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comparison toolbox/redheff.m @ 2:c124219d7bfa draft

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Re-add the 1995 toolbox after noticing the statement in the ~higham/mctoolbox/ webpage.
author Antonio Pino Robles <data.script93@gmail.com>
date Thu, 07 May 2015 18:36:24 +0200
parents 8f23314345f4
children
comparison
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1:e471a92d17be 2:c124219d7bfa
1 function A = redheff(n)
2 %REDHEFF A (0,1) matrix of Redheffer associated with the Riemann hypothesis.
3 % A = REDHEFF(N) is an N-by-N matrix of 0s and 1s defined by
4 % A(i,j) = 1 if j = 1 or if i divides j,
5 % A(i,j) = 0 otherwise.
6 % It has N - FLOOR(LOG2(N)) - 1 eigenvalues equal to 1,
7 % a real eigenvalue (the spectral radius) approximately SQRT(N),
8 % a negative eigenvalue approximately -SQRT(N),
9 % and the remaining eigenvalues are provably ``small''.
10 % Barrett and Jarvis (1992) conjecture that
11 % ``the small eigenvalues all lie inside the unit circle
12 % ABS(Z) = 1'',
13 % and a proof of this conjecture, together with a proof that some
14 % eigenvalue tends to zero as N tends to infinity, would yield
15 % a new proof of the prime number theorem.
16 % The Riemann hypothesis is true if and only if
17 % DET(A) = O( N^(1/2+epsilon) ) for every epsilon > 0
18 % (`!' denotes factorial).
19 % See also RIEMANN.
20
21 % Reference:
22 % W.W. Barrett and T.J. Jarvis,
23 % Spectral Properties of a Matrix of Redheffer,
24 % Linear Algebra and Appl., 162 (1992), pp. 673-683.
25
26 i = (1:n)'*ones(1,n);
27 A = ~rem(i',i);
28 A(:,1) = ones(n,1);