Editing Gram–Schmidt process (section)

== Alternatives ==
Other [[orthogonalization]] algorithms use [[Householder transformation]]s or [[Givens rotation]]s. The algorithms using Householder transformations are more stable than the stabilized Gram–Schmidt process. On the other hand, the Gram–Schmidt process produces the <math>j</math>th orthogonalized vector after the <math>j</math>th iteration, while orthogonalization using [[Householder reflection]]s produces all the vectors only at the end. This makes only the Gram–Schmidt process applicable for [[iterative method]]s like the [[Arnoldi iteration]].

Yet another alternative is motivated by the use of [[Cholesky decomposition]] for [[Ordinary least squares|inverting the matrix of the normal equations in linear least squares]]. Let <math>V</math> be a [[full rank|full column rank]] matrix, whose columns need to be orthogonalized. The matrix <math>V^* V </math> is [[Hermitian matrix|Hermitian]] and [[Positive definite matrix|positive definite]], so it can be written as <math> V^* V = L L^*, </math> using the [[Cholesky decomposition]]. The lower triangular matrix <math>L </math> with strictly positive diagonal entries is [[invertible]]. Then columns of the matrix <math>U = V\left(L^{-1}\right)^*</math> are [[orthonormal]] and [[linear span|span]] the same subspace as the columns of the original matrix <math>V</math>. The explicit use of the product <math>V^* V </math> makes the algorithm unstable, especially if the product's [[condition number]] is large. Nevertheless, this algorithm is used in practice and implemented in some software packages because of its high efficiency and simplicity.

In [[quantum mechanics]] there are several orthogonalization schemes with characteristics better suited for certain applications than original Gram–Schmidt. Nevertheless, it remains a popular and effective algorithm for even the largest electronic structure calculations.<ref>{{cite book|last1=Pursell|first1=Yukihiro|title=Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis |chapter=First-principles calculations of electron states of a silicon nanowire with 100,000 atoms on the K computer | display-authors=etal|date=2011|pages=1:1–1:11| doi=10.1145/2063384.2063386 | isbn=9781450307710|s2cid=14316074}}</ref>