Manuals/calci/LUDECOMPOSITION

LUDECOMPOSITION (Matrix)

$Matrix$ is the set of values.

Description

This function gives the value of LU Decomposition of a given matrix.
In $LUDECOMPOSITION(Matrix)$ , $Matrix$ is any square matrix.
LU Decomposition is the procedure for decomposing any square matrix in to a product of Lower Triangular matrix and Upper Triangular matrix.
In LU Decomposition, L stands for Lower Triangular matrix and U stands for Upper Triangular matrix.
So A=LU.But sometimes the product includes Permutation Matrix also(PA=LU).
LU Decomposition is also called LU Factorization.Here given matrix is split in to lower triangular and Upper triangular matrix.
- For 2x2 matrix,

${\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}={\begin{bmatrix}l_{11}&0\\l_{21}&l_{22}\end{bmatrix}}{\begin{bmatrix}u_{11}&u_{12}\\0&u_{22}\end{bmatrix}}$

For 3x3 matrix,

${\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}}={\begin{bmatrix}l_{11}&0&0\\l_{21}&l_{22}&0\\l_{31}&l_{32}&l_{33}\end{bmatrix}}{\begin{bmatrix}u_{11}&u_{12}&u_{13}\\0&u_{22}&u_{23}\\0&0&u_{33}\end{bmatrix}}$