NLA MT25, Linear systems
Flashcards
What are the pros and cons of solving (consistent) linear systems with QR factorisation compared to LU factorisation?
- Pros: The QR-based method is guaranteed to be backward-stable
- Cons: It is around twice as slow as the LU-based method
@State a fact about the stability of a certain class of linear systems.
Triangular linear systems can be solved in a backward stable manner, i.e. if $R$ is an upper or lower triangular linear system $Rx = b$, then the computed solution satisfies
\[(R + \Delta R) \hat x = b, \quad \vert \vert \Delta R \vert \vert = O(\epsilon \vert \vert R \vert \vert )\]Stability analysis
How are linear systems solved in a backward stable manner? You may appeal to a useful result.
Triangular linear systems can be solved in a backward stable manner, so we can solve $Ax = b$ via $Ly = b$, then $Ux = y$.