| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
IB |
61 |
IB |
2012 |
Numerical Analysis |
|
Paper 3, Section II, D |
2012 |
Define the QR factorization of an $m \times n$ matrix $A$ and explain how it can be used to solve the least squares problem of finding the vector $x^{} \in \mathbb{R}^{n}$ which minimises $\left|A x^{}-b\right|$, where $b \in \mathbb{R}^{m}, m>n$, and the norm is the Euclidean one.
Define a Householder transformation $H$ and show that it is an orthogonal matrix.
Using a Householder transformation, solve the least squares problem for
$$A=\left[\begin{array}{rrr}
1 & -1 & 5 \\
0 & 1 & 5 \\
0 & 0 & 3 \\
0 & 0 & 4
\end{array}\right], \quad b=\left[\begin{array}{r}
1 \\
2 \\
-1 \\
2
\end{array}\right]$$
giving both $x^{}$ and $\left|A x^{}-b\right|$.