Numerical Linear Algebra Midterm Exam
Takehome
Exam
Open Notes, Textbook, Homework Solutions Only
Calculators Allowed
Tuesday 19 October, 2010
Question
Points
Points
Possible
Awarded
1. LU
25
2. Structured
30
Factorizations
3. Symmetric Tridiagonal
25
Eigenvalue Problem
4. SVD
30
Total
110
Points
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Problem 1
(25 points)
Let
A
∈
R
n
×
n
be a diagonally dominant matrix with
A
=
LU
.
Suppose you have computed the first
i

1 rows of
L
and the first
i

1 rows of
U
by
reading and processing the first
i

1 rows of
A
, i.e., you have not touched rows
i
to
n
of
A
so the algorithm is a delayed update version.
1.a
.
(15 points)
Derive the
i
th step of the algorithm where you read row
i
of
A
and
compute row
i
of
L
and row
i
of
U
.
1.b
.
(5 points)
Identify the computational primitives used and the level of BLAS in
which they appear.
1.c
.
(5 points)
Suppose you want to add pivoting to guarantee stability and existence
of the factorization for any nonsingular matrix
A
. What form of pivoting can be
introduced into this algorithm?
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 Fall '06
 gallivan
 Linear Algebra, Matrices, Diagonal matrix

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