% V03 of generalized_Fibonacci, written by Wei-Cheng Wang. % to demonstrate stability and instability of recurrence formula % this is the case of multiple characteristic root format long; n = 100; % n = 73; a = zeros(1,n); ar = zeros(1,n); a_true = zeros(1,n); % Example in page 33 a(1) = 1/3; a(2) = -1/3; b = 2; c=-1; c1 = 1; c2 = -2/3; for i = 3:n a(i) = b*a(i-1) + c*a(i-2); end % true solution: for i = 1:n a_true(i) = c1 + c2*i; end % double check if the above calculation is correct. check_a1 = a(1)-a_true(1) check_a2 = a(2)-a_true(2) % relative error for a(n) check_an = ( a(n)-a_true(n) ) check_an_rel_err = abs( ( a(n)-a_true(n) )/a_true(n) ) rel_err = abs( (a-a_true)./a_true ); figure(1), plot(1:n,abs(a-a_true),'x'), title('true error') figure(2), plot(1:n,rel_err,'x'), title('realtive error') return