O. Nevanlinna - Academia.edu (original) (raw)

Papers by O. Nevanlinna

Linear Algebra and its Applications, 1989

Consider the iteration r k+l=~k+H(b-A~k) for solving Ax=b (A is non nonsingular). We discuss rank... more Consider the iteration r k+l=~k+H(b-A~k) for solving Ax=b (A is non nonsingular). We discuss rank-one updates to improve H as an approximation to A-' during the iteration. The update kills and reduces singular values of I-AH and thus speeds up the convergence. The algorithm terminates after at most n sweeps, and if all n sweeps are needed, then A-' has been computed. 1. DERIVATION OF THE SCHEME In this note we propose an acceleration scheme for iteration methods for solving linear systems of equations. Let A be a nonsingular n X n matrix. Assume given a nonsingular H which approximates the inverse of A. Then the usual iteration for solving Ax = b (1.1) can be written as (1.2a) where r, = b-Ax,.

In the paper by U. Ascher, [1], there is an empirical observat ion of high frequency oscillations... more In the paper by U. Ascher, [1], there is an empirical observat ion of high frequency oscillations, which start to appear in the long-term, when solving the one dimensional cubic non -linear Schrödinger equation with Strang splitting and when the space variable is discretized with the midpoint met hod, suggesting that choosing the time step k be smaller thanh2, the space step squared, prevented oscillations from emerg ing. In this work we provide theoretical support for this evidence and derive it by using wave train analysis. The non-linear Schrödinger equation has infinitely many conservation laws, but the numerical method used here c onserves only thel2-norm, and is symplectic. The Hamiltonian is not preserved by the method and the numerical examples show that the Hamiltonian can be used as an indicator when the high frequency oscillations start to eme rge.

Mathematical Proceedings of the Royal Irish Academy, 2009

We study the relation between the growth of sequences T n and (n + 1)(I − T)T n for operators T ∈... more We study the relation between the growth of sequences T n and (n + 1)(I − T)T n for operators T ∈ L(X) satisfying weak variants of the Ritt resolvent condition (λ − T) −1 ≤ C |λ−1| for various sets of |λ| > 1.

Numerische Mathematik, 1982

This paper continues earlier work by the same authors concerning the shape and size of the stabil... more 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.

In multicentric calculus one takes a polynomial p with distinct roots as a new variable and repre... more In multicentric calculus one takes a polynomial p with distinct roots as a new variable and represents complex valued functions by C-valued functions, where d is the degree of p. An application is e.g. the possibility to represent a piecewise constant holomorphic function as a convergent power series, simultaneously in all components of |p(z)| ≤ ρ. In this paper we study the necessary modifications needed, if we take a rational function r = p/q as the new variable instead. This allows to consider functions defined in neighborhoods of any compact set as opposed to the polynomial case where the domains |p(z)| ≤ ρ are always polynomially convex. Two applications are formulated. One giving a convergent power series expression for Sylvester equations AX − XB = C in the general case of A,B being bounded operators in Banach spaces with distinct spectra. The other application formulates a K-spectral result for bounded operators in Hilbert spaces.

Operators and Matrices, 2019

Sylvester equations AX − XB = C have unique solutions for all C when the spectra of A and B are d... more Sylvester equations AX − XB = C have unique solutions for all C when the spectra of A and B are disjoint. Here A and B are bounded operators in Banach spaces. We discuss the existence of polynomials p such that the spectra of p(A) and p(B) are well separated, either inside and outside of a circle or separated into different half planes. Much of the discussion is based on the following inclusion sets for the spectrum: Vp(T) = {λ ∈ C : |p(λ)| ≤ p(T) } where T is a bounded operator. We also give an explicit series expansion for the solution in terms of p(M), where M = A C B , in the case where the spectra of A and B lie in different components of Vp(M) .

We give an iteration scheme for finding zeros of maximal monotone operators in Hilbert spaces. We... more We give an iteration scheme for finding zeros of maximal monotone operators in Hilbert spaces. We assume that the operator is defined in the whole space. The iterates converge strongly to a solution if there exists any, otherwise they tend to infinity. As an application we get a strongly convergent minimization scheme for convex functionals in Hilbert spaces.

Computational Methods and Function Theory, 2011

We show how for any bounded operator or an element of a Banach algebra one can construct a practi... more We show how for any bounded operator or an element of a Banach algebra one can construct a practical power series calculus.

Banach Center Publications, 2017

We display methods that allow for computations of spectra, pseudospectra and resolvents of linear... more We display methods that allow for computations of spectra, pseudospectra and resolvents of linear operators on Hilbert spaces and also elements in unital Banach algebras. The paper considers two different approaches, namely, pseudospectral techniques and polynomial numerical hull theory. The former is used for Hilbert space operators whereas the latter can handle the general case of elements in a Banach algebra. This approach leads to multicentric holomorphic calculus. We also discuss some new types of pseudospectra and the recently defined Solvability Complexity Index.

A the resolvent operator is defined outside the spectrum of A while the cosolvent operator is def... more A the resolvent operator is defined outside the spectrum of A while the cosolvent operator is defined outside the proper values of A. In this paper these two functions are studied. Series expansions are given. A new characterization for the eigenvalues of real matrices is obtained. The cosolvent operator is used to define and analyze analytic functions of A. An application of this leads to a decomposition of R-linear operators. Classes of structured R-linear operators are considered.

Banach Center Publications, 1997

This paper addresses and establishes some of the fundamental barriers in the theory of computatio... more This paper addresses and establishes some of the fundamental barriers in the theory of computations and finally settles the long standing computational spectral problem. Due to the barriers presented in this paper, there are many problems, some of them at the heart of computational theory, that do not fit into the classical frameworks of complexity theory. Hence, we are in need for a new extended theory of complexity, capable of handling these new issues. Such a theory is presented in this paper. Many computational problems can be solved as follows: a sequence of approximations is created by an algorithm, and the solution to the problem is the limit of this sequence (think about computing eigenvalues of a matrix for example). However, as we demonstrate, for several basic problems in computations (computing spectra of infinite dimensional operators, solutions to linear equations or roots of polynomials using rational maps) such a procedure based on one limit is impossible. Yet, one can compute solutions to these problems, but only by using several limits. This may come as a surprise, however, this touches onto the definite boundaries of computational mathematics. To analyze this phenomenon we use the Solvability Complexity Index (SCI). The SCI is the smallest number of limits needed in order to compute a desired quantity. In several cases (spectral problems, inverse problems) we provide sharp results on the SCI, thus we establish the absolute barriers for what can be achieved computationally. For example, we show that the SCI of spectra and essential spectra of infinite matrices is equal to three, and that the SCI of spectra of self-adjoint infinite matrices is equal to two, thus providing the lower bound barriers and the first algorithms to compute such spectra in two and three limits. This finally settles the long standing computational spectral problem. We also show that the SCI of solutions to infinite linear systems is two. Moreover, we establish barriers on error control. We prove that no algorithm can provide error control on the computational spectral problem or solutions to infinite-dimensional linear systems. In particular, one can get arbitrarily close to the solution, but never knowing when one is "epsilon" away. This is universal for all algorithms regardless of operations allowed. In addition, we provide bounds for the SCI of spectra of classes of Schrödinger operators, thus we affirmatively answer the long standing question on whether or not these spectra can actually be computed. Finally, we show how the SCI provides a natural framework for understanding barriers in computations. It has a direct link to the Arithmetical Hierarchy, and in particular, we demonstrate how the impossibility result of McMullen on polynomial root finding with rational maps in one limit, and the framework of Doyle and McMullen on solving the quintic in several limits, can be put in the SCI framework.

Annales Academiae Scientiarum Fennicae Mathematica, 2000

We discuss two characteristic functions, T ∞ and T 1 , to measure the growth of operator valued m... more We discuss two characteristic functions, T ∞ and T 1 , to measure the growth of operator valued meromorphic functions. The smaller one, T ∞ , is based on the operator norm, while T 1 works within finitely trace class meromorphic functions and is preserved in inversion. Applications to perturbation analysis of linear operators are given.

This paper establishes some of the fundamental barriers in the theory of computations and finally... more This paper establishes some of the fundamental barriers in the theory of computations and finally settles the long standing computational spectral problem. Due to these barriers, there are problems at the heart of computational theory that do not fit into classical complexity theory. Many computational problems can be solved as follows: a sequence of approximations is created by an algorithm, and the solution to the problem is the limit of this sequence. However, as we demonstrate, for several basic problems in computations (computing spectra of operators, inverse problems or roots of polynomials using rational maps) such a procedure based on one limit is impossible. Yet, one can compute solutions to these problems, but only by using several limits. This may come as a surprise, however, this touches onto the boundaries of computational mathematics. To analyze this phenomenon we use the Solvability Complexity Index (SCI). The SCI is the smallest number of limits needed in the computa...

We develop a microspectral theory for quasinilpotent linear operators Q (i.e., those with σ(Q) = ... more We develop a microspectral theory for quasinilpotent linear operators Q (i.e., those with σ(Q) = {0}) in a Banach space. When such Q is not compact, normal, or nilpotent, the classical spectral theory gives little information, and a somewhat deeper structure can be recovered from microspectral sets in C. Such sets describe, e.g., semigroup generation, resolvent properties, power boundedness as well as Tauberian properties associated to zQ for z ∈ C.

In multicentric holomorphic calculus one represents the function φ using a new polynomial variabl... more In multicentric holomorphic calculus one represents the function φ using a new polynomial variable w = p(z) in such a way that when evaluated at the operator p(A) is small in norm. Here it is assumed that p has distinct roots. In this paper we discuss two related problems, separating a compact set, such as the spectrum, into different components by a polynomial lemniscate, and then applying the calculus for computation and estimation of the Riesz spectral projection. It may then be desirable to move to using p(z)^n as a new variable and we develop the necessary modifications to incorporate the multiplicities in the roots.

1988., IEEE International Symposium on Circuits and Systems

Work on establishing a framework for discussing the convergence of waveform-relaxation-type proce... more Work on establishing a framework for discussing the convergence of waveform-relaxation-type processes is surveyed. Methods for accelerating the iteration are examined, with the discussion limited to the linear autonomous problem. Consideration is given to continuous iteration, discretized iteration, finite windows, and differential/algebraic equation systems

Studia Mathematica, 2001

We discuss the relation between the growth of the resolvent near the unit circle and bounds for t... more We discuss the relation between the growth of the resolvent near the unit circle and bounds for the powers of the operator. Resolvent conditions like those of Ritt and Kreiss are combined with growth conditions measuring the resolvent as a meromorphic function. 0. Introduction. In this paper we discuss powers of bounded operators whose spectrum lies in the closed unit disc. The general theme is to relate growth conditions on the resolvent near the unit disc to bounds for the powers and their differences. Much of my present interest in these questions originated from a question of J. Zemánek who asked whether there are quasinilpotent operators Q such that A = 1 + Q would satisfy the Ritt resolvent condition, i.e. We should also mention that Strikwerda and Wade [SW] have shown

Perspectives in Operator Theory, 2007