LU Decomposition Definition/Meaning:
A method used in numerical linear algebra in order to solve a
set of linear equations,
Ax = b
where A is a square matrix and b is a column vector. In this method, a lower
triangular matrix L and an upper
triangular matrix U are sought such that
LU = A
For definiteness, the diagonal elements of L may be taken to be 1. The elements
of
successive rows of U and L may easily be calculated from the defining equations.
Once L and U have been determined, so that
LUx = b,
the equation
Ly = b
is found by forward substitution.
Thereafter the equation
Ux = y
is found by backward substitution, x is then the solution to the original
problem. A variant of the method, the method of LDU decomposition, seeks lower
and upper triangular matrices with unit diagonal and a diagonal matrix D, such
that
A = LDU
If the matrix A is symmetric and positive definite, there is an advantage in
finding a lower triangular matrix L such that
A = LLT
This method is known as Cholesky decomposition; the diagonal elements of L are
not, in general, unity.
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