An iterative procedure containing two parameters for linear algebraic systems originating from the domain decomposition technique is proposed. The optimization of the parameters is investigated. A numeric example is given as an illustration.
The Lanczos method for solving systems of linear equations is implemented by using some recurrence relationships between polynomials of a family of formal orthogonal polynomials or between those of two adjacent families of formal orthogonal polynomials. A division by zero can occur in these relations, thus producing a breakdown in the algorithm which has to be stopped. In this paper, three strategies to avoid this drawback are discussed: the MRZ and its variants, the normalized and unnormalized...
In this paper we consider the eigenvalue problem for positive definite symmetric matrices. Convergence properties for the zero shift method and the shift Cholesky method both in restoring and in non restoring version are deduced from the convergence properties of triangular matrices sequences. For general matrices we obtain some results on the convergence speed of the Cholesky method as a function of the chosen shift. These results follow from the absolute convergence of numerical series associated...
The construction of a well-conditioned integral equation for iterative solution of scattering problems with a variable Leontovitch boundary condition is proposed. A suitable parametrix is obtained by using a new unknown and an approximation of the transparency condition. We prove the well-posedness of the equation for any wavenumber. Finally, some numerical comparisons with well-tried method prove the efficiency of the new formulation.
Abelovu cenu získala v roce 2019 matematička Karen Uhlenbecková. Její práce mají důležitý dopad hned na několik oborů matematiky - geometrii, analýzu i matematickou fyziku. Zásadním způsobem ovlivnila moderní pojetí geometrické analýzy. V článku se pomocí relativně jednoduchých příkladů snažíme čtenáře seznámit se dvěma z oblastí, kterými se doposud zabývala. Na závěr též velmi stručně zmiňujeme hlavní výsledky několika jejích prací.
We consider the absolute value equations (AVEs) with a certain tensor product structure. Two aspects of this kind of AVEs are discussed in detail: the solvability and approximate solution. More precisely, first, some sufficient conditions are provided which guarantee the unique solvability of this kind of AVEs. Furthermore, a new iterative method is constructed for solving AVEs and its convergence properties are investigated. The validity of established theoretical results and performance of the...
A two-level algebraic algorithm is introduced and its convergence is proved. The restriction as well as prolongation operators are defined with the help of aggregation classes. Moreover, a particular smoothing operator is defined in an analogical way to accelarate the convergence of the algorithm. A model example is presented in conclusion.
The technique for accelerating the convergence of the algebraic multigrid method is proposed.
The aim of this paper is to discuss the acceleration properties of the hybrid procedure for solving a system of linear equations. These properties are studied in a general case and in two particular cases which are illustrated by numerical examples.