Editing Singular value decomposition (section)

=== One-sided Jacobi algorithm ===
One-sided Jacobi algorithm is an iterative algorithm,<ref>{{cite journal|first1=P.P.M. de|last1=Rijk|title=A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer|journal=SIAM J. Sci. Stat. Comput.|volume=10|pages=359–371|year=1989|issue=2 |doi=10.1137/0910023 }}</ref>
where a matrix is iteratively transformed into a matrix with orthogonal columns. The elementary iteration is given as a [[Jacobi rotation]],

<math display=block>
M\leftarrow MJ(p, q, \theta),
</math>

where the angle <math>\theta</math> of the Jacobi rotation matrix <math>J(p,q,\theta)</math> is chosen such that after the rotation the columns with numbers <math>p</math> and <math>q</math> become orthogonal. The indices <math>(p,q)</math> are swept cyclically, <math>(p=1\dots m,q=p+1\dots m)</math>, where <math>m</math> is the number of columns.

After the algorithm has converged, the singular value decomposition <math>M=USV^T</math> is recovered as follows: the matrix <math>V</math> is the accumulation of Jacobi rotation matrices, the matrix <math>U</math> is given by [[norm_(mathematics)|normalising]] the columns of the transformed matrix <math>M</math>, and the singular values are given as the norms of the columns of the transformed matrix <math>M</math>.