MATH 417: Numerical Analysis
Instructor: Prof. Wolfgang Bangerth
Teaching Assistants: Dukjin Nam
Homework assignment 5 – due 3/8/2007
Problem 1 (LU decomposition). Write a program that implements the LU
decomposition algorithm for general n×nmatrices and outputs the Land U
factors. Apply it to the matrix of the linear system
11
2
1
3
1
4
1
2
1
3
1
4
1
5
1
3
1
4
1
5
1
6
1
4
1
5
1
6
1
7
x1
x2
x3
x4
=
1
2
3
4
.
In a second step, implement the backward and forward substitution solves
with the upper and lower triangular factors Land Ufor any given vector. Apply
it to the right hand side above and verify that your solution is correct.
(6 points)
Problem 2 (Norms on Rn). In the analysis of iterative solution methods
for linear systems, we will come across different vector norms. A functional
k·k:Rn→Ris called a norm if it satisfies the following three conditions:
(a) kxk ≥ 0 for all vectors x∈Rnand kxk= 0 if and only if x= 0 (positive
definiteness);
(b) kλxk=|λ|kxkfor all λ∈Rand all vectors x∈Rn(scalability);
(c) kx+yk ≤ kxk+kykfor all vectors x, y ∈Rn(triangle inequality).
Determine which of the following are norms on Rnby proving or disproving
that they satisfy the three conditions above:
a) max1≤i≤n|xi|
b) max2≤i≤n|xi|
c) Pn
i=1 |xi|3
d) Pn
i=1 |xi|1/22
e) max{|x1−x2|,|x1+x2|,|x3|,|x4|, . . . , |xn|}
f) Pn
i=1 2−i|xi|(6 points)
(please turn over)
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