function [a,b] = bkfchsg(x,j,Theta)

%
%    [a,b] = bkfchsg(x,j,Theta)
%
%  find the endpoints of the n-j disjoint open intervals with total length
%  j/(n-1)*diam(Theta) contained within conv(Theta) mentioned in the paper of
%  Shadrin. He predicts the Birkhoff spline bkfspmak(x,j,Theta) has exactly one
%  zero when x is in one of these intervals and otherwise is of constant sign.
%  This behaviour is indeed observed when using bkfplt to view the spline.
%
%  S. Waldron October 1994
%

Theta=sort(Theta); n=length(Theta);

p=poly(diag(Theta(2:n)));
for i=1:j, p=polyder(p); end
a=sort(roots(p))';
a=[min(Theta),a];  % the left endpoints of the intervals

p=poly(diag(Theta(1:n-1)));
for i=1:j, p=polyder(p); end
b=sort(roots(p))';
b=[b,max(Theta)];  % the right endpoints of the intervals