A process of searching on the sphere for the best (in a sense) cubature formulas that are invariant under the transformations of the icosahedral rotation group is described. The parameters of the best cubature formulas of this symmetry type up to the 30th order of accuracy are given to 16 significant digits. A table which contains the main characteristics of all the best to date cubature formulas of the icosahedral rotation group up to the 79th order of accuracy is given.
A review of the relatively little-known matrix class, called coninvolutions, is given. The properties of these matrices are compared with those of the well studied involutory matrices or, briefly, involutions.
In this paper, iterative stochastic simulation algorithms for the Lamé equation describing the displacements of an isotropic elastic body are constructed. Three different stochastic methods are proposed: the first one is based on a global algorithm of random walk on spheres to compute the solution and its derivatives for an anisotropic diffusion equation. It does not use grids and does not require large amounts of RAM. The second method is based on a randomized algorithm for solving large systems of linear equations and requires the introduction of a grid. The third method is also grid-based and uses a random walk algorithm. All three methods implement an iterative process, at each step of which anisotropic diffusion equations are solved. The paper provides a comparative analysis of the proposed methods and discusses the limits of applicability of each of them.
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