www.gusucode.com > elmat工具箱matlab源码程序 > elmat/private/redheff.m
function A = redheff(n,classname) %REDHEFF Redheffer matrix. % A = GALLERY('REDHEFF',N) is an N-by-N matrix of 0s and 1s % defined by % A(i,j) = 1, if j = 1 or if i divides j, % = 0 otherwise. % % Properties: % A has N-FLOOR(LOG2(N))-1 eigenvalues equal to 1, % a real eigenvalue (the spectral radius) approximately SQRT(N), % a negative eigenvalue approximately -SQRT(N), % and the remaining eigenvalues are provably "small". % % The Riemann hypothesis is true if and only if % DET(A) = O( N^(1/2+epsilon) ) for every epsilon > 0. % % Note: % Barrett and Jarvis [1] conjecture that "the small eigenvalues % all lie inside the unit circle ABS(Z) = 1". A proof of this % conjecture, together with a proof that some eigenvalue tends to % zero as N tends to infinity, would yield a new proof of the % prime number theorem. % % See also PRIVATE/RIEMANN % Reference: % [1] W. W. Barrett and T. J. Jarvis, Spectral Properties of a % Matrix of Redheffer, Linear Algebra and Appl., % 162 (1992), pp. 673-683. % % Nicholas J. Higham % Copyright 1984-2015 The MathWorks, Inc. i = 1:n; A = cast(rem(i,i')==0,classname); A(:,1) = 1;