Infinite systems of equations. The effect of truncation

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Abstract
We consider an infinite system of equations of the form or in matrix notation where I denotes the infinite unit matrix, and where the elements kmn of the matrix K and the components an of the column vector a are given. With the solution {xn} of this system we associate the sum In practice this system is replaced by the finite truncated system involving N unknowns, or in matrix notation with the associated sum where the symbol [a]N denotes a column vector of dimension N and the symbol [A]N denotes a matrix of dimension N×N. We wish to find how S(N) approaches the limit S, that is, we wish to find the truncation error as a function of the large parameter N. The elements of the inverse matrices (I+K)−1 and ([I+K]N)−1 cannot in general be found explicitly even when the elements of (I+K) are known explicitly. It will be seen (see Theorem 2.1 below) that an explicit bound can always be found for Δ(N) but this bound may not give the true order of magnitude. There are infinite systems for which the form of the asymptotic expansion of the components {xM} of the infinite-dimensional solution vector x can readily be found for large suffixes M. It is shown that for such systems this expansion for xM can be used to improve the bound for Δ(M), and in suitable cases to find the form of an asymptotic expression for Δ(N), through not necessarily the explicit coefficients. The theory is illustrated by means of the example with the associated sum It is shown that where α and β are certain constants; the bound of Theorem 2.1 gives The methods used in the present paper are elementary, involving at most Schwarz's Inequality (∑XmYm)2≤(∑X2m)(∑Y2m).

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