naginterfaces.library.quad.md_​gauss

naginterfaces.library.quad.md_gauss(nptvec, weight, abscis, f, data=None)

md_gauss computes an estimate of a multidimensional integral (from $1$ to $20$ dimensions), given the analytic form of the integrand and suitable Gaussian weights and abscissae.

For full information please refer to the NAG Library document for d01fb

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/d01/d01fbf.html

Parameters

nptvecint, array-like, shape $\left(\text{ndim}\right)$

$nptvec[\text{j}-1]$ must specify the number of points in the $\text{j}$th dimension of the summation, for $\text{j}=1,2,\dots ,n$.

weightfloat, array-like, shape $\left(\text{lwa}\right)$

Must contain in succession the weights for the various dimensions, i.e., $weight[k-1]$ contains the $i$th weight for the $j$th dimension, with

$$k=nptvec\left[0\right]+nptvec\left[1\right]+\cdots +nptvec[j-2]+i\text{.}$$

abscisfloat, array-like, shape $\left(\text{lwa}\right)$

Must contain in succession the abscissae for the various dimensions, i.e., $abscis[k-1]$ contains the $i$th abscissa for the $j$th dimension, with

$$k=nptvec\left[0\right]+nptvec\left[1\right]+\cdots +nptvec[j-2]+i\text{.}$$

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 $f\left(x\right)$ evaluated at $x$.

dataarbitrary, optional

User-communication data for callback functions.

Returns

mdintfloat

The estimate of the integral.

Raises

NagValueError

(errno $1$)

On entry, $\text{lwa}$ is too small. $\text{lwa}=⟨value⟩$. Minimum possible dimension: $⟨value⟩$.

(errno $1$)

On entry, $\text{ndim}=⟨value⟩$.

Constraint: $\text{ndim}\le 20$.

(errno $1$)

On entry, $\text{ndim}=⟨value⟩$.

Constraint: $\text{ndim}\ge 1$.

Notes

md_gauss approximates a multidimensional integral by evaluating the summation

$$\sum _{i_1=1}^{l_1}{w}_{1,i_1}\sum _{i_2=1}^{l_2}{w}_{2,i_2}\cdots \sum _{i_n=1}^{l_n}{w}_{n,i_n}f(x_{1,i_1},x_{2,i_2},\dots ,x_{n,i_n})$$

given the weights $w_{j,i_j}$ and abscissae $x_{j,i_j}$ for a multidimensional product integration rule (see Davis and Rabinowitz (1975)). The number of dimensions may be anything from $1$ to $20$.

The weights and abscissae for each dimension must have been placed in successive segments of the arrays $weight$ and $abscis$; for example, by calling dim1_gauss_wres() or dim1_gauss_wgen() once for each dimension using a quadrature formula and number of abscissae appropriate to the range of each $x_j$ and to the functional dependence of $f$ on $x_j$.

If normal weights are used, the summation will approximate the integral

$$\int w_1\left(x_1\right)\int w_2\left(x_2\right)\cdots \int w_n\left(x_n\right)f(x_1,x_2,\dots ,x_n)d{x}_n\cdots d{x}_{2}d{x}_1$$

where $w_j\left(x\right)$ is the weight function associated with the quadrature formula chosen for the $j$th dimension; while if adjusted weights are used, the summation will approximate the integral

$$\int \int \cdots \int f(x_1,x_2,\dots ,x_n)d{x}_n\cdots d{x}_{2}d{x}_1\text{.}$$

You must supply a function to evaluate

$$f(x_1,x_2,\dots ,x_n)$$

at any values of $x_1,x_2,\dots ,x_n$ within the range of integration.

References

Davis, P J and Rabinowitz, P, 1975, Methods of Numerical Integration, Academic Press