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92 4 ⋅ random alpha pagerank with values Only √√√ 131 FR(⋅)= 1+ −r; −i −1; 1+ . rrr r2 + 2r − 1 αR = 2r is inside the region of integration, and thus There is one more step: √ 1 ω=1+. r GQ(N) dα GQ(N) ∣E [xi (A)] − xi ∣ ≤ ∣ ∣ ∣E [xi (z)] − xi ∣ . The initial bound (4.41) now applies to the second expression with ρa + ρb = √√ 1+ 1 −1andω= 1+1.24 24 We avoided much of the tedious algebra in this proof by valiantly employing the computer algebra packages Maple and Mathematica. X X X X XX rr2 r dz Thus, the quadrature codes converge to the exact solutions as N → ∞ when r < 1. When r = 1, the story is much more complicated. Using Tre- fethen’s bound on the convergence of quadrature, and bounds on analytic functions in the complex plane, it suffices to show that x(α) is analytic in an ellipse that encloses the integration region [0, 1]. This follows because x(α) is analytic at α = 1 and has poles at 1 where λ is an eigenvalue of P that is λii X different from 1. All of the other eigenvalues have ∣λi ∣ < 1 and thus, we must be able to fit an ellipse (potentially a small ellipse) between these eigenvalues and the integration region [0, 1]. Figure 4.10 illustrates and shows a hypo- thetical ellipse with semi-major and semi-minor axes that sum to more than 1. This result gives us convergence when r = 1, but at an unknown rate. 4.7.4 Implementation correctness and convergence In this section, we present empirical results pertaining to the accuracy and convergence of our implementations. This type of analysis is important because numerical experimentation allows us to explore broader ranges of parameter values than may be feasible in the theoretical analysis. Addition- ally, it provides strong evidence that we have correctly implemented all the algorithms in this chapter. To begin, we use three experiments to verify that our algorithms are convergent when implemented with and without approxi- mate solutions of the linear algebra problems. Each of our algorithms has a X X X Figure 4.10 – Quadrature conver- gence with extreme endpoint. This figure illustrates the ellipse of an- alyticity with semi-major and semi-minor axis sum larger than 1 used to show that Gauss quadrature converges. The red x’s are singu- larities of the PageRank function with ∣α∣ ≥ 1. The circle shows the boundary ∣α∣ = 1.PDF Image | Instagram Cheat Sheet
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