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LU decomposition

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LU Solvers

For the system of equations A\cdot\phi=B.
The solvers based on factorization are widely popular. The factorization of a non singular matrix A in two matrices L and U, called lower and upper matrices, leads to a direct procedure for the evaluation of the inverse of the matrix. Thus making it possible to calculate the solution vector with given source vector. It now remains a two step process: a forward substitution followed by a backward substitution.


Calculate LU factors of A
 A\cdot \phi = (LU)\cdot \phi = L(U\cdot\phi)=LY=B
Solve LY=B by forward substitution
 Y = L^{-1} B
Solve U\phi =Y by backward substitution
 \phi = U^{-1}Y

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