Notes - Numerical Analysis HT24, Givens rotations
Flashcards
Suppose we are using Givens rotations. A typical problem boils down to solving
\[\begin{bmatrix}
c & -s \\\\
s & c
\end{bmatrix}
\begin{bmatrix}
a \\\\
b
\end{bmatrix}
=
\begin{bmatrix}
r \\\\
0
\end{bmatrix}\]
where $r = \sqrt{a^2 + b^2}$ ($\pm r$ are the only possible choices since Givens rotations are orthogonal and so preserve the length).
What values of $c = \cos \theta$ and $s = \sin \theta$ can be used here?
\[\cos\theta = \frac{a}{r}, \quad \sin\theta = -\frac{b}{r}\]
How can you convert an $n \times n$ tridiagonal matrix into an upper triangular matrix using $(n-1)$ Givens rotations?
Iteratively introduce zeroes below the main diagonal.