y - Institute for Building Materials - ETH Zürich

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Lagged gg Fibbonacci RNG Typically one uses, since it is easy to implement: x = x ⊕x ≡ x + x ( ) mod 2 i i−a i−b i−a i−b a < b Theorem of fA A. C Compagner (1992): If If ( (a,b ( (a,b a,b) ) Zierler trinomial then then sequence sequence has has maximal period 2 2b – 1 and : 2 x ⋅x − x = 0 ∀ k < b i i−k i Zierler trinomials 1+ x a + x b primitive on Z Z2[x] (Zierler, 1969) (a, ( , b) ) (103, 250) (Kirkpatrick and Stoll, 1981) (1689, 4187) (54454, 132049) (J.R. Heringa et al, 1992) (3037958 (3037958, 6972592) (R.P.Brent et al., 2003) 31 32
Making 64 64-bit bit integers ..… z zi = (01101….101110) (01101 101110) z i+1 +1 = (10001….010111) zi+2 i+2 2 = (00101….111001) ( ) zi+3 +3 = (01011….011100) zi+4 +4 = (00011….011011) ..... Tests <strong>for</strong> random numbers • Check distribution • Average is 0.5 • Average of each bit is 0.5 • n-cube-test n cube test • Correlations should vanish • SSpectral t lt test: t no peaks k iin Fourier F i trans<strong>for</strong>m t f • χ2 χ test: partial p sums follow a Gaussian • Kolmogorov – Smirnov test → „Diehard Di h d battery“ b tt “ of f M Marsaglia li (1995) 33 34
Page 1 and 2: Autumn term 2008 I Introduction t d
Page 3 and 4: Plan of this course • 19.09. Intr
Page 5 and 6: Sedimentation Sedimentation compari
Page 7 and 8: Book Series • „Monte Carlo Meth
Page 9 and 10: Computer tools • Object oriented
Page 11 and 12: Distribution of random numbers +∞
Page 13 and 14: Mersenne prime p numbers n M = 2
Page 15: n-cube cube-test test One One can c
Page 19 and 20: Transformation method We want rando
Page 21: Rejection method Generate random nu

y - Institute for Building Materials - ETH Zürich

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