NAG Library Function Document

1Purpose

nag_quad_1d_inf_exp_wt (d01ubc) returns the Gaussian quadrature approximation for the specific problem . 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.

2Specification

 #include #include
 void (*fun)(const double x[], double f[], Integer n, Nag_Comm *comm, Integer *istop),
Integer n, double *ans, Nag_Comm *comm, NagError *fail)

3Description

nag_quad_1d_inf_exp_wt (d01ubc) uses the weights ${w}_{i}$ and the abscissae ${x}_{i}$ such that $\underset{0}{\overset{\infty }{\int }}\mathrm{exp}\left({-x}^{2}\right)f\left(x\right)$ is approximated by $\sum _{\mathit{i}=1}^{n}{w}_{i}f\left({x}_{i}\right)$ to maximum precision i.e., it is exact when $f\left(x\right)$ is a polynomial of degree $2n-1$.
Golub G H and Welsch J H (1969) Calculation of Gauss quadrature rules Math. Comput. 23 221–230

5Arguments

1:    $\mathbf{fun}$function, supplied by the userExternal Function
fun must return the integrands $f\left({x}_{i}\right)$ in ${\mathbf{f}}\left(\mathit{i}\right)$ for each ${x}_{i}$ in ${\mathbf{x}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$ at a given point.
The specification of fun is:
 void fun (const double x[], double f[], Integer n, Nag_Comm *comm, Integer *istop)
1:    $\mathbf{x}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the points at which the integrand function $f$ must be evaluated.
2:    $\mathbf{f}\left[{\mathbf{n}}\right]$doubleOutput
On exit: ${\mathbf{f}}\left(\mathit{i}\right)$ must contain the value of the integrand $f\left({x}_{i}\right)$ evaluated at the point ${\mathbf{x}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:    $\mathbf{n}$IntegerInput
On entry: n specifies the number of weights and abscissae to be used.
4:    $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to fun.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_quad_1d_inf_exp_wt (d01ubc) you may allocate memory and initialize these pointers with various quantities for use by fun when called from nag_quad_1d_inf_exp_wt (d01ubc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
5:    $\mathbf{istop}$Integer *Input/Output
On entry: ${\mathbf{istop}}=0$.
On exit: 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 nag_quad_1d_inf_exp_wt (d01ubc) halts with fail set to the value of istop and the value returned in ans will be that of a non-signalling NaN.
2:    $\mathbf{n}$IntegerInput
On entry: n specifies the number of weights and abscissae to be used.
Constraint: ${\mathbf{n}}=1$, $2$, $3$, $4$, $5$, $10$, $15$, $20$ or $25$.
3:    $\mathbf{ans}$double *Output
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=0$, ans contains an approximation to the integral. Otherwise, ans will be a non-signalling NaN.
4:    $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{n}}\le 25$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
n is not one of the allowed values.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_USER_STOP
The user has halted the calculation.

7Accuracy

The weights and abscissae have been calculated using quadruple precision arithmetic.

None.

10Example

This example computes an approximation to .

10.1Program Text

Program Text (d01ubce.c)

None.

10.3Program Results

Program Results (d01ubce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017