# triangle01_monte_carlo

triangle01_monte_carlo, a FORTRAN90 code which uses the Monte Carlo method to estimate the integral of a function F(X,Y) over the interior of the unit triangle in 2D.

The interior of the unit triangle in 2D is defined by the constraints:

```        0 <= X
0 <= Y
X + Y <= 1
```
The functions F(X,Y) are monomials, having the form
```        F(X,Y) = X^E(1) * Y^E(2)
```
where the exponents are nonnegative integers.

### Languages:

triangle01_monte_carlo is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

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TRIANGLE_NCC_RULE, a FORTRAN90 code which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_NCO_RULE, a FORTRAN90 code which defines Newton-Cotes open (NCO) quadrature rules over the interior of the triangle in 2D.

TRIANGLE_SYMQ_RULE, a FORTRAN90 code 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.

TRIANGLE_WANDZURA_RULE, a FORTRAN90 code which sets up a quadrature rule of exactness 5, 10, 15, 20, 25 or 30 over the interior of a triangle in 2D.

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### Reference:

1. Claudio Rocchini, Paolo Cignoni,
Generating Random Points in a Tetrahedron,
Journal of Graphics Tools,
Volume 5, Number 4, 2000, pages 9-12.
2. Reuven Rubinstein,
Monte Carlo Optimization, Simulation and Sensitivity of Queueing Networks,
Krieger, 1992,
ISBN: 0894647644,
LC: QA298.R79.
3. Greg Turk,
Generating Random Points in a Triangle,
in Graphics Gems I,
edited by Andrew Glassner,
AP Professional, 1990,
ISBN: 0122861663,
LC: T385.G697

### Source Code:

Last revised on 08 September 2020.