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Understanding Gaussian Elimination and the Role of Pivoting in Numerical Stability

Gaussian elimination is a fundamental algorithm in numerical linear algebra, but it has notable limitations, particularly regarding round-off errors. This presentation explores the naive implementation's issues, such as dividing by zero and small numbers, and introduces pivoting techniques to enhance stability. It covers partial pivoting, that mitigates errors by rearranging rows based on specific criteria, and discusses various pivoting strategies, including complete and threshold pivoting. Additionally, it highlights the implications of pivoting in LU factorization for solving linear systems effectively.

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Understanding Gaussian Elimination and the Role of Pivoting in Numerical Stability

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  1. Linear SystemsPivoting in Gaussian Elim. CSE 541 Roger Crawfis

  2. Limitations of Gaussian Elimation • The naïve implementation of Gaussian Elimination is not robust and can suffer from severe round-off errors due to: • Dividing by zero • Dividing by small numbers and adding. • Both can be solved with pivoting

  3. Factored Portion Row i Row j Partial Pivoting • What if at step i, Aii = 0? • Simple Fix: If Aii = 0 Find Aji 0 j > i Swap Row j with i

  4. Forward Elimination Example – Partial Pivoting

  5. Forward Elimination Rounded to 3 digits Example – Partial Pivoting

  6. Better Pivoting • Partial Pivoting to mitigate round-off error • Adds an O(n) search. Avoids Small Multipliers

  7. Partial Pivoting swap Forward Elimination Rounded to 3 digits

  8. k k k k Pivoting strategies • Partial Pivoting: • Only row interchange • Complete (Full) Pivoting • Row and Column interchange • Threshold Pivoting • Only if prospective pivot is found to be smaller than a certain threshold

  9. Pivoting With Permutations • Adding permutation matrices in the mix: • However, in Gaussian Elimination we will only swap rows or columns below the current pivot point. This implies a global reordering of the equations will work:

  10. Pivoting • Again, the pivoting is strictly a function of the matrix A, so once we determine P it is trivial to apply it to many problems bk. • For LU factorization we have: • LU = PA • Ly = Pb • Ux = y

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