Cholesky decomposition
\gdef\delim#1#2#3{\mathopen{}\mathclose{\left#1 #2 \right#3}} \gdef\p#1{\delim({#1})} \gdef\ps#1{\delim\{{#1}\}} \gdef\N{\mathcal N} \gdef\vec#1{\bm #1} \gdef\mat#1{\mathrm #1} \gdef\k{\mathrm k} \gdef\u{\mathrm u} \gdef\c{\mathrm c} \gdef\T{\mathrm T}
Positive definite matrices occur as covariance matrices. Given a positive definite matrix \mat M it can be factored into a triangular matrix \mat L such that
\mat M = \mat L ⋅ \mat L^\T
Computing the Cholesky decompostion
https://en.wikipedia.org/wiki/Cholesky_decomposition#The_Cholesky_algorithm
https://en.wikipedia.org/wiki/Cholesky_decomposition#The_Cholesky%E2%80%93Banachiewicz_and_Cholesky%E2%80%93Crout_algorithms
A reasonably efficient way to compute it is:
=
-= @ .
/=
return
But it's more efficient to use the LAPACK dpotrf
function, available as np.linalg.cholesky
.
https://netlib.org/lapack/explore-html/d2/d09/group__potrf_ga84e90859b02139934b166e579dd211d4.html#ga84e90859b02139934b166e579dd211d4
=
= /
= /
=
= /
= * - *
=
return
Subset of Cholesky
References
https://algowiki-project.org/en/Cholesky_decomposition
https://www.cs.utexas.edu/users/flame/Notes/NotesOnCholReal.pdf
https://www.cs.utexas.edu/users/flame/pubs/flawn41.pdf
https://christian-igel.github.io/paper/AMERCMAUfES.pdf
https://github.com/scipy/scipy/issues/8188
https://link.springer.com/article/10.1007/BF01933218