SobolRsg Class Reference

#include <ql/RandomNumbers/sobolrsg.hpp>

List of all members.


Detailed Description

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Bratley and Fox published coefficients of the free direction integers up to dimension 40, crediting unpublished work of Sobol' and Levitan.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

Tests:
  • 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.


Public Types

typedef Sample< Arraysample_type
enum  DirectionIntegers { Unit, Jaeckel, SobolLevitan, SobolLevitanLemieux }

Public Member Functions

 SobolRsg (Size dimensionality, unsigned long seed=0, DirectionIntegers directionIntegers=Jaeckel)
const std::vector< unsigned
long > & 
nextInt32Sequence () const
const SobolRsg::sample_typenextSequence () const
const sample_typelastSequence () const
Size dimension () const


Constructor & Destructor Documentation

SobolRsg Size  dimensionality,
unsigned long  seed = 0,
DirectionIntegers  directionIntegers = Jaeckel
 

Precondition:
dimensionality must be <= PPMT_MAX_DIM