S. P. Kulikov

Questions of optimum convergence of the modified method of simple iteration (MMPI) for the decision of great systems of the linear algebraic equations with complex nonermit matrix are considered. The complex optimum parameter of convergence which allows to achieve convergence MMPI and minimizes spectral radius of a matrix of transition, essentially depends on a configuration of area of an arrangement of a spectrum of this matrix. Algorithms of definition of optimum parameter in cases when the convex environment of a spectrum of an initial matrix of system does not contain its point of singularity - a point 0 of complex plane are developed and proved and coincides with figures of the simple form, such as a circle, a piece, a triangle, a convex polygon. Thus value is the center of the certain circle which are surrounding a convex environment of a spectrum and not containing a point of singularity. The denominator of convergence of a method of optimum simple iteration (МОSI) in the considered cases is certain. МОSI is applied to the decision of the great systems arising at digitization of the integral equation of scattering of electromagnetic waves on locally non-uniform transparent dielectric. In 2D case of polarization the integral operator of a problem is quite continuous and has a discrete spectrum with a unique point of accumulation of a spectrum - p.0. Received as a result of application of a method of the final sums and digitization of the integral operator the matrix is a matrix of transition in a method of simple iteration. In cases of Relay scattering by area of an arrangement of a spectrum the small area near this point is and optimum parameter of convergence is value , where is maximal on the module the isolated own number which can be effectively found by means of an iterative degree method. Similarly optimum parameter is, if the area of scattering has the Relay size only on one coordinate. In case of resonant dispersion (length of a wave is of the order of the size of area) the spectrum of a matrix of transition and an optimum circle can approach rather close to a point of singularity, that worsens convergence МОSI. Value thus is defined by means of algorithms for a triangle or a polygon which tops are a point 0, the points of a spectrum nearest to a point of singularity and greatest on the module own number. It is important, that a spectral picture together with optimum parameter are established with sufficient accuracy at even weak physical accuracy of approximation of a problem in 2-3 units of a settlement grid for length of a wave in the environment and do not change any more for the best approximation in 8-12 and more units. Thus, in considered problems of scattering the optimum parameter is defined for a rare grid and small matrixes of transition, and then used for greater systems with good physical accuracy of approximation of a problem for which definition of a spectral picture would demand is an unreal greater machine resources. In this paper, based on the theory MOSI spectral images and numerical solutions of two-dimensional scalar problems and Rayleigh scattering in the resonance wave lengths are presented. Scalar case allows numerically significant move deeper into the resonance region, even for large values of permittivity medium with losses. Numerical solutions of scalar problems of absorption of a plane wave in biological tissues in microwave range are presented.