Mercurial > matrix-functions
comparison toolbox/riemann.m @ 2:c124219d7bfa draft
Re-add the 1995 toolbox after noticing the statement in the ~higham/mctoolbox/ webpage.
author | Antonio Pino Robles <data.script93@gmail.com> |
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date | Thu, 07 May 2015 18:36:24 +0200 |
parents | 8f23314345f4 |
children |
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1:e471a92d17be | 2:c124219d7bfa |
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1 function A = riemann(n) | |
2 %RIEMANN A matrix associated with the Riemann hypothesis. | |
3 % A = RIEMANN(N) is an N-by-N matrix for which the | |
4 % Riemann hypothesis is true if and only if | |
5 % DET(A) = O( N! N^(-1/2+epsilon) ) for every epsilon > 0 | |
6 % (`!' denotes factorial). | |
7 % A = B(2:N+1, 2:N+1), where | |
8 % B(i,j) = i-1 if i divides j and -1 otherwise. | |
9 % Properties include, with M = N+1: | |
10 % Each eigenvalue E(i) satisfies ABS(E(i)) <= M - 1/M. | |
11 % i <= E(i) <= i+1 with at most M-SQRT(M) exceptions. | |
12 % All integers in the interval (M/3, M/2] are eigenvalues. | |
13 % | |
14 % See also REDHEFF. | |
15 | |
16 % Reference: | |
17 % F. Roesler, Riemann's hypothesis as an eigenvalue problem, | |
18 % Linear Algebra and Appl., 81 (1986), pp. 153-198. | |
19 | |
20 n = n+1; | |
21 i = (2:n)'*ones(1,n-1); | |
22 j = i'; | |
23 A = i .* (~rem(j,i)) - ones(n-1); |