Geology Reference
In-Depth Information
In general, after the call to the subroutine BIDIAG, we have the real, positive,
upper bidiagonal matrix
B
and the accumulated unitary matrices
U
H
and
V
from
the transformation of
C
,
⎝
⎠
d
1
e
1
d
2
e
2
.
.
.
.
.
.
d
M
−
1
U
H
B
=
·
C
·
V
=
.
(2.225)
e
M
−
1
d
M
The product
⎝
⎠
d
1
d
1
e
1
d
1
e
1
d
2
+
e
1
d
2
e
2
.
.
.
.
.
.
.
.
.
d
M
−
2
e
M
−
2
d
2
M
−
1
+
B
T
B
=
,
(2.226)
e
2
M
−
2
d
M
−
1
e
M
−
1
d
2
M
+
e
2
M
−
1
d
M
−
1
e
M
−
1
is then a real, symmetric tridiagonal matrix. The singular value decomposition of
B
itself leads to the representation
Y
T
B
=
X
·
W
·
,
(2.227)
where
X
and
Y
are orthogonal matrices with column vectors
x
1
,
x
2
,...,
x
M
and
y
1
,
y
2
,...,
y
M
, respectively.
W
is again the diagonal matrix with the singular val-
ues
s
1
,
s
2
,...,
s
M
down its diagonal, appearing in the original factorisation (2.199).
Multiplying (2.227) on the right by
Y
yields the relation
B
y
i
=
s
i
x
i
.
(2.228)
Taking the transpose of (2.227) gives
B
T
=
Y
·
W
·
X
T
,
(2.229)
and multiplying this relation on the right by
X
yields
B
T
x
i
=
s
i
y
i
.
(2.230)
Multiplying (2.228) on the left by
B
T
gives
B
T
B
y
i
=
s
i
B
T
x
i
=
s
i
y
i
.
(2.231)
The eigenvalues of the tridiagonal matrix
B
T
B
are then the squares of the singular
values of
B
and
C
.
The calculation of the eigenvalues of a symmetric tridiagonal matrix is a classical
problem solved by the QR algorithm (Wilkinson, 1968), in which the matrix is
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