naginterfaces.library.quad.md_​simplex

naginterfaces.library.quad.md_simplex(ndim, vert, f, minord, finvls, data=None)

md_simplex returns a sequence of approximations to the integral of a function over a multidimensional simplex, together with an error estimate for the last approximation.

For full information please refer to the NAG Library document for d01pa

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d01/d01paf.html

Parameters

ndimint

$n$, the number of dimensions of the integral.

vertfloat, array-like, shape $(ndim+1,2\times (ndim+1))$

$vert[\text{i}-1,\text{j}-1]$ must be set to the $\text{j}$th component of the $\text{i}$th vertex for the simplex integration region, for $\text{j}=1,2,\dots ,n$, for $\text{i}=1,2,\dots ,n+1$. If $minord>0$, $vert$ must be unchanged since the previous call of md_simplex.

fcallable retval = f(x, data=None)

$f$ must return the value of the integrand at a given point.

Parameters

xfloat, ndarray, shape $\left(\text{ndim}\right)$

The coordinates of the point at which the integrand $f$ must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

The value of the integrand $f$ at the given point.

minordint

Must specify the highest order of the approximations currently available in the array $finvls$.

$minord=0$

Indicates an initial call.

$minord>0$

Indicates that $finvls\left[0\right],finvls\left[1\right],\dots ,finvls[minord-1]$ have already been computed in a previous call of md_simplex.

finvlsfloat, array-like, shape $\left(\text{maxord}\right)$

If $minord>0$, $finvls\left[0\right],finvls\left[1\right],\dots ,finvls[minord-1]$ must contain approximations to the integral previously computed by md_simplex.

dataarbitrary, optional

User-communication data for callback functions.

Returns

vertfloat, ndarray, shape $(ndim+1,2\times (ndim+1))$

These values are unchanged. The rest of the array $vert$ is used for workspace and contains information to be used if another call of md_simplex is made with $minord>0$. In particular $vert[n,2n+1]$ contains the volume of the simplex.

minordint

$minord=\text{maxord}$.

finvlsfloat, ndarray, shape $\left(\text{maxord}\right)$

Contains these values unchanged, and the newly computed values $finvls\left[minord\right],finvls[minord+1],\dots ,finvls[\text{maxord}-1]$. $finvls[j-1]$ is an approximation to the integral of polynomial degree $2j-1$.

esterrfloat

An absolute error estimate for $finvls[\text{maxord}-1]$.

Raises

NagValueError

(errno $1$)

On entry, $\text{maxord}=⟨value⟩$ and $minord=⟨value⟩$.

Constraint: $\text{maxord}>minord$.

(errno $1$)

On entry, $minord=⟨value⟩$.

Constraint: $minord\ge 0$.

(errno $1$)

On entry, $ndim=⟨value⟩$.

Constraint: $ndim\ge 2$.

(errno $2$)

The volume of the simplex integration region is too large or too small to be represented on the machine.

Notes

md_simplex computes a sequence of approximations $finvls[\text{j}-1]$, for $\text{j}=minord+1,\dots ,\text{maxord}$, to an integral

$${\int }_{S}f(x_1,x_2,\dots ,x_n)d{x}_{1}d{x}_2\cdots d{x}_n$$

where $S$ is an $n$-dimensional simplex defined in terms of its $n+1$ vertices. $finvls[j-1]$ is an approximation which will be exact (except for rounding errors) whenever the integrand is a polynomial of total degree $2j-1$ or less.

The type of method used has been described in Grundmann and Moller (1978), and is implemented in an extrapolated form using the theory from de Doncker (1979).

References

de Doncker, E, 1979, New Euler–Maclaurin Expansions and their application to quadrature over the $s$ -dimensional simplex, Math. Comput. (33), 1003–1018

Grundmann, A and Moller, H M, 1978, Invariant integration formulas for the $n$ -simplex by combinatorial methods, SIAM J. Numer. Anal. (15), 282–290