Homework 6 Numerical Linear Algebra 1 Fall 2010
Solutions will be posted on Monday, 11/22/10
Problem 6.1
Define the random matrix
A
∈
R
50
×
10
via
Udiag
(1
,
10

1
, . . . ,
10

9
)
V
T
where
U
∈
R
50
×
50
and
V
∈
R
10
×
10
are random orthogonal matrices. The singular values of
A
are therefore 1
,
10

1
, . . . ,
10

9
and the condition number is
κ
(
A
)
2
≈
10
9
.
Let
A
k
be the matrix consisting of the first
k
columns of
A
and let
κ
(
A
k
)
2
,F
be the condi
tion number of
A
k
using either the matrix 2norm or the matrix Frobenius norm. Implement
both Classical and Modified GramSchmidt and assess their relative loss of orthogonality
over the various ranges of columns by evaluating for each
k κ
(
A
k
),
I
k

Q
T
k
Q
k
2
, and
I
k

V
T
k
V
k
2
, where
Q
k
is computed via classical GramSchmidt and
V
k
is computed by
modified GramSchmidt. Of course, you should examine these values for several samples of
U
and
V
. Does the loss of orthogonality for classical occur more rapidly and severely when
compared to the modified algorithm? Is the loss of orthogonality for the modified algorithm
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 Fall '06
 gallivan
 Linear Algebra, projection method, GMRES, general projection framework, Hessenberg matrix Hi

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