomial as n h → ∞ with step size h fixed. Often the trigonometric polynomial vanishes identically. [ABSTRACT FROM AUTHOR]

Basic concepts of asymptotic theory for discretization methods have been difined so that convergence is always equivalent to consistency and stanility. The theory is then applied to a class of discretization methods for the initial value problems of ordinary differential equations, the so-called weakly nonlinear methods. [ABSTRACT FROM AUTHOR]

Let A be an m-accretive operator in a Banach space E. Suppose that A 0 is not empty and that both E and E are uniformly convex. We study a general condition on A that guarantees the strong convergence of the semigroup generated by- A and of related implicit and explicit iterative schemes to a zero of A. Rates of convergence are also obtained. In Hilbert space this condition has been recently introduced by A. Pazy. We also establish strong convergence under the assumption that the interior of A 0 is not empty. In Hilbert space this result is due to H. Brezis. [ABSTRACT FROM AUTHOR]

The possibility of balancing the iteration and discretization errors in iterative solution of large systems of initial value problems is discussed. The main result answers the question affirmatively by stating that in the convergence process, in a model situation, the decay rate of iteration errors is essentially independent of the step size refinement process. [ABSTRACT FROM AUTHOR]

The possibility to accelerate the Picard-Lindelöf process by taking linear combinations of the iterates is discussed. The convergence is studied on infinitely long intervals. [ABSTRACT FROM AUTHOR]