Quadrature formulas of the Gaussian type with a diagonal weight matrix for matrix-valued functions
Abstract
This paper in the field of matrix analysis is devoted to the problem of approximate calculation of functional matrix integrals. In particular, questions of constructing and studying quadrature formulas of the highest algebraic degree of accuracy for matrix-valued functions, which would be generalisations of the corresponding (Gaussian type) quadrature rules in the case of scalar functions, are considered. Quadrature formulas of the highest algebraic degree of accuracy of various form are constructed for the approximate integration of matrix-valued functions of the second order and, as a generalisation, of an arbitrary fixed order. Particular cases of quadrature rules are considered, when a scalar or diagonal functional matrix acts as a weight function. The convergence of the proposed quadrature process to the exact value of the matrix integral is investigated. The obtained results are based on the application of certain known facts of the theory of interpolation and approximate integration of scalar functions. The presentation of the material is illustrated by some examples.
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