We revisit the two standard dense methods for matrix Sylvester and Lyapunov equations: the Bartels-Stewart method for ${\mathbf{A}}_1\mathbf{X}+\mathbf{X}{\mathbf{A}}_2+\mathbf{D}=\mathbf{0}$ and Hammarling's method for $\mathbf{A}\mathbf{X}+\mathbf{X}{\mathbf{A}}^T+\mathbf{B}{\mathbf{B}}^T=\mathbf{0}$ with $\mathbf{A}$ stable. We construct three schemes for solving the unitarily reduced quasitriangular systems. We also construct a new rank-1 updating scheme in Hammarling's method. This new scheme is able to accommodate a $\mathbf{B}$ with more columns than rows as well as the usual case of a $\mathbf{B}$ with more rows than columns, while Hammarling's original scheme needs to separate these two cases. We compared all of our schemes with the Matlab Sylvester and Lyapunov solver lyap.m; the results show that our schemes are much more efficient. We also compare our schemes with the Lyapunov solver sllyap in the currently possibly the most efficient control library package SLICOT; numerical results show our scheme to be competitive.

We consider the integral equation arising as a result of heat radiation exchange in both convex and nonconvex enclosures of diffuse grey surfaces. For nonconvex geometries, the visibility function must be taken into consideration. Therefore, a geometrical algorithm has been developed to provide an efficient detection of the shadow zones. For the numerical realization of the Fredholm integral equation, a boundary element method based on Galerkin-Bubnov discretization scheme is implemented. Consequently, multigrid iteration methods, which are closely related to two-grid methods, are used to solve the system of linear equations. To demonstrate the high efficiency of these iterations, we construct some numerical experiments for different enclosure geometries.

The purpose of this paper is to present an algorithm for evaluating Hankel transform of the null and the first kind. The result is the exact analytical representation as the series of the Bessel and Struve functions multiplied by the wavelet coefficients of the input function. Numerical evaluation of the test function with known analytical Hankel transform illustrates the proposed algorithm.