Geology Reference
In-Depth Information
where we have abbreviated cosθ as
c
,andsinθ as
s
. Such an orthogonal trans-
formation is generally referred to as a
Givens rotation
. In order to bring the (1,2)
element to zero, we require that
t
12
t
11
+
t
11
t
11
+
s
=−
t
12
,
c
=
t
12
,
(2.235)
d
1
−
σ
2
and
t
12
=
where
t
11
=
d
1
e
1
. Instead of applying the Givens rotation to
the shifted matrix
T
, it is applied directly to the positive, upper bidiagonal matrix
B
by post-multiplication by
P
12
. This transformation generates a non-zero
rogue
element
in the (2,1) position, destroying the upper bidiagonal form of
B
.Bypre-
multiplying by an orthogonal matrix with the form of
P
12
,the(2,1) rogue element
can be brought to zero, generating a new non-zero rogue element in the (1,3) pos-
ition. In turn, this element can be brought to zero by post-multiplying by an ortho-
gonal rotation matrix for the (2,3) axes, producing a new non-zero rogue element
in the (3,2) position. The process of post- and pre-multiplication by orthogonal
rotation matrices is continued until the non-zero rogue elements are chased from
the matrix and it is returned to its original upper bidiagonal form.
The Wilkinson shift is computed from the trailing 2
×
2 submatrix of
B
written as
⎝
⎠
.
qr
0
Z
=
(2.236)
p
The eigenvalues of the matrix
⎝
⎠
,
q
2
qr
Z
T
Z
=
(2.237)
p
2
r
2
qr
+
are the squares of the singular values of
Z
. The characteristic equation for the
eigenvalues of
Z
T
Z
is
p
2
r
2
λ
+
2
q
2
p
2
q
2
λ
−
+
+
=
0.
(2.238)
If σ
min
is the smallest singular value of
Z
and σ
max
is the largest, then the charac-
teristic equation for the eigenvalues of
Z
T
Z
becomes
λ
−
σ
min
λ
−
σ
2
max
σ
2
max
λ
+
σ
2
2
2
min
2
2
max
=
λ
−
+
σ
min
σ
=
0.
(2.239)
Thus,
2
min
2
max
p
2
q
2
r
2
2
2
max
p
2
q
2
σ
+
σ
=
+
+
and σ
min
σ
=
.
(2.240)
Search WWH ::
Custom Search