Editing Orthogonalization (section)

==Orthogonalization algorithms==

Methods for performing orthogonalization include:
*[[Gram–Schmidt process]], which uses [[Projection (linear algebra)|projection]]
*[[Householder transformation]], which uses [[Reflection (mathematics)|reflection]]
*[[Givens rotation]]
*Symmetric orthogonalization, which uses the [[Singular value decomposition]]
When performing orthogonalization on a computer, the Householder transformation is usually preferred over the Gram–Schmidt process since it is more [[numerical stability|numerically stable]], i.e. rounding errors tend to have less serious effects.

On the other hand, the Gram–Schmidt process produces the jth orthogonalized vector after the jth iteration, while orthogonalization using Householder reflections produces all the vectors only at the end. This makes only the Gram–Schmidt process applicable for [[iterative method]]s like the [[Arnoldi iteration]].

The Givens rotation is more easily [[parallel computing|parallelized]] than Householder transformations.

Symmetric orthogonalization was formulated by [[Per-Olov Löwdin]].<ref>{{Cite book| publisher = Elsevier| volume = 5| pages = 185–199| last = Löwdin| first = Per-Olov| title = Advances in quantum chemistry| chapter = On the nonorthogonality problem| date = 1970| doi = 10.1016/S0065-3276(08)60339-1| isbn = 9780120348053|chapter-url=https://www.sciencedirect.com/science/article/pii/S0065327608603391}}</ref>