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,

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

Examples

Spreadsheet

	A	B
1	4	3
2	6	3

=LUDECOMPOSITION(A1:B2)

1 0	0.6666666666666666 1
6 3	0 1
0 1	1 0

Spreadsheet

	A	B	C
1	1	2	3
2	4	5	6
3	7	8	9

=LUDECOMPOSITION(A1:C3)

1 0 0	0.5714285714285714 1 0	0.14285714285714285 1.999999999999999 1
7 8 9	0 0.4285714285714288 0.8571428571428577	0 0 0
0 0 1	0 1 0	1 0 0

See Also

• CHOLESKYFACTORIZATION
• CONFERENCE
• PASCAL

References

• LU Decomposition

• List of Main Z Functions

• Z3 home