# LU decomposition

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(towards a uniform notation for linear systems : A*Phi = B) |
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== LU Solvers == | == LU Solvers == | ||

- | For the system of equations | + | For the system of equations <math>A\cdot\phi=B</math>. <br> |

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. | 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. | It now remains a two step process: a forward substitution followed by a backward substitution. | ||

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: Calculate '''LU''' factors of '''A''' | : Calculate '''LU''' factors of '''A''' | ||

- | :: | + | ::<math> A\cdot \phi = (LU)\cdot \phi = L(U\cdot\phi)=LY=B</math> |

- | : Solve | + | : Solve <math>LY=B</math> by <i>forward substitution</i> |

- | :: | + | :: <math> Y = L^{-1} B</math> |

- | : Solve | + | : Solve <math>U\phi =Y</math> by <i>backward substitution</i> |

- | :: | + | :: <math> \phi = U^{-1}Y</math> |

+ | {{stub}} | ||

---- | ---- | ||

## Revision as of 20:42, 15 December 2005

## LU Solvers

For the system of equations .

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.

## Algorithm

- Calculate
**LU**factors of**A** - Solve by
*forward substitution* - Solve by
*backward substitution*

* Return to Numerical Methods *