Mercurial > matrix-functions
comparison toolbox/minij.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 = minij(n) | |
2 %MINIJ Symmetric positive definite matrix MIN(i,j). | |
3 % A = MINIJ(N) is the N-by-N symmetric positive definite matrix with | |
4 % A(i,j) = MIN(i,j). | |
5 % Properties, variations: | |
6 % A has eigenvalues .25*sec^2(r*PI/(2*N+1)), r=1:N, and the eigenvectors | |
7 % are also known explicitly. | |
8 % INV(A) is tridiagonal: it is minus the second difference matrix | |
9 % except its (N,N) element is 1. | |
10 % 2*A-ONES(N) (Givens' matrix) has tridiagonal inverse and | |
11 % eigenvalues .5*sec^2((2r-1)PI/4N), r=1:N. | |
12 % (N+1)*ONES(N)-A also has a tridiagonal inverse. | |
13 % FLIPUD(TRIW(N,1)) is a square root of A. | |
14 | |
15 % References: | |
16 % J. Fortiana and C. M. Cuadras, A family of matrices, the discretized | |
17 % Brownian bridge, and distance-based regression, Linear Algebra | |
18 % Appl., 264 (1997), 173-188. (For the eigensystem of A.) | |
19 % J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, | |
20 % Birkhauser, Basel, and Academic Press, New York, 1977, p. 158. | |
21 % D.E. Rutherford, Some continuant determinants arising in physics and | |
22 % chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241. | |
23 % (For the eigenvalues of Givens' matrix.) | |
24 | |
25 A = min( ones(n,1)*(1:n), (1:n)'*ones(1,n) ); |