Homework 4 Foundations of Computational Math 1 Fall
2011
The solutions will be posted on Friday, 10/7/11
Problem 4.1
Recall that an elementary reflector has the form
Q
=
I
+
αxx
T
∈
R
n
×
n
with
bardbl
x
bardbl
2
negationslash
= 0.
4.1.a
. Show that
Q
is orthogonal if and only if
α
=
−
2
x
T
x
or
α
= 0
4.1.b
. Given
v
∈
R
n
, let
γ
=
±bardbl
v
bardbl
and
x
=
v
+
γe
1
. Assuming that
x
negationslash
=
v
show that
x
T
x
x
T
v
= 2
4.1.c
. Using the definitions and results above show that
Qv
=
−
γe
1
Problem 4.2
4.2.a
This part of the problem concerns the computational complexity question of operation count.
For both
LU
factorization and Householder reflectorbased orthogonal factorization, we
have used elementary transformations,
T
i
, that can be characterized as rank1 updates to
the identity matrix, i.e.,
T
i
=
I
+
x
i
y
T
i
,
x
i
∈
R
n
and
y
i
∈
R
n
Gauss transforms and Householder reflectors differ in the definitions of the vectors
x
i
and
y
i
. Maintaining computational efficiency in terms of a reasonable operation count usually
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 Fall '09
 gallivan
 Multiplication, Rank, Computational complexity theory, Invertible matrix, computational complexity question

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