Mercurial > matrix-functions

annotate toolbox/minij.m @ 2:c124219d7bfa draft

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
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
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
0
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
1 function A = minij(n)
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
2 %MINIJ Symmetric positive definite matrix MIN(i,j).
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
3 % A = MINIJ(N) is the N-by-N symmetric positive definite matrix with
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
4 % A(i,j) = MIN(i,j).
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
5 % Properties, variations:
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
6 % A has eigenvalues .25*sec^2(r*PI/(2*N+1)), r=1:N, and the eigenvectors
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
7 % are also known explicitly.
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
8 % INV(A) is tridiagonal: it is minus the second difference matrix
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
9 % except its (N,N) element is 1.
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
10 % 2*A-ONES(N) (Givens' matrix) has tridiagonal inverse and
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
11 % eigenvalues .5*sec^2((2r-1)PI/4N), r=1:N.
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
12 % (N+1)*ONES(N)-A also has a tridiagonal inverse.
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
13 % FLIPUD(TRIW(N,1)) is a square root of A.
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
14
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
15 % References:
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
16 % J. Fortiana and C. M. Cuadras, A family of matrices, the discretized
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
17 % Brownian bridge, and distance-based regression, Linear Algebra
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
18 % Appl., 264 (1997), 173-188. (For the eigensystem of A.)
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
19 % J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra,
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
20 % Birkhauser, Basel, and Academic Press, New York, 1977, p. 158.
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
21 % D.E. Rutherford, Some continuant determinants arising in physics and
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
22 % chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241.
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
23 % (For the eigenvalues of Givens' matrix.)
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
24
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff changeset
25 A = min( ones(n,1)*(1:n), (1:n)'*ones(1,n) );