niederreiter2

niederreiter2, an Octave code which produces elements of the Niederreiter quasirandom sequence, using a base of 2.

A quasirandom or low discrepancy sequence, such as the Faure, Halton, Hammersley, Niederreiter or Sobol sequences, is "less random" than a pseudorandom number sequence, but more useful for such tasks as approximation of integrals in higher dimensions, and in global optimization. This is because low discrepancy sequences tend to sample space "more uniformly" than random numbers. Algorithms that use such sequences may have superior convergence.

NIEDERREITER2 is an adapation of the INLO2 and GOLO2 routines in ACM TOMS Algorithm 738. The original code can only compute the "next" element of the sequence. The revised code allows the user to specify the index of any desired element.

The original, true, correct version of ACM TOMS Algorithm 738 is available in the TOMS subdirectory of the NETLIB web site.

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

Languages:

niederreiter2 is available in a C++ version and a Fortran90 version and a MATLAB version and an Octave version.

Related Data and Programs:

niederreiter2_test

cvt, an Octave code which computes elements of a Centroidal Voronoi Tessellation.

faure, an Octave code which computes elements of a Faure quasirandom sequence.

halton, an Octave code which computes elements of a Halton quasirandom sequence.

hammersley, an Octave code which computes elements of a Hammersley quasirandom sequence.

latin_center, an Octave code which computes elements of a Latin Hypercube dataset, choosing center points.

latin_edge, an Octave code which computes elements of a Latin Hypercube dataset, choosing edge points.

latin_random, an Octave code which computes elements of a Latin Hypercube dataset, choosing points at random.

sobol, an Octave code which computes elements of a Sobol quasirandom sequence.

uniform, an Octave code which computes elements of a uniform pseudorandom sequence.

van_der_corput, an Octave code which computes elements of a 1D van der Corput sequence.

Author:

MATLAB version by John Burkardt; performance enhancements by Jeremy Dewar, Tulane University.

Reference:

1. Paul Bratley, Bennett Fox,
   Algorithm 659: Implementing Sobol's Quasirandom Sequence Generator,
   ACM Transactions on Mathematical Software,
   Volume 14, Number 1, pages 88-100, 1988.
2. Paul Bratley, Bennett Fox, Harald Niederreiter,
   Algorithm 738: Programs to Generate Niederreiter's Low-Discrepancy Sequences,
   ACM Transactions on Mathematical Software,
   Volume 20, Number 4, pages 494-495, 1994.
3. Paul Bratley, Bennett Fox, Harald Niederreiter,
   Implementation and Tests of Low Discrepancy Sequences,
   ACM Transactions on Modeling and Computer Simulation,
   Volume 2, Number 3, pages 195-213, 1992.
4. Bennett Fox,
   Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators,
   ACM Transactions on Mathematical Software,
   Volume 12, Number 4, pages 362-376, 1986.
5. R Lidl, Harald Niederreiter,
   Finite Fields,
   Cambridge University Press, 1984, page 553.
6. Harald Niederreiter,
   Low-discrepancy and low-dispersion sequences,
   Journal of Number Theory,
   Volume 30, 1988, pages 51-70.
7. Harald Niederreiter,
   Random Number Generation and quasi-Monte Carlo Methods,
   SIAM, 1992.

Source Code:

• calcc2.m, computes the constants C(I,J,R).
• calcv2.m, computes the constants V(J,R).
• exor.m, computes the bitwise exclusive OR of two integers.
• niederreiter2.m, returns an element of the Niederreiter sequence base 2.
• niederreiter2_generate.m, generates a set of Niederreiter values.
• plymul2.m, multiplies two polynomials in the field of order 2.
• r8mat_write.m, writes an R8MAT file.
• setfld2.m, sets up arithmetic tables for the finite field of order 2.