Givens rotation
Let be an matrix with and full rank (viz. rank ). An orthogonal matrix triangularization (QR Decomposition) consists of determining an orthogonal matrix such that
with the upper triangular matrix . One only has then to solve the triangular system , where consists of the first rows of .
Householder transformations clear whole columns except for the first element of a vector. If one wants to clear parts of a matrix one element at a time, one can use Givens rotation, which is particularly practical for parallel implementation .
A matrix
with properly chosen and for some rotation angle can be used to zero the element . The elements can be zeroed column by column from the bottom up in the following order:
is then the product of Givens matrices .
To annihilate the bottom element of a vector:
the conditions and give:
For “Fast Givens”, see [Golub89].
References
-
•
Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.htmlhttp://rkb.home.cern.ch/rkb/titleA.html)
-
Golub89
Gene H. Golub and Charles F. van Loan: Matrix Computations, 2nd edn., The John Hopkins University Press, 1989.
Title | Givens rotation |
---|---|
Canonical name | GivensRotation |
Date of creation | 2013-03-22 12:06:10 |
Last modified on | 2013-03-22 12:06:10 |
Owner | akrowne (2) |
Last modified by | akrowne (2) |
Numerical id | 8 |
Author | akrowne (2) |
Entry type | Algorithm |
Classification | msc 15A57 |
Classification | msc 65F25 |
Related topic | GramSchmidtOrthogonalization |