Rybicki Press algorithm: Difference between revisions

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{{Short description|An algorithm for inverting a matrix}}

{{Technical|date=August 2021}}

[[File:Extended_Sparse_MatrixExtended Sparse Matrix.png|thumb|Extended Sparse Matrix arising from a <math>10 \times 10</math> semi-separable matrix whose semi-separable rank is <math>4</math>.]]

The '''Rybicki–Press algorithm''' is a fast [[algorithm]] for inverting a [[Matrix (mathematics)|matrix]] whose entries are given by <math>A(i,j) = \exp(-a \vert t_i - t_j \vert)</math>, where <math>a \in \mathbb{R}</math><ref name=":2">{{citation

The '''Rybicki–Press algorithm''' is a fast [[algorithm]] for inverting a [[Matrix (mathematics)|last1matrix]] whose entries are given by <math>A(i,j) = \exp(-a \vert t_i - t_j \vert)</math>, where <math>a \in \mathbb{R}</math><ref name=":2">{{cite journal |last1=Rybicki |first1 = George B. |last2 =Press Press|first2 = William H. |arxiv = comp-gas/9405004 |doi = 10.1103/PhysRevLett.74.1060 |journal = Physical Review Letters |title = Class of fast methods for processing Irregularly sampled or otherwise inhomogeneous one-dimensional data |volume =74 74|issue =7 7|pages =1060–1063 1060–1063|year date=1995 1995|bibcode = 1995PhRvL..74.1060R|pmid=10058924|s2cid = 17436268}} {{Open access}}</ref> and where the <math>t_i</math> are sorted in order.<ref name=":3" /> The key observation behind the Rybicki-Press observation is that the [[matrix inverse]] of such a matrix is always a [[tridiagonal matrix]] (a matrix with nonzero entries only on the main diagonal and the two adjoining ones), and [[Tridiagonal matrix algorithm|tridiagonal systems of equations]] can be solved efficiently (to be more precise, in linear time).<ref name=":2" /> It is a computational optimization of a general set of statistical methods developed to determine whether two noisy, irregularly sampled data sets are, in fact, dimensionally shifted representations of the same underlying function.<ref>{{Cite journal|title = Interpolation, realization, and reconstruction of noisy, irregularly sampled data|last1 = Rybicki|first1 = George B.|date = October 1992|journal = The Astrophysical Journal|doi = 10.1086/171845|last2 = Press|first2 = William H.|bibcode = 1992ApJ...398..169R|volume=398|page=169}}{{Open access}}</ref><ref name=":0">{{Cite journal|last1=MacLeod|first1=C. L.|last2=Brooks|first2=K.|last3=Ivezic|first3=Z.|last4=Kochanek|first4=C. S.|last5=Gibson|first5=R.|last6=Meisner|first6=A.|last7=Kozlowski|first7=S.|last8=Sesar|first8=B.|last9=Becker|first9=A. C.|date=2011-02-10|title=Quasar Selection Based on Photometric Variability|journal=The Astrophysical Journal|volume=728|issue=1|pages=26|doi=10.1088/0004-637X/728/1/26|issn=0004-637X|arxiv=1009.2081|bibcode=2011ApJ...728...26M|s2cid=28219978}}</ref> The most common use of the algorithm is in the detection of periodicity in astronomical observations{{Verify source|date=October 2021}}, such as for detecting [[Quasar|quasars]].<ref name=":0" />

The method has been extended to the '''Generalized Rybicki-Press algorithm''' for inverting matrices with entries of the form <math>A(i,j) = \sum_{k=1}^p a_k \exp(-\beta_k \vert t_i - t_j \vert)</math>.<ref name=":3">{{Cite journal|last=Ambikasaran|first=Sivaram|date=2015-12-01|title=Generalized Rybicki Press algorithm|journal=Numerical Linear Algebra with Applications|language=en|volume=22|issue=6|pages=1102–1114|doi=10.1002/nla.2003|issn=1099-1506|arxiv=1409.7852|s2cid=1627477}}</ref> The key observation in the Generalized Rybicki-Press (GRP) algorithm is that the matrix <math>A</math> is a [[semi-separable matrix]] with rank <math>p</math> (that is, a matrix whose upper half, not including the main diagonal, is that of some matrix with [[matrix rank]] <math>p</math> and whose lower half is also that of some possibly different rank <math>p</math> matrix<ref name=":3" />) and so can be embedded into a larger [[band matrix]] (see figure on the right), whose sparsity structure can be leveraged to reduce the computational complexity. As the matrix <math>A \in \mathbb{R}^{n\times n}</math> has a semi-separable rank of <math>p</math>, the [[computational complexity]] of solving the linear system <math>Ax=b</math> or of calculating the determinant of the matrix <math>A</math> scales as <math>\mathcal{O}\left(p^2n \right)</math>, thereby making it attractive for large matrices.<ref name=":3" />

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==See also==

* [[Invertible matrix]]

* [[Matrix decomposition]]

* [[Multidimensional signal processing]]

* [[System of linear equations]]

==References==