01 a has 3092 nonzero entries and 106 the cg

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Unformatted text preview: 1] at each off-diagonal position (maintaining the symmetry A=A ) Then we replace each off-diagonal entry with |Aij | > τ by zero, where τ is a parameter For τ close to zero, the result is a well-conditioned positive definite matrix whose density of nonzero entries is approximately τ As τ increases, both the condition number and the sparsity deteriorate 16 / 25 17 / 25 For τ = 0.01, A has 3,092 nonzero entries and κ ≈ 1.06, the CG convergence takes place in 9 steps For τ = 0.05, A has 13,062 nonzero entries with κ ≈ 1.83, and convergence takes place in 19 steps For τ = 0.1, A has 25,526 nonzero entries with κ ≈ 10.3 and the process converges in 20 steps For τ = 0.2 with 50,834 nonzero entries, there is no convergence at all For this example, the CG beats Cholesky factorization by a factor of about 700 in terms of operation counts   # §'  # !   A  !   # ! ' © ' !  ©  S  '   '•  # “ " 7B$!9§474¦–ˆ!y$H¨d%¨ 0@0$X i'¦"XP...
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This document was uploaded on 02/10/2014.

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