TEST_TRI_INT
Triangle Integration Test Functions

TEST_TRI_INT is a MATLAB library which defines test functions to be integrated over a triangle.

It is possible to invoke a particular function by number, or to try out all available functions, as demonstrated in the sample calling program.

For convenience, all the integrand functions have been scaled by a constant, so that the integral of the function over the specific domain is exactly 1.

The test functions include F(X,Y)=

1. 1 on the unit triangle.
2. x on the unit triangle.
3. y on the unit triangle.
4. x² on the unit triangle.
5. x*y on the unit triangle.
6. y² on the unit triangle.
7. x³ on the unit triangle.
8. x⁴ on the unit triangle.
9. x⁵ on the unit triangle.
10. x^(-0.2) remapped to (1,0), (5,0), (5,1).
11. (x+y)^(-0.2) on the unit triangle.
12. (1-x-y)^(-0.2) remapped to (-1,-3), (3,-2), (-1,2).
13. (x*y)^(-0.2) remapped to (0,0), (-7,0), (0,-3).
14. 1/sqrt(x) + 1/sqrt(y) + 1/sqrt(x+y) on the unit triangle.
15. 1/sqrt(1-x-y) on the unit triangle.
16. log(x*y) on the unit triangle.
17. 1/sqrt(|x-1/4|) + 1/sqrt(|y-1/2|) on the unit triangle.
18. log ( x + y ) on the unit triangle.
19. sin ( x ) cos ( 5 y ) on the unit triangle.
20. sin ( 11 x ) cos ( y ) on the unit triangle.
21. 1 / r on the unit triangle,
    r = sqrt ( x^2+y^2).
22. log ( r ) / r on the unit triangle,
    r = sqrt ( x^2+y^2).

The library includes a routines to define the integrand function, the triangle over which the integral is to be carried out, and a title for the problem. Thus, for each integrand function, four routines are supplied. For instance, for function #4, we have the routines:

• P04_FUN evaluates the integrand for problem 4.
• P04_TITLE returns the title for problem 4.
• P04_VERTICES returns the integration triangle for problem 4

Moreover, since the same interface is used for each function, if you wish to work with problem 16 instead, you simply change the "04" to "07" in your routine calls.

If you wish to call all of the functions, then you simply use the generic interface, which again has four routines, but which requires you to specify the problem number as an extra input argument:

• P00_FUN evaluates the integrand for problem PROB.
• P00_SINGULARITY warns if the integran for problem PROB has vertex, edge or interior singularities that might cause problems for some integration schemes.
• P00_TITLE returns the title for problem PROB.
• P00_VERTICES returns the integration triangle for problem PROB.

Finally, some demonstration routines are built in for simple quadrature methods. These routines include

• P00_WANDZURA05_SUB applies an order-5 accurate Wandzura rule, plus subdivision of the triangle.
• P00_MONTE_CARLO uses Monte Carlo sampling.
• P00_VERTEX_SUB averages values at vertices, and uses subdivision.

Licensing:

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

Languages:

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

Related Data and Programs:

DUNAVANT, a MATLAB library which defines Dunavant rules for quadrature on a triangle.

FEKETE, a MATLAB library which defines a Fekete rule for quadrature or interpolation over a triangle.

GM_RULE, a MATLAB library which defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

LYNESS_RULE, a MATLAB library which returns Lyness-Jespersen quadrature rules for the triangle.

NCC_TRIANGLE, a MATLAB library which defines Newton-Cotes closed quadrature rules on a triangle.

NCO_TRIANGLE, a MATLAB library which defines Newton-Cotes open quadrature rules on a triangle.

STROUD, a MATLAB library which includes some quadrature rules for triangles.

TOMS612, a FORTRAN77 library which can estimate the integral of a function over a triangle.

TRIANGLE_MONTE_CARLO, a MATLAB program which uses the Monte Carlo method to estimate integrals over a triangle.

TRIANGULATION, a MATLAB library which can be used for triangulations, and includes some quadrature rules on triangles.

WANDZURA, a MATLAB library which defines Wandzura rules for quadrature on a triangle.

Reference:

1. Elise deDoncker, Ian Robinson,
   An Algorithm for Automatic Integration Over a Triangle Using Nonlinear Extrapolation,
   ACM Transactions on Mathematical Software,
   Volume 10, Number 1, March 1984, pages 1-16.
2. Elise deDoncker, Ian Robinson,
   Algorithm 612: Integration over a Triangle Using Nonlinear Extrapolation,
   ACM Transactions on Mathematical Software,
   Volume 10, Number 1, March 1984, pages 17-22.
3. Stephen Wandzura, Hong Xiao,
   Symmetric Quadrature Rules on a Triangle,
   Computers and Mathematics with Applications,
   Volume 45, pages 1829-1840, 2003.

Source Code:

• i4_huge.m returns a "huge" I4.
• get_prob_num.m returns the number of test integration problems.
• p00_fun.m evaluates the integrand for any problem.
• p00_monte_carlo.m applies the Monte Carlo rule to integrate a function.
• p00_singularity.m warns of common singularities for any problem.
• p00_title.m returns the title for any problem.
• p00_vertex_sub.m approximates an integral in a triangle by subdivision.
• p00_vertices.m returns the vertices for any problem.
• p00_wandzura05_sub.m uses subdivision and a Wandzura rule.
• p01_fun.m evaluates the integrand for problem 1.
• p01_title.m returns the title of problem 1.
• p01_vertices.m returns the vertices for problem 1.
• p02_fun.m evaluates the integrand for problem 2.
• p02_title.m returns the title of problem 2.
• p02_vertices.m returns the vertices for problem 2.
• p03_fun.m evaluates the integrand for problem 3.
• p03_title.m returns the title of problem 3.
• p03_vertices.m returns the vertices for problem 3.
• p04_fun.m evaluates the integrand for problem 4.
• p04_title.m returns the title of problem 4.
• p04_vertices.m returns the vertices for problem 4.
• p05_fun.m evaluates the integrand for problem 5.
• p05_title.m returns the title of problem 5.
• p05_vertices.m returns the vertices for problem 5.
• p06_fun.m evaluates the integrand for problem 6.
• p06_title.m returns the title of problem 6.
• p06_vertices.m returns the vertices for problem 6.
• p07_fun.m evaluates the integrand for problem 7.
• p07_title.m returns the title of problem 7.
• p07_vertices.m returns the vertices for problem 7.
• p08_fun.m evaluates the integrand for problem 8.
• p08_title.m returns the title of problem 8.
• p08_vertices.m returns the vertices for problem 8.
• p09_fun.m evaluates the integrand for problem 9.
• p09_title.m returns the title of problem 9.
• p09_vertices.m returns the vertices for problem 9.
• p10_fun.m evaluates the integrand for problem 10.
• p10_title.m returns the title of problem 10.
• p10_vertices.m returns the vertices for problem 10.
• p11_fun.m evaluates the integrand for problem 11.
• p11_title.m returns the title of problem 11.
• p11_vertices.m returns the vertices for problem 11.
• p12_fun.m evaluates the integrand for problem 12.
• p12_title.m returns the title of problem 12.
• p12_vertices.m returns the vertices for problem 12.
• p13_fun.m evaluates the integrand for problem 13.
• p13_title.m returns the title of problem 13.
• p13_vertices.m returns the vertices for problem 13.
• p14_fun.m evaluates the integrand for problem 14.
• p14_title.m returns the title of problem 14.
• p14_vertices.m returns the vertices for problem 14.
• p15_fun.m evaluates the integrand for problem 15.
• p15_title.m returns the title of problem 15.
• p15_vertices.m returns the vertices for problem 15.
• p16_fun.m evaluates the integrand for problem 16.
• p16_title.m returns the title of problem 16.
• p16_vertices.m returns the vertices for problem 16.
• p17_fun.m evaluates the integrand for problem 17.
• p17_title.m returns the title of problem 17.
• p17_vertices.m returns the vertices for problem 17.
• p18_fun.m evaluates the integrand for problem 18.
• p18_title.m returns the title of problem 18.
• p18_vertices.m returns the vertices for problem 18.
• p19_fun.m evaluates the integrand for problem 19.
• p19_title.m returns the title of problem 19.
• p19_vertices.m returns the vertices for problem 19.
• p20_fun.m evaluates the integrand for problem 20.
• p20_title.m returns the title of problem 20.
• p20_vertices.m returns the vertices for problem 20.
• p21_fun.m evaluates the integrand for problem 21.
• p21_title.m returns the title of problem 21.
• p21_vertices.m returns the vertices for problem 21.
• p22_fun.m evaluates the integrand for problem 22.
• p22_title.m returns the title of problem 22.
• p22_vertices.m returns the vertices for problem 22.
• r8vec_uniform_01.m returns a unit pseudorandom R8VEC.
• subtriangle_next.m computes the next subtriangle of a triangle.
• timestamp.m prints the current YMDHMS date as a time stamp.
• triangle_area.m computes the area of a triangle in 2D.
• triangle_sample.m returns random points in a triangle.

Examples and Tests:

• test_tri_int_test.m, a sample calling program.
• test_tri_int_test01.m, tests GET_PROB_NUM and P00_TITLE.
• test_tri_int_test02.m, tests P00_MONTE_CARLO.
• test_tri_int_test03.m, tests P00_VERTEX_SUB.
• test_tri_int_test04.m, tests P00_WANDZURA05_SUB.
• test_tri_int_test_output.txt, the output from a run of the sample program.

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