Page 257 - MATLAB an introduction with applications

242 ——— MATLAB: An Introduction with Applications

The MATLAB program for this presented below:

%Given the diagonals c and d of A = [c\d\c] and the value of l, this function returns
the sturn sequence P (l),P (l),P (l),……..P (l). Note that P (l) = | A −λ 1 |
1
n
0
n
2
A = [2 –1 0 0;–1 2 –2 0;0 –2 2 –1;0 0 –1 2];
d = diag(A)’;
c = [–1 –2 –1];
lam = input(‘Enter guess value lambda\n’);
n = length(d)+1;
p = ones(n,1);
p(2) = d(1)–lam;
for i = 2:n–1
p(i+1) = (d(i)–lam)*p(i)–(c(i–1)^2)*p1(i–1);
end
fprintf(‘sturn sequence p(%f) is\n’,lam);
disp(p);
%%% number of sign changes in the sturn sequence p is
%%% number eigenvalues of matrix A that are smaller than
%%% lam_min
n = length(p);
oldsign = 1;
num_eval = 0;
for i = 2:n
psign = sign(p(i));
if psign = 0 psign = –oldsign;
end
if psign*oldsign <0
num_eval = num_eval+1;
end
oldsign = psign;
end
fprintf(‘Number of eigenvalues less than lamda = %f are %d\n’,lam,num_eval);
Output of the program is as follows (in two runs)
>>Enter guess value lambda
0
sturn sequence p(0.000000) is
1
2
4
0
–4