Geology Reference
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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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