/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */ /* Copyright (C) 2003, 2004 Ferdinando Ametrano This file is part of QuantLib, a free-software/open-source library for financial quantitative analysts and developers - http://quantlib.org/ QuantLib is free software: you can redistribute it and/or modify it under the terms of the QuantLib license. You should have received a copy of the license along with this program; if not, please email <quantlib-dev@lists.sf.net>. The license is also available online at <http://quantlib.org/reference/license.html>. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the license for more details. */ /*! \file sobolrsg.hpp \brief Sobol low-discrepancy sequence generator */ #ifndef quantlib_sobol_ld_rsg_hpp #define quantlib_sobol_ld_rsg_hpp #include <ql/Math/array.hpp> #include <ql/MonteCarlo/sample.hpp> #include <vector> namespace QuantLib { //! Sobol low-discrepancy sequence generator /*! A Gray code counter and bitwise operations are used for very fast sequence generation. The implementation relies on primitive polynomials modulo two from the book "Monte Carlo Methods in Finance" by Peter Jäckel. 21 200 primitive polynomials modulo two are provided by default in QuantLib. Jäckel has calculated 8 129 334 polynomials, also available in a different file that can be downloaded from http://quantlib.org. If you need that many dimensions you must replace the default version of the primitivepolynomials.c file with the extended one. The choice of initialization numbers (also know as free direction integers) is crucial for the homogeneity properties of the sequence. Sobol defines two homogeneity properties: Property A and Property A'. The unit initialization numbers suggested in "Numerical Recipes in C", 2nd edition, by Press, Teukolsky, Vetterling, and Flannery (section 7.7) fail the test for Property A even for low dimensions. Bratley and Fox published coefficients of the free direction integers up to dimension 40, crediting unpublished work of Sobol' and Levitan. See Bratley, P., Fox, B.L. (1988) "Algorithm 659: Implementing Sobol's quasirandom sequence generator," ACM Transactions on Mathematical Software 14:88-100. These values satisfy Property A for d<=20 and d = 23, 31, 33, 34, 37; Property A' holds for d<=6. Jäckel provides in his book (section 8.3) initialization numbers up to dimension 32. Coefficients for d<=8 are the same as in Bradley-Fox, so Property A' holds for d<=6 but Property A holds for d<=32. The implementation of Lemieux, Cieslak, and Luttmer includes coefficients of the free direction integers up to dimension 360. Coefficients for d<=40 are the same as in Bradley-Fox. For dimension 40<d<=360 the coefficients have been calculated as optimal values based on the "resolution" criterion. See "RandQMC user's guide - A package for randomized quasi-Monte Carlo methods in C," by C. Lemieux, M. Cieslak, and K. Luttmer, version January 13 2004, and references cited there (http://www.math.ucalgary.ca/~lemieux/randqmc.html). The values up to d<=360 has been provided to the QuantLib team by Christiane Lemieux, private communication, September 2004. For more info on Sobol' sequences see also "Monte Carlo Methods in Financial Engineering," by P. Glasserman, 2004, Springer, section 5.2.3 \test - the correctness of the returned values is tested by reproducing known good values. - the correctness of the returned values is tested by checking their discrepancy against known good values. */ 00093 class SobolRsg { public: typedef Sample<Array> sample_type; enum DirectionIntegers { Unit, Jaeckel, SobolLevitan, SobolLevitanLemieux }; /*! \pre dimensionality must be <= PPMT_MAX_DIM */ SobolRsg(Size dimensionality, unsigned long seed = 0, DirectionIntegers directionIntegers = Jaeckel); const std::vector<unsigned long>& nextInt32Sequence() const; const SobolRsg::sample_type& SobolRsg::nextSequence() const { const std::vector<unsigned long>& v = nextInt32Sequence(); // normalize to get a double in (0,1) for (Size k=0; k<dimensionality_; k++) sequence_.value[k] = v[k] * normalizationFactor_; return sequence_; } const sample_type& lastSequence() const { return sequence_; } Size dimension() const { return dimensionality_; } private: static const int bits_; static const double normalizationFactor_; Size dimensionality_; mutable unsigned long sequenceCounter_; mutable bool firstDraw_; mutable sample_type sequence_; mutable std::vector<unsigned long> integerSequence_; std::vector<std::vector<unsigned long> > directionIntegers_; }; } #endif

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