Rybicki Press algorithm

[2] 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 systems of equations can be solved efficiently (to be more precise, in linear time).

[1] 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.

[3][4] The most common use of the algorithm is in the detection of periodicity in astronomical observations[verification needed], such as for detecting quasars.

[4] The method has been extended to the Generalized Rybicki-Press algorithm for inverting matrices with entries of the form

[2] The key observation in the Generalized Rybicki-Press (GRP) algorithm is that the matrix

is a semi-separable matrix with rank

and whose lower half is also that of some possibly different rank

matrix[2]) 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.

, the computational complexity of solving the linear system

or of calculating the determinant of the matrix

, thereby making it attractive for large matrices.

is a semi-separable matrix also forms the basis for celerite[5] library, which is a library for fast and scalable Gaussian process regression in one dimension[6] with implementations in C++, Python, and Julia.

The celerite method[6] also provides an algorithm for generating samples from a high-dimensional distribution.

The method has found attractive applications in a wide range of fields,[which?]

especially in astronomical data analysis.