Overdetermined System (numerical analysis)¶
Definition¶
An overdetermined system is one where there are more equations than unknowns when solving for the linear equation:
\[
\mathbf{Ax = b}
\]
Here, \(\mathbf{A} \in \mathbb{R}^{m \times n}\), where \(m > n\), \(\mathbf{x}\in \mathbb{R}^{n}\) and \(\mathbf{b} \in \mathbb{R}^{m}\).
In general, with only \(n\) parameters in the vector \(\mathbf{x}\), we would not expect to be able to reproduce the \(m\)-vector \(\mathbf{b}\) as a linear combination of the \(n\) columns of \(\mathbf{A}\). In other words, for an overdetermined system, there is no exact solution.