--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/toolbox/minij.m	Thu May 07 18:36:24 2015 +0200
@@ -0,0 +1,25 @@
+function A = minij(n)
+%MINIJ   Symmetric positive definite matrix MIN(i,j).
+%        A = MINIJ(N) is the N-by-N symmetric positive definite matrix with
+%        A(i,j) = MIN(i,j).
+%        Properties, variations:
+%        A has eigenvalues .25*sec^2(r*PI/(2*N+1)), r=1:N, and the eigenvectors
+%                    are also known explicitly.
+%        INV(A) is tridiagonal: it is minus the second difference matrix
+%                    except its (N,N) element is 1.
+%        2*A-ONES(N) (Givens' matrix) has tridiagonal inverse and
+%                    eigenvalues .5*sec^2((2r-1)PI/4N), r=1:N.
+%        (N+1)*ONES(N)-A also has a tridiagonal inverse.
+%        FLIPUD(TRIW(N,1)) is a square root of A.
+
+%        References:
+%        J. Fortiana and C. M. Cuadras, A family of matrices, the discretized
+%           Brownian bridge, and distance-based regression, Linear Algebra
+%           Appl., 264 (1997), 173-188.  (For the eigensystem of A.)
+%        J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra,
+%           Birkhauser, Basel, and Academic Press, New York, 1977, p. 158.
+%        D.E. Rutherford, Some continuant determinants arising in physics and
+%           chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241.
+%           (For the eigenvalues of Givens' matrix.)
+
+A = min( ones(n,1)*(1:n), (1:n)'*ones(1,n) );