--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/toolbox/triw.m	Wed May 06 14:56:53 2015 +0200
@@ -0,0 +1,41 @@
+function t = triw(n, alpha, k)
+%TRIW   Upper triangular matrix discussed by Wilkinson and others.
+%       TRIW(N, ALPHA, K) is the upper triangular matrix with ones on
+%       the diagonal and ALPHAs on the first K >= 0 superdiagonals.
+%       N may be a 2-vector, in which case the matrix is N(1)-by-N(2) and
+%       upper trapezoidal.
+%       Defaults: ALPHA = -1,
+%                 K = N - 1     (full upper triangle).
+%       TRIW(N) is a matrix discussed by Kahan, Golub and Wilkinson.
+%
+%       Ostrowski (1954) shows that
+%         COND(TRIW(N,2)) = COT(PI/(4*N))^2,
+%       and for large ABS(ALPHA),
+%         COND(TRIW(N,ALPHA)) is approximately ABS(ALPHA)^N*SIN(PI/(4*N-2)).
+%
+%       Adding -2^(2-N) to the (N,1) element makes TRIW(N) singular,
+%       as does adding -2^(1-N) to all elements in the first column.
+
+%       References:
+%       G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the
+%          computation of the Jordan canonical form, SIAM Review,
+%          18(4), 1976, pp. 578-619.
+%       W. Kahan, Numerical linear algebra, Canadian Math. Bulletin,
+%          9 (1966), pp. 757-801.
+%       A.M. Ostrowski, On the spectrum of a one-parametric family of
+%          matrices, J. Reine Angew. Math., 193 (3/4), 1954, pp. 143-160.
+%       J.H. Wilkinson, Singular-value decomposition---basic aspects,
+%          in D.A.H. Jacobs, ed., Numerical Software---Needs and Availability,
+%          Academic Press, London, 1978, pp. 109-135.
+
+m = n(1);              % Parameter n specifies dimension: m-by-n.
+n = n(max(size(n)));
+
+if nargin < 3, k = n-1; end
+if nargin < 2, alpha = -1; end
+
+if max(size(alpha)) ~= 1
+   error('Second argument must be a scalar.')
+end 
+
+t = tril( eye(m,n) + alpha*triu(ones(m,n), 1), k);