md_gauss computes an estimate of a multidimensional integral (from to 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

must specify the number of points in the th dimension of the summation, for .

weightfloat, array-like, shape

Must contain in succession the weights for the various dimensions, i.e., contains the th weight for the th dimension, with

abscisfloat, array-like, shape

Must contain in succession the abscissae for the various dimensions, i.e., contains the th abscissa for the th dimension, with

fcallable retval = f(x, data=None)

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

Parameters
xfloat, ndarray, shape

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

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
retvalfloat

The value of evaluated at .

dataarbitrary, optional

User-communication data for callback functions.

Returns
mdintfloat

The estimate of the integral.

Raises
NagValueError
(errno )

On entry, is too small. . Minimum possible dimension: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

Notes

md_gauss approximates a multidimensional integral by evaluating the summation

given the weights and abscissae for a multidimensional product integration rule (see Davis and Rabinowitz (1975)). The number of dimensions may be anything from to .

The weights and abscissae for each dimension must have been placed in successive segments of the arrays and ; 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 and to the functional dependence of on .

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

where is the weight function associated with the quadrature formula chosen for the th dimension; while if adjusted weights are used, the summation will approximate the integral

You must supply a function to evaluate

at any values of within the range of integration.

References

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