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We wish to decompose the matrix A into the product LU, where L is a lower triangular matrix with 1's on the main diagonal, and U is an upper triangular matrix.
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We assume A is a full M by M matrix.
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In general, pivoting is necessary to ensure numerical stability.
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LU decomposition is often used in the solution of systems of linear equations, Ax = b. The equations can be written as two triangular systems,
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The first equation is solved for y by forward reduction, and the solution x is then obtained from the second equation by back substitution.
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