NIEDERREITER2
The Niederreiter Quasirandom Sequence [Base 2]

NIEDERREITER2 is a MATLAB library 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 GNU LGPL license.

Languages:

NIEDERREITER2 is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CVT, a MATLAB library which computes elements of a Centroidal Voronoi Tessellation.

FAURE, a MATLAB library which computes elements of a Faure quasirandom sequence.

GRID, a MATLAB library which computes elements of a grid sequence.

HALTON, a MATLAB library which computes elements of a Halton quasirandom sequence.

HAMMERSLEY, a MATLAB library which computes elements of a Hammersley quasirandom sequence.

HEX_GRID, a MATLAB library which computes elements of a hexagonal grid dataset.

HEX_GRID_ANGLE, a MATLAB library which computes elements of an angled hexagonal grid dataset.

IHS, a MATLAB library which computes elements of an improved distributed Latin hypercube dataset.

LATIN_CENTER, a MATLAB library which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_EDGE, a MATLAB library which computes elements of a Latin Hypercube dataset, choosing edge points.

LATIN_RANDOM, a MATLAB library which computes elements of a Latin Hypercube dataset, choosing points at random.

LCVT, a MATLAB library which computes a latinized Centroidal Voronoi Tessellation.

NIEDERREITER2_DATASET, a MATLAB program which creates a Niederreiter quasirandom dataset with base 2;

SOBOL, a MATLAB library which computes elements of a Sobol quasirandom sequence.

TOMS738, a FORTRAN77 library which evaluates Niederreiter's quasirandom sequence;
this is ACM TOMS algorithm 738;

UNIFORM, a MATLAB library which computes elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT, a MATLAB library which computes elements of a 1D van der Corput sequence.

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.
• get_seed.m, returns a seed for the uniform random number generator.
• i4_uniform.m, returns a random integer in a given range.
• 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.
• timestamp.m, prints the current YMDHMS date as a time stamp.

Examples and Tests:

• niederreiter2_test.m, runs all the tests;
• niederreiter2_test01.m, tests setfld2;
• niederreiter2_test02.m, tests plymul2;
• niederreiter2_test03.m, tests exor;
• niederreiter2_test04.m, tests niederreiter2;
• niederreiter2_test05.m, shows how the seed may be used to manipulate the output of niederreiter2;
• niederreiter2_test_output.txt, output from the tests;

You can go up one level to the MATLAB source codes.