Multidimensional integration over simplices.

This library allows to evaluate integrals over Euclidean and spherical simplices.

scubature

Pure Haskell implementation of simplicial cubature (integration over a simplex).

This library is a port of a part of the R package SimplicalCubature, written by John P. Nolan, and which contains R translations of some Matlab and Fortran code written by Alan Genz. It is also a port of a part of the R package SphericalCubature, also written by John P. Nolan. In addition it provides a function for the exact computation of the integral of a polynomial over a simplex.

Integral of a function on a simplex

integrateOnSimplex
    :: (VectorD -> VectorD)   -- integrand
    -> Simplices              -- domain of integration (union of the simplices)
    -> Int                    -- number of components of the integrand
    -> Int                    -- maximum number of evaluations
    -> Double                 -- desired absolute error
    -> Double                 -- desired relative error
    -> Int                    -- integration rule: 1, 2, 3 or 4
    -> IO Results             -- values, error estimates, evaluations, success

Example

Define the integrand:

import Data.Vector.Unboxed as V
:{
f :: Vector Double -> Vector Double
f v = singleton $ exp (V.sum v)
:}

Define the simplex (tetrahedron in dimension 3) by the list of its vertices:

simplex = [[0, 0, 0], [1, 1, 1], [0, 1, 1], [0, 0, 1]]

Integrate:

import Numeric.Integration.SimplexCubature
integrateOnSimplex f [simplex] 1 100000 0 1e-10 3
-- Results { values = [0.8455356852954488]
--         , errorEstimates = [8.082378899762402e-11]
--         , evaluations = 8700
--         , success = True }

For a scalar-valued integrand, it's more convenient to define... a scalar-valued integrand! That is:

:{
f :: Vector Double -> Double
f v = exp (V.sum v)
:}

and then to use integrateOnSimplex':

integrateOnSimplex' f [simplex] 100000 0 1e-10 3
-- Result { value         = 0.8455356852954488
--        , errorEstimate = 8.082378899762402e-11
--        , evaluations   = 8700
--        , success       = True }

Exact integral of a polynomial on a simplex

integratePolynomialOnSimplex
  :: (C a, Fractional a, Ord a) -- `C a` means that `a` must be a ring
  => Spray a -- ^ polynomial to be integrated
  -> [[a]]   -- ^ simplex to integrate over
  -> a

Example

We take as an example the rational numbers for a. Thus we must take a polynomial with rational coefficients and a simplex whose vertices have rational coordinates. Then the integral will be a rational number.

Our polynomial is

It must be defined in Haskell with the hspray library.

import Numeric.Integration.IntegratePolynomialOnSimplex
import Data.Ratio
import Math.Algebra.Hspray 

:{
simplex :: [[Rational]]
simplex = [[1, 1, 1], [2, 2, 3], [3, 4, 5], [3, 2, 1]]
:}

x = lone 1 :: Spray Rational
y = lone 2 :: Spray Rational
z = lone 3 :: Spray Rational

:{
poly :: Spray Rational
poly = x^**^4 ^+^ y ^+^ 2.^(x ^*^ y^**^2) ^-^ 3.^z
:}

integratePolynomialOnSimplex poly simplex
-- 1387 % 42

Integration on a spherical triangle

The library also allows to evaluate an integral on a spherical simplex on the unit sphere (in dimension 3, a spherical triangle).

Example

For example take the first orthant in dimension 3:

import Numeric.Integration.SphericalSimplexCubature
o1 = orthants 3 !! 0
o1
-- [ [1.0, 0.0, 0.0]
-- , [0.0, 1.0, 0.0]
-- , [0.0, 0.0, 1.0] ]

And this integrand:

:{
integrand :: [Double] -> Double
integrand x = (x!!0 * x!!0 * x!!2) + (x!!1 * x!!1 * x!!2) + (x!!2 * x!!2 * x!!2)
:}

Compute the integral (the exact result is pi/4 ≈ 0.7853981634):

integrateOnSphericalSimplex integrand o1 20000 0 1e-7 3
-- Result { value         = 0.7853981641913279
--        , errorEstimate = 7.71579524444753e-8
--        , evaluations   = 17065
--        , success       = True }

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).