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Understanding Partial Pivoting and the PA=LU Factorization in Matrix Equations

Learn about Partial Pivoting to compare values and LU Factorization for solving systems of equations. Discover how to apply P to A and find LU Factorization to ensure all matrices comply.

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Understanding Partial Pivoting and the PA=LU Factorization in Matrix Equations

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  1. Partial Pivoting and the PA=LU Factorization Grayson Ishihara Math 480 April 15, 2013

  2. What is Partial Pivoting? • What is the PA=LU Factorization? • What kinds of things can we use these tools for? Topics at Hand

  3. Used to solve matrix equations • Compare absolute values of entries in and below pivot position • If |ap1|≥|ai1|, switch rows 1 and p • Repeat for each pivot position Partial Pivoting

  4. Writing a matrix A as the product of a lower triangular matrix L and an upper triangular matrix U LU Factorization

  5. LU Factorization

  6. Row reduce A into upper triangular U • Keep track of steps in lower triangular L LU Factorization

  7. Not all matrices allow such a factorization… ac Ab LU Factorization aa ad b = 0, ab = 1 ?

  8. Apply permutation matrix P to A • Find LU Factorization of the resulting matrix • Ensures all matrices have LU Factorization PA=LU Factorization

  9. Solving systems of equations: • Matrix form Ax=b • Multiply both sides by P: PAx=PbLUx=Pb • Two-step solution: Ly=Pb, Ux=y • Easy: back substitution Why Is It Useful?

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