naginterfaces.library.quad.dim1_​inf_​exp_​wt

naginterfaces.library.quad.dim1_inf_exp_wt(f, n, data=None)

dim1_inf_exp_wt returns the Gaussian quadrature approximation for the specific problem ${\int }_0^{\infty }exp\left({-x}^2\right)f\left(x\right)dx$. The degrees of precision catered for are: $1$, $3$, $5$, $7$, $9$, $19$, $29$, $39$ and $49$, corresponding to values of $n=1$, $2$, $3$, $4$, $5$, $10$, $15$, $20$ and $25$, where $n$ is the number of weights.

For full information please refer to the NAG Library document for d01ub

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/d01/d01ubf.html

Parameters

fcallable (fv, istop) = f(x, istop, data=None)

$f$ must return the integrand function values $f\left(x_i\right)$ for the given $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

Parameters

xfloat, ndarray, shape $\left(n\right)$

The points at which the integrand function $f$ must be evaluated.

istopint

$istop=0$.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

fvfloat, array-like, shape $\left(n\right)$

$fv[\text{i}-1]$ must contain the value of the integrand $f\left(x_i\right)$ evaluated at the point $x[\text{i}-1]$, for $\text{i}=1,2,\dots ,n$.

istopint

You may set $istop$ to a negative number if at any time it is impossible to evaluate the function $f\left(x\right)$. In this case dim1_inf_exp_wt halts with $\text{errno}$ set to the value of $istop$ and the value returned in $ans$ will be that of a non-signalling NaN.

nint

$n$ specifies the number of weights and abscissae to be used.

dataarbitrary, optional

User-communication data for callback functions.

Returns

ansfloat

If no exception or warning is raised, $ans$ contains an approximation to the integral. Otherwise, $ans$ will be a non-signalling NaN.

Raises

NagValueError

(errno $i<0$)

The user has halted the calculation.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $1\le n\le 25$.

(errno $2$)

On entry, $n=⟨value⟩$.

$n$ is not one of the allowed values.

Notes

dim1_inf_exp_wt uses the weights $w_i$ and the abscissae $x_i$ such that ${\int }_0^{\infty }exp\left({-x}^2\right)f\left(x\right)$ is approximated by ${\sum }_1^n{w}_{i}f\left(x_i\right)$ to maximum precision i.e., it is exact when $f\left(x\right)$ is a polynomial of degree $2n-1$.

References

Golub, G H and Welsch, J H, 1969, Calculation of Gauss quadrature rules, Math. Comput. (23), 221–230