Features Of Inverse to Block - Tridiagonal Matrices with Zero Leading Block Angular Minors

Аннотация

The article reports about construction of a previously unknown structure and shows the results of the study of direct representations of inverse to block-tridiagonal matrices with zero (or close to zero) leading block- angular minors.The study of the structure of the inverse matrix are related, firstly, to the solution of a system of linear algebraic equations, and secondly, to the inversion of a matrix of the indicated type. The task of inversing a block-tridiagonal type matrix arises while solving the problem of processing and analyzing experimental data of high-energy physics. At the same time, the special-type likelihood function is minimized, the parameters of which are the kinematic parameters of the secondary particles. The experimental data obtained by measurements contain errors. Therefore, stringent requirements are set to numerical methods for the stability of intermediate computer calculations. INTRODUCTION linear algebraic equations. It is known that solutions of poorly conditioned systems of linear algebraic equations with block - tridiagonal matrices arise when This article is devoted to the construction and the following problems are formulated: in the process analysis of the structure of direct representations of of numerical solution of boundary value problems of inverse to block-tridiagonal matrices with zero (or mathematical physics; processing and analysis of close to zero) leading block-angular minors. The experimental data in high-energy physics; modeling study of the properties of the explicit type of inverse and numerical solution of geophysics problems,etc. matrices and knowledge of their matrix structure is (see, for example, [1 - 6]). necessary when constructing direct (non-iteration) So, let 𝐶 be a block-three-diagonal matrix of the methods for solving poorly conditioned systems of general type 𝑞 𝑟 1 2 𝑝 𝑞 𝑟 2 2 3 𝑝 𝑞 𝑟 𝐶 = 3 3 4 , (1) 𝑝 𝑞 𝑟 𝑚−1 𝑚−1 𝑚 [ 𝑝 𝑞 ] 𝑚 𝑚 𝑚 𝑚 𝑚 where {𝑞 } - diagonal elements - blocks of the various dimensions, at {𝑝 } and {𝑟 } - under 𝜉 𝜉 𝜉 𝜉=1 𝜉=2 𝜉= 2 matrix 𝐶, which are generally square matrices of (above) diagonal elements - blocks of the matrix 𝐶, which are generally rectangular matrices, the problems of mathematical physics. The main task in dimensions of which are determined by the their development is to minimize the number of dimensions of the corresponding square matrices arithmetic operations taking into account the specifics {𝑞 𝑎𝑛𝑑 𝑞 } 𝑚 . Tridiagonal and block - tridiagonal of the equations or to ensure reliability, i.e. stability 𝜉−1 𝜉 𝜉=2 to the accumulation of rounding errors when matrices play an independent role in solving many implemented on a computer. problems of computational mathematics and mathematical physics. Indeed, the numerical solution of boundary value problems of mathematical physics