Introduction to Numerical Analysis I MATH/CMPSC 455 PA=LU.

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Introduction to Numerical Analysis I MATH/CMPSC 455 PA=LU

P ARTIAL P IVOTING Why? (1) zeros pivot; (2) swapping rows reduces possibility of rounding errors and poor scaling contaminating the solution. How? Swap rows at each elimination step to place the largest element on the diagonal.

P ERMUTATION M ATRICES Definition: A permutation matrix is a matrix consisting of all zeros, except for a single 1 in every row and column. Fundamental Theorem of Permutation Matrices: Let P be the permutation matrix formed by a particular set of row exchanges applied to the identity matrix. Then, for any matrix A, PA is the matrix obtained by applying exactly the same set of row exchanges to A.

PA=LU FACTORIZATION Definition: PA=LU factorization is simply the LU factorization of a row-exchanged version of A. The algorithm consists of two parts: Factorization phase: apply to A only and is designed to produce the LU decomposition of PA. Solution phase: update the right hand side Pb, and then solve LU = Pb by forward and backward substitution.

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