Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8252Research ArticleApplication of Functional Polynomials to Approximation of Matrix-Valued Functional IntegralsAyryanE ALaboratory of Information Technologiesayrjan@jinr.ruMalyutinV BThe National Academy of Sciences of Belarusmalyutin@im.bas-net.byJoint Institute for Nuclear ResearchInstitute of Mathematics150120141434608092016Copyright © 2014,2014The 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 × 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. This method is based on the expansion of functional in a series. 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.functional integralsmatrix-valued integralsfunctional polynomialsapproximation of integralsфункциональные интегралыматричнозначные интегралыфункциональные полиномыаппроксимация интегралов