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Lecture Slides 14

# Lecture Slides 14 - AMS 210 Applied Linear Algebra AMS 210...

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Unformatted text preview: AMS 210: Applied Linear Algebra October 27, 2009 AMS 210 Topics Today Problem Set 5 Midterm 2 Computational Complexity of Solving Systems of Linear Equations Solving Tridiagonal Systems Sparse Matrix Fill-in Stable Elimination and Roundoff Error Condition Number of a Matrix Condition Number Examples AMS 210 Problem Set 5 Grades and solutions available on Blackboard. Mean of 93, median of 96, and standard deviation of 8. AMS 210 Midterm 2 November 5 during class. Scientific calculators, plus TI-83/84/86, allowed. TI-89-class calculators (TI-89/92, the HP-48/49/50s/etc) disallowed. Example problems on Blackboard by end of today. AMS 210 Computational Complexity of Solving Systems of Linear Equations Gaussian Elimination: ( n- 1) 2 + ( n- 2) 2 + ··· + (1) 2 ≈ n 3 3 multiplications. Gauss-Jordan Pivoting: ( n- i + 1) n multiplications to eliminate x i from other equations, so approximately n 3 2 multiplications. Theorem: A system of n equations in n unknowns requires approimately n 3 3 multiplications (and subtractions) to solve by Gaussian elimination and n 3 2 multiplications to solve by pivoting. Either method requires approximately n 3 multiplications to invert an n-by- n matrix. For solving Ax = b for multiple right-hand sides, LU generally best: n 2 for each new right-hand-side. Jacobi iteration requires n 2 multiplications per iteration. Can be better or worse than LU depending on sparsity and convergence rate....
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Lecture Slides 14 - AMS 210 Applied Linear Algebra AMS 210...

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