Page 147 - Numerical methods for chemical engineering

The QR method for computing all eigenvalues 133

below the principal one. These lower nonzero elements can be removed, and convergence
accelerated for the case of complex eigenvalues, by using double-shift QR iterations.Ifat
iteration k, there appears to be a 2 × 2 diagonal submatrix R jj with approximate eigenvalues
[k] [k]
¯
λ and λ , we perform the two-step iteration
j j
[k]
Q [k] R [k] = A [k] − λ I
j
[k]
[k]
A [k+1] = A Q [k] + λ I
j (3.169)
[k]
Q [k+1] R [k+1] = A [k+1] − λ I
j
[k]
A [k+2] = R [k+1] Q [k+1] + λ I
j
and then resume single-shift iterations. For discussion of more complex, and efficient,
shifting strategies and algorithms for special classes of matrices (e.g. Hermitian), consult
Quateroni et al. (2000) and Stoer & Bulirsch (1993).
Example. QR method for a real 4 × 4 matrix
We now demonstrate the QR method for the real, anti-symmetric matrix
4 0 −1 1
 
0 4 2 −1
 
A =   (3.170)
 1 −2 4 1 
−1 1 −1 4
that has the complex eigenvalues
λ 1,2 = 4 ± 2.8059i λ 3,4 = 4 ± 0.3564i (3.171)

and is generated (along with a working copy) by
A = [40-11;042-1;1-241;-11-14];N= size(A,1);
A w=A;
A single-shift QR iteration is performed by
mu=A w(N,N); [Q,R] = qr(A w - mu*eye(N));
A w = R*Q +mu*eye(N);
[k]
We have the following first few values of A ,
4.0 −1.0 −2.5 −0.3 4.0 −2.7 −0.8 0.0
   
1.0 4.0 0.6 −0.3 2.7 4.0 0.1 0.1
   
   
2.5 −0.6 4.0 −0.0 0.9 −0.1 4.0 −0.3
  →   →
0.3 0.3 0.0 4.0 −0.0 −0.1 0.3 4.0
A [1] A [2]
4.0 −2.8 −0.1 −0.0
 
2.8 4.0 0.0 −0.0
 
 
 0.1 −0.0 4.0 0.4  (3.172)
0.0 0.0 −0.4 4.0
A [3]