ⓘ Symbolic Cholesky decomposition. In the mathematical subfield of numerical analysis the symbolic Cholesky decomposition is an algorithm used to determine the no ..

                                     

ⓘ Symbolic Cholesky decomposition

In the mathematical subfield of numerical analysis the symbolic Cholesky decomposition is an algorithm used to determine the non-zero pattern for the L {\displaystyle L} factors of a symmetric sparse matrix when applying the Cholesky decomposition or variants.

                                     

1. Algorithm

Let A = a i j ∈ K n × n {\displaystyle A=a_{ij}\in \mathbb {K} ^{n\times n}} be a sparse symmetric positive definite matrix with elements from a field K {\displaystyle \mathbb {K} }, which we wish to factorize as A = L T {\displaystyle A=LL^{T}\,}.

In order to implement an efficient sparse factorization it has been found to be necessary to determine the non zero structure of the factors before doing any numerical work. To write the algorithm down we use the following notation:

  • Use a parent function π i {\displaystyle \pi i\,\!} to define the elimination tree within the matrix.
  • Let A i {\displaystyle {\mathcal {A}}_{i}} and L j {\displaystyle {\mathcal {L}}_{j}} be sets representing the non-zero patterns of columns i and j below the diagonal only, and including diagonal elements of matrices A and L respectively.
  • Take min L j {\displaystyle \min {\mathcal {L}}_{j}} to mean the smallest element of L j {\displaystyle {\mathcal {L}}_{j}}.

The following algorithm gives an efficient symbolic factorization of A:

π i:= 0 for all i For i:= 1 to n L i:= A i For all j such that π j = i L i:= L i ∪ L j ∖ { j } π i:= min L i ∖ { i } {\displaystyle {\begin{aligned}&\pi i:=0~{\mbox{for all}}~i\\&{\mbox{For}}~i:=1~{\mbox{to}}~n\\&\qquad {\mathcal {L}}_{i}:={\mathcal {A}}_{i}\\&\qquad {\mbox{For all}}~j~{\mbox{such that}}~\pi j=i\\&\\qquad {\mathcal {L}}_{i}:={\mathcal {L}}_{i}\cup {\mathcal {L}}_{j}\setminus \{j\}\\&\qquad \pi i:=\min{\mathcal {L}}_{i}\setminus \{i\}\end{aligned}}}
Free and no ads
no need to download or install

Pino - logical board game which is based on tactics and strategy. In general this is a remix of chess, checkers and corners. The game develops imagination, concentration, teaches how to solve tasks, plan their own actions and of course to think logically. It does not matter how much pieces you have, the main thing is how they are placement!

online intellectual game →