caa a22 4500 397572654 CHVBK 20180308164846.0 cr unu---uuuuu 161202e199605 xx s 000 0 eng 10.1093/qjmam/49.2.217 doi (NATIONALLICENCE)oxford-10.1093/qjmam/49.2.217 URSELL F. Department of Mathematics, University of Manchester, Manchester Ml3 9PL INFINITE SYSTEMS OF EQUATIONS. THE EFFECT OF TRUNCATION [Elektronische Daten] [F. URSELL] We consider an infinite system of equations of the form xm+∑n=1∞kmnxn=am,1≤m&lt;∞, or in matrix notation (1+k)x=a 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 S=∑n=1∞bnxn=b′x In practice this system is replaced by the finite truncated system involving N unknowns, ξm(N)+∑n=1Nkmnξn(N)=am,1≤m≤N, or in matrix notation [ (1+k) ]N[ ξ(N) ]N=[ a ]N, with the associated sum S(N)=∑n=1Nbnξn(N)=[ b ]N'[ ξ(N) ]N, 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 S(N)=∑n=1Nbnξn(N)=[ b ]N'[ ξ(N) ]N, 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 xm-∑n=1∞xnmn(m+n+1)=1m2, 1≤m&lt;∞, with the associated sum S=∑ xn/n2 It is shown that Δ(N)∼αN3+βN4+O(1N92), where α and β are certain constants; the bound of Theorem 2.1 gives Δ(N)=O(1N32) The methods used in the present paper are elementary, involving at most Schwarz's Inequality (∑XmYm)2≤(∑X2m)(∑Y2m). © Oxford University Press Articles nationallicence The Quarterly Journal of Mechanics and Applied Mathematics Oxford University Press 49/2(1996-05), 217-233 0033-5614 49:2<217 1996 49 qjmamj https://doi.org/10.1093/qjmam/49.2.217 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1093/qjmam/49.2.217 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- URSELL F. Department of Mathematics, University of Manchester, Manchester Ml3 9PL NATIONALLICENCE 773 0- The Quarterly Journal of Mechanics and Applied Mathematics Oxford University Press 49/2(1996-05), 217-233 0033-5614 49:2<217 1996 49 qjmamj Metadata rights reserved CC BY-NC-4.0 http://creativecommons.org/licenses/by-nc/4.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-oxford