HYPERBALL_VOLUME_QUAD
Quadrature Estimate of Hyperball Volume

HYPERBALL_VOLUME_QUAD is a FORTRAN90 program which investigates the behavior of a quadrature rule when it is applied to the characteristic function of the unit M-dimensional hyperball, which is a discontinuous function.

The program integrates the function over the unit hypercube [-1,+1]^M, where the spatial dimension M is arbitrary. The value of M is inferred from the user input quadrature rule information.

The problem used as example input to the program works in M = 6 dimensions. In that case, the volume of the hypercube is 64; the volume of the hyperball is pi^3/6, or about 5.16771.

Because the integrand is discontinuous, any quadrature rule based on the idea of interpolation will probably be unable to do a good job. A family of quadrature rules, which rely on increasing the order of interpolation to improve accuracy, will probably get increasingly bad answers.

By contrast, a basic Monte Carlo rule, which assumes nothing about the function, integrates this function just as well as it integrates most any other square-integrable function. (That's both the strength and weakness of the blunt instrument we call Monte Carlo integration.)

The program assumes that the quadrature rule is defined by three text files.

Usage:

hyperball_volume_quad prefix

• prefix_r.txt contains the coordinates of two points that define the quadrature region.
• prefix_w.txt contains the weights;
• prefix_x.txt contains the points;

Licensing:

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

Languages:

HYPERBALL_VOLUME_QUAD is available in a FORTRAN90 version.

Related Data and Programs:

HYPERBALL_INTEGRALS, a FORTRAN90 library which defines test functions for integration over the interior of the unit hyperball in M dimensions.

HYPERBALL_VOLUME_MONTE_CARLO, a FORTRAN90 program which applies a Monte Carlo method to estimate the volume of the unit hyperball in M dimensions;

NINTLIB, a FORTRAN90 library which numerically estimates integrals in multiple dimensions.

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

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

TEST_NINT, a FORTRAN90 library which defines integrand functions for testing multidimensional quadrature routines.

TESTPACK, a FORTRAN90 library which defines a set of integrands used to test multidimensional quadrature.

Reference:

1. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
2. Gerald Folland,
   How to Integrate a Polynomial Over a Sphere,
   American Mathematical Monthly,
   Volume 108, Number 5, May 2001, pages 446-448.

Source Code:

• hyperball_volume_quad.f90, the source code.

Examples and Tests:

• hyperball_volume_quad.txt, the output file.

List of Routines:

• MAIN is the main program for HYPERBALL_VOLUME_QUAD.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• GET_UNIT returns a free FORTRAN unit number.
• HYPERBALL01_INDICATOR evaluates the unit hyperball indicator function.
• HYPERBALL01_VOLUME computes the volume of the unit hyperball.
• R8MAT_DATA_READ reads data from an R8MAT file.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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