Symmetric Matrix Eigenvalue Problems - Numerical Methods in Engineering with MATLAB - page 373

Computation of Eigenvalues

Once the desired eigenvalues are bracketed, they can befound by determining the
roots of P n ( λ ) = 0 with bisection orBrent's method.

eigenvals3

The function eigenvals3 computes the m smallest eigenvalues of a symmetric
tridiagonal matrix with the method of Brent.

function eVals = eigenvals3(C,D,m)
%ComputesthesmallestmeigenvaluesofA=[C\D\C].
% USAGE: eVals = eigenvals3(C,D,m).
% C and D must be delared 'global' in calling program.

eVals = zeros(m,1);
r = eValBrackets(C,D,m); % Bracket eigenvalues
for i=1:m
% Solve | |

for eVal by Brent's method
eVals(i) = brent(@func,r(i),r(i+1));

A - eVal*I

end

functionf=func(eVal);
% Returns

|

A - eVal*I

|

(last element of Sturm seq.)

global C D
p = sturmSeq(C,D,eVal);
f = p(length(p));

EXAMPLE 9.12
Determine the three smallest eigenvalues of the 100

×

100 matrix

⎡
⎣

⎤
⎦

2 − 1 0 ··· 0

− 12 − 1 ··· 0

0 − 12 ··· 0

A =

. . .

. . . . . .

··· −

00 12