The Condition of Certain Matrices

By reve rsing the usual d irection of appl ication, a co mmon procedure for solv ing integr a l eq ua tion s numeri call y is used to obtain th e asymptotic P-condition numbe rs of two weU-known tes t matrices. Todd [1]1 has recently sugges ted th e matrix AI7 collection of test matrices [2]. The P-co ndition numbers of the matrices are used as a measure of th eir difficulty for num e rical purposes. Where th e co ndi-tion numbers are not explicitly known, the asym ptotic behavior in 11 is give n. Lehmer's matrix A7 I is exceptional in that the correct order in /J is not known_ A si mple id ea will allow us to obtain the asymptotic cond ition number of A 17 and the correc t order for A 7-Hilbert 's first method for integral equation s [3] approximates the e ige nvalues of the kern el K(x, y), o ,s; x, y ,s; 1, by t hose of t he matrix If K is bounded and Riemann integrable, then the eigen-values of th e matrices tend to those of the integral equation as n tends to 00. We reverse this procedure. We wish to es timate the behavior of the eigenvalues of a set of matri ces as n tends to 00. If we can regard them as arising from the application of Hilbert's first method to a fixed ke rnel, then we may hope for an asymptotic res ult. To estimate the largest eigenvalue of A 17 , let us form *This work was SUI)por'!cd by the United States Atomic Energy Commission. Rcp rodu c-I lion in whole or in part is pe rmitt ed for any purpose of the U.S. Government. ] Fi gures in brac ke ts ind ica te the literature references at the end of this paper. We regard them as arising from the approximation of the kernel K(x , y)= Ix-yl. A simple co mputation gives the largest eigenvalue of K as A =! Z-2 I 0 where Zo is the unique real root of coth z = z. This equation has bee n studied [4] and Zo is approximately Zo = 1.9967864. We conclude that the largest eigen-value of A 17 is asymptotically It is easy to es timate the accuracy of the approximation using the bounds of [5] but we s …