Cannabaceae

October 12, 2016 thcscience_admin

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

Algorithm

Let $A=(a_{ij})\in \mathbb {K} ^{n\times n}$ be a sparse symmetric positive definite matrix with elements from a field $\mathbb {K}$ , which we wish to factorize as $A=LL^{T}\,$ .

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:

Let ${\mathcal {A}}_{i}$ and ${\mathcal {L}}_{j}$ 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 $\min {\mathcal {L}}_{j}$ to mean the smallest element of ${\mathcal {L}}_{j}$ .
Use a parent function $\pi (i)\,\!$ to define the elimination tree within the matrix.

The following algorithm gives an efficient symbolic factorization of $A$ :

{\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 \qquad {\mathcal {L}}_{i}:=({\mathcal {L}}_{i}\cup {\mathcal {L}}_{j})\setminus \{j\}\\&\qquad \pi (i):=\min({\mathcal {L}}_{i}\setminus \{i\})\end{aligned}}

source: https://en.wikipedia.org/wiki/Symbolic_Cholesky_decomposition

One thought on “Cannabaceae”

Iam Hassan says:

July 25, 2018 at 9:40 am

Well, that’s interesting to know that Psilotum nudum are known as whisk ferns. Psilotum nudum is the commoner species of the two. While the P. flaccidum is a rare species and is found in the tropical islands. Both the species are usually epiphytic in habit and grow upon tree ferns. These species may also be terrestrial and grow in humus or in the crevices of the rocks.
View the detailed Guide of Psilotum nudum: Detailed Study Of Psilotum Nudum (Whisk Fern), Classification, Anatomy, Reproduction

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