# pysimplicialcubature
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[](http://pysimplicialcubature.readthedocs.io)
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This package is a port of a part of the R package **SimplicialCubature**,
written by John P. Nolan, and which contains R translations of
some Matlab and Fortran code written by Alan Genz. In addition it
provides a function for the exact computation of the integral of a
polynomial over a simplex.
___
A simplex is a triangle in dimension 2, a tetrahedron in dimension 3.
This package provides two main functions: `integrateOnSimplex`, to integrate
an arbitrary function on a simplex, and `integratePolynomialOnSimplex`, to
get the exact value of the integral of a multivariate polynomial on a
simplex.
Suppose for example you want to evaluate the following integral:
$$\int\_0^1\int\_0^x\int\_0^y \exp(x + y + z) \text{d}z \text{d}y \text{d}x.$$
```python
from pysimplicialcubature.simplicialcubature import integrateOnSimplex
from math import exp
# simplex vertices
v1 = [0.0, 0.0, 0.0]
v2 = [1.0, 1.0, 1.0]
v3 = [0.0, 1.0, 1.0]
v4 = [0.0, 0.0, 1.0]
# simplex
S = [v1, v2, v3, v4]
# function to integrate
f = lambda x : exp(x[0] + x[1] + x[2])
# integral of f on S
I_f = integrateOnSimplex(f, S)
I_f["integral"]
```
Now let's turn to a polynomial example. You have to define the polynomial with
`sympy.Poly`.
```python
from pysimplicialcubature.simplicialcubature import integratePolynomialOnSimplex
from sympy import Poly
from sympy.abc import x, y, z
# simplex vertices
v1 = [1.0, 1.0, 1.0]
v2 = [2.0, 2.0, 3.0]
v3 = [3.0, 4.0, 5.0]
v4 = [3.0, 2.0, 1.0]
# simplex
S = [v1, v2, v3, v4]
# polynomial to integrate
P = Poly(x**4 + y + 2*x*y**2 - 3*z, x, y, z, domain = "RR")
# integral of P on S
integratePolynomialOnSimplex(P, S)
```
## References
- A. Genz and R. Cools.
*An adaptive numerical cubature algorithm for simplices.*
ACM Trans. Math. Software 29, 297-308 (2003).
- Jean B. Lasserre.
*Simple formula for the integration of polynomials on a simplex.*
BIT Numerical Mathematics 61, 523-533 (2021).
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"description": "# pysimplicialcubature\n\n<!-- badges: start -->\n[](http://pysimplicialcubature.readthedocs.io)\n<!-- badges: end -->\n\nThis package is a port of a part of the R package **SimplicialCubature**, \nwritten by John P. Nolan, and which contains R translations of \nsome Matlab and Fortran code written by Alan Genz. In addition it \nprovides a function for the exact computation of the integral of a \npolynomial over a simplex.\n\n___\n\nA simplex is a triangle in dimension 2, a tetrahedron in dimension 3. \nThis package provides two main functions: `integrateOnSimplex`, to integrate \nan arbitrary function on a simplex, and `integratePolynomialOnSimplex`, to \nget the exact value of the integral of a multivariate polynomial on a \nsimplex.\n\nSuppose for example you want to evaluate the following integral:\n\n$$\\int\\_0^1\\int\\_0^x\\int\\_0^y \\exp(x + y + z) \\text{d}z \\text{d}y \\text{d}x.$$\n\n```python\nfrom pysimplicialcubature.simplicialcubature import integrateOnSimplex\nfrom math import exp\n\n# simplex vertices\nv1 = [0.0, 0.0, 0.0] \nv2 = [1.0, 1.0, 1.0] \nv3 = [0.0, 1.0, 1.0] \nv4 = [0.0, 0.0, 1.0]\n# simplex\nS = [v1, v2, v3, v4]\n# function to integrate\nf = lambda x : exp(x[0] + x[1] + x[2])\n# integral of f on S\nI_f = integrateOnSimplex(f, S)\nI_f[\"integral\"]\n```\n\nNow let's turn to a polynomial example. You have to define the polynomial with \n`sympy.Poly`.\n\n```python\nfrom pysimplicialcubature.simplicialcubature import integratePolynomialOnSimplex\nfrom sympy import Poly\nfrom sympy.abc import x, y, z\n\n# simplex vertices\nv1 = [1.0, 1.0, 1.0] \nv2 = [2.0, 2.0, 3.0] \nv3 = [3.0, 4.0, 5.0] \nv4 = [3.0, 2.0, 1.0]\n# simplex\nS = [v1, v2, v3, v4]\n# polynomial to integrate\nP = Poly(x**4 + y + 2*x*y**2 - 3*z, x, y, z, domain = \"RR\")\n# integral of P on S\nintegratePolynomialOnSimplex(P, S)\n```\n\n\n## References\n\n- A. Genz and R. Cools. \n*An adaptive numerical cubature algorithm for simplices.* \nACM Trans. Math. Software 29, 297-308 (2003).\n\n- Jean B. Lasserre.\n*Simple formula for the integration of polynomials on a simplex.* \nBIT Numerical Mathematics 61, 523-533 (2021).",
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