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 in QuantLib.
Jäckel has calculated 8 129 334 polynomials: if you need that many
dimensions you can replace the primitivepolynomials.cpp file included
in QuantLib with the one provided in the CD of the "Monte Carlo
Methods in Finance" book.
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integers up to dimension 40, crediting unpublished work of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
The Joe--Kuo numbers and the Kuo numbers are due to Stephen Joe
and Frances Kuo.
S. Joe and F. Y. Kuo, Constructing Sobol sequences with better
two-dimensional projections, preprint Nov 22 2007
The Joe-Kuo numbers are available under a BSD-style license
available at the above link.
Note that the Kuo numbers were generated to work with a
different ordering of primitive polynomials for the first 40
or so dimensions which is why we have the Alternative
Primitive Polynomials.
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.
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 in QuantLib. Jäckel has calculated 8 129 334 polynomials: if you need that many dimensions you can replace the primitivepolynomials.cpp file included in QuantLib with the one provided in the CD of the "Monte Carlo Methods in Finance" book.
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
The Joe--Kuo numbers and the Kuo numbers are due to Stephen Joe and Frances Kuo.
S. Joe and F. Y. Kuo, Constructing Sobol sequences with better two-dimensional projections, preprint Nov 22 2007
See http://web.maths.unsw.edu.au/~fkuo/sobol/ for more information.
The Joe-Kuo numbers are available under a BSD-style license available at the above link.
Note that the Kuo numbers were generated to work with a different ordering of primitive polynomials for the first 40 or so dimensions which is why we have the Alternative Primitive Polynomials.
test