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A procedure for decomposing an matrix
into a product of a lower triangular matrix
and an upper triangular matrix
,
![]() | (1) |
LU decomposition is implemented in Mathematica as LUDecomposition[m].
Written explicitly for a matrix, the decomposition is
![]() | (2) |
![]() | (3) |
This gives three types of equations
![]() | (4) |
![]() | (5) |
![]() | (6) |
This gives equations for
unknowns (the decomposition is not unique), and can be solved using Crout's method. To solve the matrix equation
![]() | (7) |
first solve for
. This can be done by forward substitution
![]() | ![]() | ![]() | (8) |
![]() | ![]() | ![]() | (9) |
for , ...,
. Then solve
for
. This can be done by back substitution
![]() | ![]() | ![]() | (10) |
![]() | ![]() | ![]() | (11) |
for , ..., 1.
Related Keyword:Gaussian Elimination, Lower Triangular Matrix, Matrix Decomposition, Cholesky Decomposition, QR Decomposition, Triangular Matrix, Upper Triangular Matrix
Labels: Numerical Method