% preliminary root search for a free-free beam
betaL= zeros(1,0);
min_root  = zeros(1,0);
%search from beta l =0 to 30;

for x =0:0.01:30,
	
s = sin(x);  c  = cos(x); sh=sinh(x);  ch = cosh(x); 
% characteristic matrix: equation (23) of lecture 12 notes
A  = [
 0  -1   0   1;
-1   0   1   0;
-s  -c  sh  ch;
-c   s  ch  sh];

          [U,D,V] = svd(full(A));
		  val = diag(D(4,4)); % store lowest eigenvalue!

min_root =[min_root  abs(val)];

betaL =[betaL x];

end;

figure(1);
semilogy(betaL, min_root );
xlabel('beta*L');
ylabel('Caracteristics roots  of free-free beam');
title('Search for initial root locations');
axis([0 20 10^(-5) 1.0]);
grid on