LATTICE
Lattice Rules for Multiple Integration

LATTICE is a MATLAB 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.

Related Programs:

CLENSHAW_CURTIS is a MATLAB library which can set up a Clenshaw Curtis quadrature grid in multiple dimensions.

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

GRID is a MATLAB library which computes elements of a grid dataset.

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

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

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

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

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

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

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

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

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

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

NIEDERREITER2 is a MATLAB library which computes elements of a Niederreiter quasirandom sequence with base 2.

NINT_EXACTNESS is a MATLAB program which measures the polynomial exactness of a multidimensional quadrature rule.

NINTLIB is a MATLAB library which contains routines for numerical estimation of integrals in multiple dimensions.

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

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

QUADRULE is a MATLAB library which defines a number of quadrature rules.

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

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

TEST_NINT is a MATLAB library which defines test functions for multi-dimensional quadrature routines.

UNIFORM is a MATLAB 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:

• e_01_2d.m, the exact integral of 2d test function #1.
• f_01_2d.m, is the 2D test function #1.
• f2.m, evaluates a function of a scalar used in defining P2(Q).
• f20_s.m, evaluates a function of a vector used in defining P2(Q).
• fibonacci.m, returns the Fibonacci number of given index.
• fibonacci_lattice_b.m, applies an optimal Fibonacci lattice integration rule in 2D.
• fibonacci_lattice_q.m, applies a Fibonacci lattice integration rule in 2D.
• fibonacci_lattice_q_nodes.m, returns Fibonacci lattice nodes in 2D.
• fibonacci_lattice_q1.m, applies a Fibonacci lattice integration rule in 2D.
• fibonacci_lattice_q2.m, applies a Fibonacci lattice integration rule in 2D.
• fibonacci_lattice_q3.m, applies a Fibonacci lattice integration rule in 2D.
• fibonacci_lattice_t.m, applies a symmetric Fibonacci lattice integration rule in 2D.
• find_z20.m, finds the appropriate Z vector to minimize P2(QS).
• gray_next.m, generates the next Gray code by switching one item at a time.
• i4_huge.m, returns a "huge" I4.
• i4vec_print.m, prints an I4VEC.
• lattice.m, applies a lattice integration rule.
• lattice_np0.m, applies a lattice integration rule to a nonperiodic function.
• lattice_np1.m, applies a lattice integration rule to a nonperiodic function.
• lattice_print.m, prints the points in a lattice rule.
• monte_carlo.m, applies a Monte Carlo integration rule.
• prime.m, returns any of the first PRIME_MAX prime numbers.
• r8_huge.m, returns a "huge" R8.
• r8mat_transpose_print.m, prints a R8MAT, transposed.
• r8mat_transpose_print_some.m, prints some of a R8MAT, transposed.
• r8vec_uniform_01.m, returns a random R8VEC.
• s_len_trim.m, returns the length of a string to the last nonblank.
• timestamp.m, prints the current YMDHMS date as a time stamp.
• tuple_next.m, computes the next element of a tuple space.

Examples and Tests:

• lattice_test.m, calls all the tests.
• lattice_prb_output.txt, the output file.
• lattice_test01.m, tests FIBONACCI_LATTICE_Q;
• lattice_test02.m, tests FIBONACCI_LATTICE_T.
• lattice_test03.m, tests tests FIBONACCI_LATTICE_B;
• lattice_test04.m, tests FIBONACCI_LATTICE_Q1;
• lattice_test05.m, tests FIBONACCI_LATTICE_Q2;
• lattice_test06.m, tests tests FIBONACCI_LATTICE_Q3;
• lattice_test07.m, tests tests LATTICE.
• lattice_test08.m, tests tests LATTICE.
• lattice_test085.m, tests tests LATTICE.
• lattice_test09.m, tests tests LATTICE_NP0;
• lattice_test10.m, tests LATTICE_NP1;
• lattice_test11.m, tests MONTE_CARLO;
• lattice_test12.m, tests LATTICE_PRINT.
• lattice_test13.m, tests tests FIND_Z20.
• lattice_test14.m, tests FIBONACCI_LATTICE_Q_NODES.
• 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.

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