It also significantly outperforms steepest descent Convergence Guarantees We

# It also significantly outperforms steepest descent

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It also significantly outperforms steepest descent. Convergence Guarantees We can actually talk intelligently about how many iterations we need for steepest descent and CG to converge to within a certain precision. Here we present (but do not prove) two “worst case” bounds that depend on the condition number κ of H : κ = λ max ( H ) λ min ( H ) = max eigenvalue min eigenvalue . For steepest descent, we will have k e k k H δ k e 0 k H 45 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 2:06, November 18, 2019

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in at most 1 k 1 2 κ log 1 δ iterations. For CG, we need at most k 1 2 κ log 2 δ . There are nice derivations for both of these bounds in the Shewchuk manuscript mentioned at the beginning of these notes. Example: Say the condition number of H is κ = 100. How many iterations do you need to get 6 digits of precision ( δ = 10 - 6 )? SD : 1 2 · 100 · log(10 6 ) = 691 , CG : 1 2 · 10 · log(2 · 10 6 ) = 73 . Again, these are worst-case bounds, and performance in both cases is typically better. 1 These are natural logarithms. 46 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 2:06, November 18, 2019
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