NIEDERREITER2
The Niederreiter Quasirandom Sequence [Base 2]

NIEDERREITER2 is a FORTRAN90 library which implements 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 the desired element.

The original, true, correct version of ACM TOMS Algorithm 738 is available in the TOMS subdirectory of the NETLIB web site. The version displayed here has been converted to FORTRAN90, and other internal changes have been made to suit me.

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 FORTRAN90 library which computes elements of a Centroidal Voronoi Tessellation.

FAURE, a FORTRAN90 library which computes elements of a Faure quasirandom sequence.

HALTON, a FORTRAN90 library which computes elements of a Halton Quasi Monte Carlo (QMC) sequence, using a simple interface.

HAMMERSLEY, a FORTRAN90 library which computes elements of a Hammersley Quasi Monte Carlo (QMC) sequence, using a simple interface.

HEX_GRID, a FORTRAN90 library which computes elements of a hexagonal grid dataset.

HEX_GRID_ANGLE, a FORTRAN90 library which computes elements of an angled hexagonal grid dataset.

IEEE_UNIFORM_SAMPLE, a FORTRAN90 library which tries to uniformly sample the discrete set of values that represent the legal IEEE real numbers;

IHS, a FORTRAN90 library which computes elements of an improved distributed Latin hypercube dataset.

LATIN_CENTER, a FORTRAN90 library which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_EDGE, a FORTRAN90 library which computes elements of a Latin Hypercube dataset, choosing edge points.

LATIN_RANDOM, a FORTRAN90 library which computes elements of a Latin Hypercube dataset, choosing points at random.

LATTICE_RULE, a FORTRAN90 library which approximates multidimensional integrals using lattice rules.

LCVT, a FORTRAN90 library which computes a latinized Centroidal Voronoi Tessellation.

NIEDERREITER, a FORTRAN90 library which computes a Niederreiter sequence for an arbitary base.

NIEDERREITER2_DATASET, a FORTRAN90 program which creates a Niederreiter quasirandom dataset with base 2;

NORMAL, a FORTRAN90 library which computes elements of a sequence of pseudorandom normally distributed values.

SOBOL, a FORTRAN90 library which computes elements of a Sobol quasirandom sequence.

TOMS738, a FORTRAN90 library which is a version of ACM TOMS algorithm 738, for evaluating Niederreiter sequences.

UNIFORM, a FORTRAN90 library which computes elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT, a FORTRAN90 library which computes elements of a van der Corput pseudorandom 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. Rudolf Lidl, Harald Niederreiter,
   Finite Fields,
   Second Edition,
   Cambridge University Press, 1997,
   ISBN: 0521392314,
   LC: QA247.3.L53.
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,
   ISBN13: 978-0-898712-95-7.

Source Code:

• niederreiter2.f90, the source code;

Examples and Tests:

• niederreiter2_test.f90, the sample test code;
• niederreiter2_test.txt, the output file;

List of Routines:

• CALCC2 computes the constants C(I,J,R).
• CALCV2 calculates the constants V(J,R).
• GET_UNIT returns a free FORTRAN unit number.
• NIEDERREITER2 returns an element of the Niederreiter sequence base 2.
• NIEDERREITER2_GENERATE generates a set of Niederreiter values.
• NIEDERREITER_WRITE writes a set of Niederreiter values to a file.
• PLYMUL2 multiplies two polynomials in the field of order 2.
• SETFLD2 sets up arithmetic tables for the finite field of order 2.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING writes the current YMDHMS date into a string.

You can go up one level to the FORTRAN90 source codes.