On an inequality of Bogar and Gustafson

Journal of Mathematical Analysis and Applications1461207-21

A note on some improvements of the simultaneous methods for determination of polynomial zeros

AbstractApplying Gauss-Seidel approach to the improvements of two simultaneous methods for finding polynomial zeros, presented in [9], two iterative methods with faster convergence are obtained. The lower bounds of the R-order of convergence for the accelerated methods are given. The improved methods and their accelerated modifications are discussed in view of the convergence order and the number of numerical operations. The considered methods are illustrated numerically in the example of an algebraic equation

Positive Solutions of a Class of Operator Equations

Positive solutions of a class of matrix equations were studied by Bhatia, et al., Bull. London Math. Soc., 32, 214 (2000), SIAM J. Matrix Anal. Appl., 14, 132 (1993) and 27, 103–114 (2005), by Kwong, Linear Algebra Appl., 108, 177–197 (1988), and by Cvetkovi? and Milovanovi?, [Linear Algebra Appl., 429, 2401–2414 (2008)]. Following the idea used in the last paper, we study a class of operator equations in infinite-dimensional spaces and prove that the positivity of solutions can be established for this class of equations under the condition that a certain rational function is positive semidefinite.Додатні розв'язки деякого класу матричних рівнянь було нещодавно вивчено в роботах Бхатіа та ін. [Bull. London Math. Soc. - 2000. - 32. - P. 214-228], [SIAM J. Matrix Anal. and Appl. - 1993. - 14. - P. 132-136; 2005. - 27. -P. 103 -114], Квонга [Linear Algebra and Appl. - 1988. - 108. -P. 177-197] та Цветковича та Міловановича [Linear Algebra and Appl. - 2008. - 429. - P. 2401 -2414]. З використанням ідеї, запропонованої в останній роботі, вивчено клас операторних рівнянь в нескінченновимірних просторах, для якого доведено, що додатність розв'язку можна встановити за умови, що деяка раціональна функція є позитивно напіввизначеною

Gaussian quadratures for oscillatory integrands

AbstractWe consider a Gaussian type quadrature rule for some classes of integrands involving highly oscillatory functions of the form f(x)=f1(x)sinζx+f2(x)cosζx, where f1(x) and f2(x) are smooth, ζ∈R. We find weights σν and nodes xν,ν=1,2,…,n, in a quadrature formula of the form ∫−11f(x)dx≈∑ν=1nσνf(xν) such that it is exact for all polynomials f1(x) and f2(x) from Pn−1. We solve the existence question, partially