Återorthogonalisering

Återorthogonalisering is a technique used in numerical linear algebra, particularly in the context of algorithms for computing eigenvalues and eigenvectors, such as the QR algorithm. It is a refinement process that aims to improve the orthogonality of a set of vectors that are intended to be orthonormal.

In many iterative numerical methods, especially those involving orthogonal transformations, a set of vectors is generated at each step, which ideally should remain orthogonal throughout the process. However, due to the accumulation of floating-point errors, these vectors can gradually lose their orthogonality. This loss of orthogonality can degrade the accuracy and convergence properties of the algorithm.

Återorthogonalisering addresses this issue by periodically re-orthogonalizing the computed vectors. This typically involves applying a process like the Gram-Schmidt process to the vectors. While Gram-Schmidt is a standard method for producing an orthonormal set from a linearly independent set, it can itself suffer from numerical instability. Therefore, modified or re-orthogonalized versions of Gram-Schmidt are often employed within the återorthogonalisering step to mitigate these problems.

The decision of when and how often to perform återorthogonalisering is a trade-off. More frequent re-orthogonalization can improve accuracy but increases computational cost. Less frequent re-orthogonalization is cheaper but risks significant loss of orthogonality. The specific implementation and frequency depend on the algorithm and the desired level of precision.