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"The production of points uniformly distributed in a multidimensional cube" Tech. (1976) "Uniformly distributed sequences with an additional uniform property". (1979) "An economic method of computing LP τ-sequences". "Low-Discrepancy and Low-Dispersion Sequences", Journal of Number Theory 30: 51–70. (1967), "Distribution of points in a cube and approximate evaluation of integrals". MATLAB also contains an implementation as part of its Statistics Toolbox. A version is available from the British-Russian Offshore Development Agency (BRODA). įinally, commercial Sobol sequence generators are available within, for example, the NAG Library. A different free/open-source implementation in up to 1111 dimensions is available for C++, Fortran 90, Matlab, and Python. A free/open-source implementation in up to 1111 dimensions, based on the Joe and Kuo initialisation numbers, is available in C, and up to 21201 dimensions in Python and Julia. Other implementations are available as C, Fortran 77, or Fortran 90 routines in the Numerical Recipes collection of software. Peter Jäckel provides initialisation numbers up to dimension 32 in his book " Monte Carlo methods in finance". Initialisation numbers for high dimensions are available on Joe and Kuo. The same set of initialisation numbers is used by Bratley and Fox. For example, Sobol provides initialisation numbers for dimensions up to 51. Good initialisation numbers for different numbers of dimensions are provided by several authors. However, such a sequence fails the test for Property A and A’ even for low dimensions and hence this initialisation is bad. This initialisation is usually called unit initialisation. A bad selection of initial numbers can considerably reduce the efficiency of Sobol sequences when used for computation.Īrguably the easiest choice for the initialisation numbers is just to have the l-th leftmost bit set, and all other bits to be zero, i.e. Therefore, it is possible to receive different realisations of the Sobol sequence for selected dimensions. There is some freedom in the selection of initial direction numbers. To construct a Sobol sequence, a set of direction numbers v i, j needs to be selected. Thus it is possible to construct the Sobol sequence that satisfies both properties A and A’ or only one of them. Tests for properties A and A’ are independent.
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Lim n → ∞ 1 n ∑ i = 1 n f ( x i ) = ∫ I s f with v k, j, m denoting the m-th digit after the binary point of the direction number v k, j = (0. The original motivation of Sobol was to construct a sequence x n in I s so that Let I s = s be the s-dimensional unit hypercube, and f a real integrable function over I s. Good distributions in the s-dimensional unit hypercube 1 Good distributions in the s-dimensional unit hypercube.The intuitive pack of options makes this hard disk tool suitable for less experienced users and professionals alike.
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