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<html>
<head>
<title>
TRIANGLE_INTEGRALS - Integrals of Polynomials over Triangles
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGLE_INTEGRALS <br> Integrals of Polynomials over Triangles
</h1>
<hr>
<p>
<b>TRIANGLE_INTEGRALS</b>
is a MATLAB library which
returns the exact value of the integral of a polynomial
over the interior of an arbitrary triangle in 2D.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>TRIANGLE_INTEGRALS</b> is available in
<a href = "../../c_src/triangle_integrals/triangle_integrals.html">a C version</a> and
<a href = "../../cpp_src/triangle_integrals/triangle_integrals.html">a C++ version</a> and
<a href = "../../f77_src/triangle_integrals/triangle_integrals.html">a FORTRAN77 version</a> and
<a href = "../../f_src/triangle_integrals/triangle_integrals.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangle_integrals/triangle_integrals.html">a MATLAB version</a> and
<a href = "../../py_src/triangle_integrals/triangle_integrals.html">a Python version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/ball_integrals/ball_integrals.html">
BALL_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit ball in 3D.
</p>
<p>
<a href = "../../m_src/circle_integrals/circle_integrals.html">
CIRCLE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the surface of the unit circle in 2D.
</p>
<p>
<a href = "../../m_src/cube_integrals/cube_integrals.html">
CUBE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit cube in 3D.
</p>
<p>
<a href = "../../m_src/disk_integrals/disk_integrals.html">
DISK_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit disk in 2D.
</p>
<p>
<a href = "../../m_src/hyperball_integrals/hyperball_integrals.html">
HYPERBALL_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit hyperball in M dimensions.
</p>
<p>
<a href = "../../m_src/hypercube_integrals/hypercube_integrals.html">
HYPERCUBE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit hypercube in M dimensions.
</p>
<p>
<a href = "../../m_src/hypersphere_integrals/hypersphere_integrals.html">
HYPERSPHERE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the surface of the unit hypersphere in M dimensions.
</p>
<p>
<a href = "../../m_src/line_integrals/line_integrals.html">
LINE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the length of the unit line in 1D.
</p>
<p>
<a href = "../../m_src/polygon_integrals/polygon_integrals.html">
POLYGON_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of a polygon in 2D.
</p>
<p>
<a href = "../../m_src/pyramid_integrals/pyramid_integrals.html">
PYRAMID_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit pyramid in 3D.
</p>
<p>
<a href = "../../m_src/simplex_integrals/simplex_integrals.html">
SIMPLEX_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit simplex in M dimensions.
</p>
<p>
<a href = "../../m_src/sphere_integrals/sphere_integrals.html">
SPHERE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the surface of the unit sphere in 3D.
</p>
<p>
<a href = "../../m_src/square_integrals/square_integrals.html">
SQUARE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit square in 2D.
</p>
<p>
<a href = "../../m_src/tetrahedron_integrals/tetrahedron_integrals.html">
TETRAHEDRON_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/triangle_dunavant_rule/triangle_dunavant_rule.html">
TRIANGLE_DUNAVANT_RULE</a>,
a MATLAB library which
sets up a Dunavant quadrature rule
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_fekete_rule/triangle_fekete_rule.html">
TRIANGLE_FEKETE_RULE</a>,
a MATLAB library which
defines Fekete rules for quadrature or interpolation
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_felippa_rule/triangle_felippa_rule.html">
TRIANGLE_FELIPPA_RULE</a>,
a MATLAB library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_lyness_rule/triangle_lyness_rule.html">
TRIANGLE_LYNESS_RULE</a>,
a MATLAB library which
returns Lyness-Jespersen quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a MATLAB library which
uses the Monte Carlo method to estimate the integral of a function
over the interior of the unit triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_ncc_rule/triangle_ncc_rule.html">
TRIANGLE_NCC_RULE</a>,
a MATLAB library which
defines Newton-Cotes closed quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_nco_rule/triangle_nco_rule.html">
TRIANGLE_NCO_RULE</a>,
a MATLAB library which
defines Newton-Cotes open quadrature
rules on a triangle.
</p>
<p>
<a href = "../../m_src/triangle_symq_rule/triangle_symq_rule.html">
TRIANGLE_SYMQ_RULE</a>,
a MATLAB library which
returns efficient symmetric quadrature rules,
with exactness up to total degree 50,
over the interior of an arbitrary triangle in 2D,
by Hong Xiao and Zydrunas Gimbutas.
</p>
<p>
<a href = "../../m_src/triangle_wandzura_rule/triangle_wandzura_rule.html">
TRIANGLE_WANDZURA_RULE</a>,
a MATLAB library which
sets up a quadrature rule of exactness 5, 10, 15, 20, 25 or 30
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle01_integrals/triangle01_integrals.html">
TRIANGLE01_INTEGRALS</a>,
a MATLAB library which
determines the exact integral of a monomial over the unit triangle in 2D.
</p>
<p>
<a href = "../../m_src/wedge_integrals/wedge_integrals.html">
WEDGE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit wedge in 3D.
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "i4_to_pascal.m">i4_to_pascal.m</a>,
converts an I4 to a pair of indices in Pascal's triangle;
</li>
<li>
<a href = "i4_to_pascal_test.m">i4_to_pascal_test.m</a>
</li>
<li>
<a href = "i4_to_pascal_degree.m">i4_to_pascal_degree.m</a>,
converts an I4 to the degree (sum) of the corresponding pair
of indices in Pascal's triangle;
</li>
<li>
<a href = "i4_to_pascal_degree_test.m">i4_to_pascal_degree_test.m</a>
</li>
<li>
<a href = "pascal_to_i4.m">pascal_to_i4.m</a>,
converts a pair of indices in Pascal's triangle to a linear index;
</li>
<li>
<a href = "pascal_to_i4_test.m">pascal_to_i4_test.m</a>
</li>
<li>
<a href = "poly_power_linear.m">poly_power_linear.m</a>,
computes a power of a linear polynomial in X and Y.
</li>
<li>
<a href = "poly_power_linear_test.m">poly_power_linear_test.m</a>
</li>
<li>
<a href = "poly_power.m">poly_power.m</a>,
computes a power of a polynomial in X and Y.
</li>
<li>
<a href = "poly_power_test.m">poly_power_test.m</a>
</li>
<li>
<a href = "poly_print.m">poly_print.m</a>,
prints a polynomial in X and Y.
</li>
<li>
<a href = "poly_print_test.m">poly_print_test.m</a>
</li>
<li>
<a href = "poly_product.m">poly_product.m</a>,
computes the product of two polynomials in X and Y.
</li>
<li>
<a href = "poly_product_test.m">poly_product_test.m</a>
</li>
<li>
<a href = "r8mat_print.m">r8mat_print.m</a>,
prints an R8MAT;
</li>
<li>
<a href = "r8mat_print_test.m">r8mat_print_test.m</a>
</li>
<li>
<a href = "r8mat_print_some.m">r8mat_print_some.m</a>,
prints some of an R8MAT;
</li>
<li>
<a href = "r8mat_print_some_test.m">r8mat_print_some_test.m</a>
</li>
<li>
<a href = "rs_to_xy_map.m">rs_to_xy_map.m</a>,
determines the map coefficients from the reference to physical triangle.
</li>
<li>
<a href = "rs_to_xy_map_test.m">rs_to_xy_map_test.m</a>
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>
prints the current YMDHMS date as a time stamp.
</li>
<li>
<a href = "triangle01_monomial_integral.m">
triangle01_monomial_integral.m</a>
computes the integral of a monomial over the reference triangle.
</li>
<li>
<a href = "triangle01_monomial_integral_test.m">
triangle01_monomial_integral_test.m</a>
</li>
<li>
<a href = "triangle01_poly_integral.m">triangle01_poly_integral.m</a>
computes the integral of a polynomial over the reference triangle.
</li>
<li>
<a href = "triangle01_poly_integral_test.m">
triangle01_poly_integral_test.m</a>
</li>
<li>
<a href = "triangle_area.m">triangle_area.m</a>
computes the area of a arbitary triangle.
</li>
<li>
<a href = "triangle_area_test.m">triangle_area_test.m</a>
</li>
<li>
<a href = "triangle_monomial_integral.m">
triangle_monomial_integral.m</a>
computes the integral of a monomial over an arbitrary triangle.
</li>
<li>
<a href = "triangle_monomial_integral_test.m">
triangle_monomial_integral_test.m</a>
</li>
<li>
<a href = "triangle_poly_integral.m">triangle_poly_integral.m</a>
computes the integral of a polynomial over an arbitrary triangle.
</li>
<li>
<a href = "triangle_poly_integral_test.m">
triangle_poly_integral_test.m</a>
</li>
<li>
<a href = "triangle_xy_integral.m">
triangle_xy_integral.m</a>
computes the integral of xy over an arbitrary triangle.
</li>
<li>
<a href = "triangle_xy_integral_test.m">
triangle_xy_integral_test.m</a>
</li>
<li>
<a href = "trinomial.m">trinomial.m</a>,
computes a trinomial coefficient;
</li>
<li>
<a href = "trinomial_test.m">trinomial_test.m</a>
</li>
<li>
<a href = "xy_to_rs_map.m">xy_to_rs_map.m</a>,
determines the map coefficients from the physical to reference triangle.
</li>
<li>
<a href = "xy_to_rs_map_test.m">xy_to_rs_map_test.m</a>
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "triangle_integrals_test.m">triangle_integrals_test.m</a>,
a sample calling program.
</li>
<li>
<a href = "triangle_integrals_test_output.txt">triangle_integrals_test_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 18 April 2015.
</i>
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