| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
IB |
62 |
IB |
2019 |
Numerical Analysis |
|
Paper 2, Section II, C |
2019 |
Define the linear least squares problem for the equation
$$A \mathbf{x}=\mathbf{b}$$
where $A$ is a given $m \times n$ matrix with $m>n, \mathbf{b} \in \mathbb{R}^{m}$ is a given vector and $\mathbf{x} \in \mathbb{R}^{n}$ is an unknown vector.
Explain how the linear least squares problem can be solved by obtaining a $Q R$ factorization of the matrix $A$, where $Q$ is an orthogonal $m \times m$ matrix and $R$ is an uppertriangular $m \times n$ matrix in standard form.
Use the Gram-Schmidt method to obtain a $Q R$ factorization of the matrix
$$A=\left(\begin{array}{lll}
1 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0 \\
1 & 0 & 0
\end{array}\right)$$
and use it to solve the linear least squares problem in the case
$$\mathbf{b}=\left(\begin{array}{l}
1 \\
2 \\
3 \\
6
\end{array}\right)$$