PDF Publication Title:
Text from PDF Page: 112
90 4 ⋅ random alpha pagerank Proof. TherearemanystatementsfortheerrorinGaussquadratureandwe begin with a modern statement from Trefethen [2008, theorem 4.5]. Consider 1 I = ∫−1 f (x) dx for an analytic function f . Let IN be the N-point Gauss quadrature approximation to I. Then 64ω ∣I−IN∣≤ −2 2N+N , (4.41) 15(1−(ρa +ρb) )(ρa +ρb) where ∣ f (z)∣ ≤ ω for all z in the ellipse with foci ±1 and semi-major axis ρa > 1 and semi-minor axis ρb. Figure 4.7 illustrates the construction. Note that we are approximating the integral r 0 Figure 4.7 – The framework for Gauss quadrature error analysis. The ellipse of analyticity provides bounds on the error in a Gauss quadrature rule. Roughly,∣error∣≤(ρa +ρB)2N. E[xi(A)] = xi(α)dα with an N-point quadrature rule. Each PageRank component is a rational function, which is a special case of an analytic function. In the remainder of the proof, we go through the details of applying the bound from (4.41) precisely. First, we transform the problem to the integration region [−1, 1] by a change of variables α to z. In this z-space, we build an ellipse in the complex plain where xi(z) is analytic. To study the function magnitude ω, we transform the ellipse back to α-space and examine the magnitude of PageRank as a function of α when α is complex. Let 2α r z= −1 ⇐⇒ α= (z+1) r2 be the change of variables between α and z. Consider z = zR + iZi for z in the ellipse with foci ±1. The ellipse satisfies z 2R + z 2I = 1 ρ2a ρb2 and the constraint on the foci implies that ρ2a = 1 + ρb2 . Both ρa and ρb live in z-space, so for 1 ρa= , r we consider an ellipse in α-space with a right end-point r/2 + 1/2—halfway between r and 1.23 The function xi (α(z)) is analytic inside this ellipse. The right endpoint (in α-space) is less than 1 and xi (α) is analytic for all complex 23 This choice of ρa may not be optimal, but other choices increase the difficulty of the computations considerably. In particular, we tried using a right endpoint of γr+(1−γ), but could only compute the upper bound ω when γ = 1/2.PDF Image | Instagram Cheat Sheet
PDF Search Title:
Instagram Cheat SheetOriginal File Name Searched:
pagerank-sensitivity-thesis-online.pdfDIY PDF Search: Google It | Yahoo | Bing
Cruise Ship Reviews | Luxury Resort | Jet | Yacht | and Travel Tech More Info
Cruising Review Topics and Articles More Info
Software based on Filemaker for the travel industry More Info
The Burgenstock Resort: Reviews on CruisingReview website... More Info
Resort Reviews: World Class resorts... More Info
The Riffelalp Resort: Reviews on CruisingReview website... More Info
CONTACT TEL: 608-238-6001 Email: greg@cruisingreview.com (Standard Web Page)