LATTICE_RULE
Lattice Rules for Multiple Integration

LATTICE_RULE is a FORTRAN90 library which approximates integrals in multiple dimensions using lattice rules.

The lattice rules are defined on the unit square, or in higher dimensions, on the unit hypercube.

Lattice rules are more suitable to integrands that are periodic, with period 1, in all variables. However, there are techniques that may be used when the integrand does not satisfy this requirement.

The performance of a lattice rule depends heavily on the choice of the generator vectors. Once the spatial dimension and the number of points have been chosen, there are techniques for finding a good generator vector.

The simplest lattice rules are called "rank 1" rules, and use a lattice generated by multiples of a single generator vector. More elaborate rules can be generated by using more generator vectors, up to the maximum, which is the dimension of the space.

Licensing:

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

Languages:

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

Related Programs:

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

GRID, a FORTRAN90 library which computes elements of a grid dataset.

HALTON, a FORTRAN90 library which computes elements of a Halton quasirandom sequence.

HAMMERSLEY, a FORTRAN90 library which computes elements of a Hammersley quasirandom sequence.

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.

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.

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

NIEDERREITER2, a FORTRAN90 library which computes elements of a Niederreiter quasirandom sequence with base 2.

PRODUCT_RULE, a FORTRAN90 program which can create a multidimensional quadrature rule as a product of one dimensional rules.

QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRULE, a FORTRAN90 library which defines a number of quadrature rules.

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

STROUD, a FORTRAN90 library which contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TEST_NINT, a FORTRAN90 library which defines test functioins for multi-dimensional quadrature routines.

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

Reference:

1. Seymour Haber,
   Parameters for Integrating Periodic Functions of Several Variables,
   Mathematics of Computation,
   Volume 41, Number 163, July 1983, pages 115-129.
2. Albert Nijenhuis, Herbert Wilf,
   Combinatorial Algorithms for Computers and Calculators,
   Second Edition,
   Academic Press, 1978,
   ISBN: 0-12-519260-6,
   LC: QA164.N54.
3. Ian Sloan, Stephen Joe,
   Lattice Methods for Multiple Integration,
   Oxford University Press, 1994,
   ISBN: 0198534728,
   LC: QA311.S56

Source Code:

• lattice_rule.f90, the source code.

Examples and Tests:

• lattice_rule_test.f90, a sample calling program.
• lattice_rule_test.txt, the output file.
• f21.txt, the 21 points of a Fibonacci lattice for K = 8, M = 21.
• f21.png, a PNG image of a Fibonacci lattice.
• f144.txt, the 144 points of a Fibonacci lattice for K = 12, M = 144.
• f144.png, a PNG image of a Fibonacci lattice.

List of Routines:

• E_01_2D is the exact integral of 2d test function #1.
• F_01_2D is the 2D test function #1.
• F2 evaluates a function of a scalar used in defining P2(Q).
• F20_S evaluates a function of a vector used in defining P2(Q).
• FIBONACCI returns the Fibonacci number of given index.
• FIBONACCI_LATTICE_B applies an optimal Fibonacci lattice integration rule in 2D.
• FIBONACCI_LATTICE_Q applies a Fibonacci lattice integration rule in 2D.
• FIBONACCI_LATTICE_Q_NODES returns Fibonacci lattice nodes in 2D.
• FIBONACCI_LATTICE_Q1 applies a Fibonacci lattice integration rule in 2D.
• FIBONACCI_LATTICE_Q2 applies a Fibonacci lattice integration rule in 2D.
• FIBONACCI_LATTICE_Q3 applies a Fibonacci lattice integration rule in 2D.
• FIBONACCI_LATTICE_T applies a symmetric Fibonacci lattice integration rule in 2D.
• FIND_Z20 finds the appropriate Z vector to minimize P2(QS).
• GRAY_NEXT generates the next Gray code by switching one item at a time.
• I4VEC_PRINT prints an I4VEC.
• LATTICE applies a lattice integration rule.
• LATTICE_NP0 applies a lattice integration rule to a nonperiodic function.
• LATTICE_NP1 applies a lattice integration rule to a nonperiodic function.
• LATTICE_PRINT prints the points in a lattice rule.
• MONTE_CARLO applies a Monte Carlo integration rule.
• PRIME returns any of the first PRIME_MAX prime numbers.
• R8MAT_TRANSPOSE_PRINT prints a R8MAT, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of a R8MAT, transposed.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TUPLE_NEXT computes the next element of a tuple space.

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