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all eigenvalues different from one achieve some reduction and the largest reduction is for eigenvectors with eigenvalue −1. For the power method, none of these eigenvalues is reduced by more than α. Thus, the inner-outer method does reduce the error associated with eigenvalues on the unit circle. 0 10 −5 10 −10 10 0123 10 10 10 10 0 10 −5 10 −10 100123 10 10 10 10 0 10 −5 10 −10 10 0123 10 10 10 10 0 10 −5 10 −10 10 0123 10 10 10 10 0 10 −5 10 −10 10 0123 10 10 10 10 0 10 −5 10 −10 10 0123 10 10 10 10 Table 5.7 – Convergence rates with inner-outer iterations. Convergence ratios from various graphs are єk+1/єk , where єk is the 1-norm change in x at the kth outer iteration of the inner-outer method. The algorithm switched to the power method after 10 iterations for all of these cases. The first eigenvalue off the unit-circle for small is λ = 0.8257; for ubc-cs-2006 it is λ = 0.9998. We observe that the convergence rate is approximately αλ. Figure 5.5 – Inner-Outer error analysis for a small graph. When α = 0.99, β = 0.5, and η = 10−2, the inner-outer iteration takes 112 matrix-vector multiplies to converge to an outer tolerance of 10−10 whereas the power method takes 2,013. The convergence of the power method is limited by the error in the eigenvec- tor with λ = −1, but the inner-outer iteration quickly eliminates the error in this component. The lines in the middle show where the inner-outer iteration switched to the power method. (The graph is the same as figure 4.2 and at left.) 5.6 ⋅ numerical results 115 −10 values are less than 10 for all iterations power inout α =0.85 Iteration graph 10 small 0.7019 ubc-cs-2006 0.8220 in-2004 0.7913 arabic-2005 0.7936 sk-2005 0.7932 uk-2007 0.7925 α=0.99 50 10 0.8472 0.8174 0.8420 0.9595 0.8354 0.9343 0.8359 0.9395 0.8386 0.9356 0.8358 0.9344 50 100 0.8174 0.8174 0.9801 0.9814 0.9736 0.9803 0.9746 0.9817 0.9773 0.9820 0.9743 0.9814 500 0.99 0.9868 0.9872 0.9875 0.9872 0.9881 λ = −0.4 λ = −0.4 λ = 0.14 + −0.28 i + 0.28 i λ = 0.83 λ = −1 λ = 1PDF Image | MODELS AND ALGORITHMS FOR PAGERANK SENSITIVITY
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