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factored iteratively into the product of a unitary matrix Q and an upper triangular
matrix R . Beginning with the matrix A s at the s th step, it is factored as Q s ·
R s .
Then, in the next step, the matrix A s + 1 is formed by the unitary transformation
Q s ·
Q s ·
=
·
Q s =
Q s ·
·
Q s =
·
A s + 1
A s
R s
R s
Q s .
(2.232)
A detailed description of this algorithm applied to the symmetric tridiagonal eigen-
value problem is given by Stewart (2001, pp. 164-167), including the implement-
ation of the Wilkinson shift of eigenvalues to produce cubic convergence to upper
triangular form, which, in the tridiagonal case, is diagonal. Since unitary trans-
formations preserve the eigenvalues of a matrix, the eigenvalues are then on the
diagonal.
While, once the eigenvalues of B T B are found, the singular values can be found
as their positive square roots, the QR iteration can be applied directly to the matrix
B (Stewart, 2001, pp. 219-225). The first step mimics the application of the QR
algorithm to the bidiagonal matrix B T B . With shift σ, a plane rotation of the (1,2)
axes is made to bring to zero the (1,2) element of the shifted matrix T
2 I .
A plane rotation of the ( x 1 , x 2 ) axes through an angle θ produces new co-ordinates
( x 1 , x 2 )givenby
B T B
=
σ
=
.
x 1
x 2
cosθ sinθ
x 1
x 2
(2.233)
sinθ cosθ
The required transformation of the matrix T is accomplished by post-multiplication
by the orthogonal matrix P 12 ,as TP 12 equal to
t 11
t 12
cs
t 12
t 22
t 23
sc
t 23
t 33
t 34
1
. . . 1
. . .
. . .
. . .
t M 2 M 1
t M 1 M 1
t M 1 M
1
t M 1 M
t MM
·
·
·
+
·
c
t 11
s
t 12
s
t 11
c
t 12
c · t 12 s · t 22
s · t 12 + c · t 22
t 2,3
s · t 23
c · t 23
t 33
t 34
=
,
. . .
. . .
. . .
t M 2 M 1
t M 1 M 1
t M 1 M
t M 1 M
t MM
(2.234)
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