Математическое моделирование UDC 517.987.4+519.6 MSC 28C20, 46G10, 46G12, 46T12, 81S40 Application of Functional Polynomials to Approximation of Matrix-Valued Functional Integrals E. A. Ayryan∗, V. B. Malyutin† ∗ Laboratory of Information Technologies Joint Institute for Nuclear Research 6, Joliot-Curie str., Dubna, Moscow region, Russia, 141980 † Institute of Mathematics, The National Academy of Sciences of Belarus, 11, Surganov str., Minsk, Belarus, 220072 The matrix-valued functional integrals, generated by solutions of the Dirac equation are considered. <...> These integrals are defined on the one-dimensional continuous path x : |s, t|→R and take values in the space of complex d Ч d matrices. <...> Matrix-valued integrals are widely used in relativistic quantum mechanics for investigation of particle in electromagnetic field. <...> Namely integrals are applied to represent the fundamental solution of the Cauchy problem for the Dirac equation. <...> The method of approximate evaluation of matrix-valued integrals is proposed. <...> Terms of a series have the form of a product of linear functionals with increasing total power. <...> Taking a finite number of terms in the series and evaluating functional integrals of a product of linear functionals we obtain approximate value of the matrix-valued functional integral. <...> Proposed method can be used for a wide class of integrals because the series converges for a large class of functionals. <...> Application of the suggested method in the case of small and large parameters included in the integral is considered. <...> Key words and phrases: functional integrals, matrix-valued integrals, functional polynomials, approximation of integrals. 1. <...> Another approach to evaluation of functional integrals is construction of approximate formulas that are exact for the class of functional polynomials given degree [1–3]. <...> These integrals are widely used in relativistic quantum mechanics for investigation of particle in electromagnetic field [4, 5]. <...> Terms of a series have the form of a product of linear functionals with increasing total power. <...> In case of Gaussian integrals the series of integrals of the product of linear functionals converges for a narrow class of functionals. <...> In case of matrix-valued integrals the series converges for a wide class of functionals. <...> Work supported by Belarusian republican foundation for fundamental research (grant No F12D001) and BRFPR-JINR No 198. 44 ∫ Bulletin of PFUR. <...> We suppose that α2 = β2 = E <...>