June 2015
Intermediate to advanced
259 pages
6h 51m
English
Content preview from Introduction to Computational Linear Algebra
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Algorithms for the Eigenvalue Problem 135
the matrix B = U
T
0
AV
0
is upper bidiagonal
B =
α
1
β
1
0 ··· 0
0 α
2
β
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. α
n−1
β
n−1
.
.
. α
n
.
.
. 0
.
.
.
.
.
.
0 0
, (5.44)
and therefore the matrix
T = B
T
B = V
T
0
A
nrm
V
0
, (5.45)
is tridiagonal. From this remark, we deduce that V
0
can be chosen as a com-
bination of Householder transformations as in Section 5.3.1. Let us consider
the technique that reaches that goal. To avoid the computation of A
T
A, the
Householder transformations will be defined on the two sides of A. At the first
step, the Householder transformation H
`
1
is applied on the left side of A so as
to eliminate all the entries of the first column located below the diagonal. At
the second step, the Householder transformation
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