Том 41, № 4 (2017)

Algorithms of the Bartels–Stewart type for the numerical solution of Sylvester matrix equations of modest size are modified for the case where the linear operators associated with these equations are normal. The superiority of the modified algorithms over the original ones is illustrated by numerical results.

A three dimensional problem of plane wave diffraction on a plasmonic nanoparticle is considered, with account for the nonlocal effect. A solution is constructed on the basis of a modified computational scheme of the discrete sources method. A numerical study of the nonlocal effect influence on the scattering properties of spherical nanoparticles upon their deformation is conducted.

A new computational algorithm based on a fast way of computing integral operators describing the coagulation process is proposed for a mathematical model of coagulating particles. Using this algorithm, the computational complexity of each timestep of an explicit difference scheme can be substantially reduced. For each step, the complexity of execution is reduced from O(NM²) arithmetic operations to O(NMRlnM), where N is the number of mesh points along the physical coordinates of particles, M is the number of mesh points in a grid corresponding to sizes of coagulating particles, and R is the rank of a matrix corresponding to the values of the function of a coagulation kernel at mesh points. Using this approach, computations can be greatly accelerated, provided that kernel rank R is small.

The Shannon complexity of a function system over a q-element finite field which contains m functions of n variables in the class of polarized polynomial forms is exactly evaluated: L_q^PPF (n,m) = qⁿ for all n ≥ 1, m ≥ 2, and all possible odd q. It has previously been known that L₂^PPF (n,m) = 2ⁿ and L₃^PPF (n,m) = 3ⁿ for all n ≥ 1 and m ≥ 2.