Igor Moret

An interpolatory approximation of the matrix exponential based on Faber polynomials

In this paper we introduce a method for the approximation of the matrix exponential obtained by interpolation in zeros of Faber polynomials. In particular, we relate this computation to the solution of linear IVPs. Numerical examples arising from practical problems are examined.

A note on Newton type iterative methods

Some particular rates of convergence, in the sense of Potra and Pták [4], are related to Newton type iterative methods which solve nonlinear operator equations in Banach spaces. This allows to obtain convergence conditions and a posteriori error estimates at the same time. The applicability of the estimates thus found is studied and their behaviour illustrated by numerical examples.ZusammenfassungEs wird ein Verfahren vorgestellt, das zugleich Konvergenzbedingungen und Fehlerschranken bei der Anwendung von Iterationsverfahren der Formxn+1=xn−A(xn)−1F(xn) zur Lösung nichlinearer GleichungenF(x)=0 in Banachräumen zu bestimmen gestattet. Die Anwendung der Abschätzungen wird besprochen und an numerischen Beispielen erläutert.

THE COMPUTATION OF FUNCTIONS OF MATRICES BY TRUNCATED FABER SERIES

In this paper we consider a method based on Faber polynomials for the approximation of functions of real nonsymmetric matrices. Particular attention is devoted to some functions that occur in practical problem, such as exp (z), . Finally we give some numerical results on a test matrix arising from the discretization of a second order partial differential operator.

On the Convergence of Krylov Subspace Methods for Matrix Mittag-Leffler Functions

In this paper we analyze the convergence of some commonly used Krylov subspace methods for computing the action of matrix Mittag-Leffler functions. As is well known, such functions find application in the solution of fractional differential equations. We illustrate the theoretical results by some numerical experiments.

A kantorovich-type theorem for inexact newton methods

Under Kantorovich-type assumptions, a general convergence theorem for inexact Newton methods (i.e., iterative procedures in which the Newton equations are solved approximately) is given. The results cover several situations already considered in the literature.

Solving the time-fractional Schrödinger equation by Krylov projection methods

The time-fractional Schrodinger equation is a fundamental topic in physics and its numerical solution is still an open problem. Here we start from the possibility to express its solution by means of the Mittag-Leffler function; then we analyze some approaches based on the Krylov projection methods to approximate this function; their convergence properties are discussed, together with related issues. Numerical tests are presented to confirm the strength of the approach under investigation.

Rational Lanczos approximations to the matrix square root and related functions

We consider restricted rational Lanczos approximations to matrix functions representable by some integral forms. A convergence analysis that stresses the effectiveness of the proposed method is developed. Error estimates are derived. Numerical experiments are presented. Copyright

The restarted shift-and-invert Krylov method for matrix functions

SUMMARY In this paper, the numerical evaluation of matrix functions expressed in partial fraction form is addressed. The shift-and-invert Krylov method is analyzed, with special attention to error estimates. Such estimates give insights into the selection of the shift parameter and lead to a simple and effective restart procedure. Applications to the class of Mittag–Leffler functions are presented. Copyright

Interpolating functions of matrices on zeros of quasi‐kernel polynomials

SUMMARY The paper deals with Krylov methods for approximating functions of matrices via interpolation. In this frame residual smoothing techniques based on quasi-kernel polynomials are considered. Theoretical

A quasi-Newton method for solving fixed point problems in Hilbert spaces

SummaryWe study the convergence properties of a projective variant of Newtons method for the approximation of fixed points of completely continuous operators in Hilbert spaces. We then discuss applications to nonlinear integral equations and we produce some numerical examples.