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harvard500 m = 0 m = 2 m = 4 m =7 m = ∞ wb-cs.stanford m = 0 m = 2 m = 4 m =7 m = ∞ Figure 5.4 – Inner-Outer preconditioner spectrum. For the matrices harvard500 and wb- cs.stanford with α = 0.99, this figure plots the eigenvalues of the preconditioned matrix (I − βP ̄ )−1 (I − αP ̄ ) and approximations based on Neumann series. Each dashed circle encloses a circle of radius 1 in the complex plane centered at λ = 1, and hence the scale is the same in each small figure. Gray lines are contours of the function pm (λ) defined in (5.26), which is the identity matrix for m = 0 (i.e. no preconditioning) and the exact inverse when m = ∞. The dots are the eigenvalues of the preconditioned system, pm (λi ). Interlacing contours of pm (λ) demonstrate that this function is not 1-1 for λ ∶ ∣1 − λ∣ ≤ 1. 5.6 ⋅ numerical results 111 β = 0.50 β = 0.70 β = 0.85 β = 0.50 β = 0.70 β = 0.85PDF Image | MODELS AND ALGORITHMS FOR PAGERANK SENSITIVITY
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