NAG CL Interface
d01rlc (dim1_​fin_​brkpts)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{f}$ – function, supplied by the user External Function

f must return the values of the integrand $f$ at a set of points.

The specification of f is:

copy

void  f (const double x[], Integer nx, double fv[], Integer *iflag, Nag_Comm *comm)

1: $\mathbf{x}\left[\mathit{dim}\right]$ – const double Input

On entry: the abscissae, $x_i$, for $\mathit{i}=1,2,\dots ,\mathbf{nx}$, at which function values are required.

2: $\mathbf{nx}$ – Integer Input

On entry: the number of abscissae at which a function value is required. nx will be of size equal to the number of Kronrod points in the quadrature rule used, in this case $21$.

3: $\mathbf{fv}\left[\mathit{dim}\right]$ – double Output

On exit: fv must contain the values of the integrand $f$. $\mathbf{fv}\left[i-1\right]=f\left(x_i\right)$ for all $i=1,2,\dots ,\mathbf{nx}$.

4: $\mathbf{iflag}$ – Integer * Input/Output

On entry: $\mathbf{iflag}=0$.

On exit: set $\mathbf{iflag}<0$ to force an immediate exit with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.

5: $\mathbf{comm}$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to f.

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling d01rlc you may allocate memory and initialize these pointers with various quantities for use by f when called from d01rlc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01rlc. If your code inadvertently does return any NaNs or infinities, d01rlc is likely to produce unexpected results.

2: $\mathbf{a}$ – double Input

On entry: $a$, the lower limit of integration.

3: $\mathbf{b}$ – double Input

On entry: $b$, the upper limit of integration. It is not necessary that $a<b$.

Note: if $\mathbf{a}=\mathbf{b}$, the function will immediately return with $\mathbf{result}=0.0$, $\mathbf{abserr}=0.0$, $\mathbf{rinfo}=0.0$ and $\mathbf{iinfo}=0$.

4: $\mathbf{npts}$ – Integer Input

On entry: the dimension of the array points. The number of user-supplied break-points within the integration interval.

Constraint: $\mathbf{npts}\ge 0$.

5: $\mathbf{points}\left[\mathbf{npts}\right]$ – const double Input

On entry: the user-specified break-points.

Constraint: the break-points must all lie within the interval of integration (but may be supplied in any order).

6: $\mathbf{epsabs}$ – double Input

On entry: ${\epsilon }_a$, the absolute accuracy required. If epsabs is negative, ${\epsilon }_a=\left|\mathbf{epsabs}\right|$. See Section 7.

7: $\mathbf{epsrel}$ – double Input

On entry: ${\epsilon }_r$, the relative accuracy required. If epsrel is negative, ${\epsilon }_r=\left|\mathbf{epsrel}\right|$. See Section 7.

8: $\mathbf{maxsub}$ – Integer Input

On entry: ${\max }_{\mathit{sdiv}}$, the upper bound on the total number of subdivisions d01rlc may use to generate new segments. This does not include the initial segments of the interval generated from the supplied break-points. If ${\max }_{\mathit{sdiv}}=1$, only the initial segments will be evaluated.

Suggested value: a value in the range $200$ to $500$ is adequate for most problems.

Constraint: $\mathbf{maxsub}\ge 1$.

9: $\mathbf{result}$ – double * Output

On exit: the approximation to the integral $I$.

10: $\mathbf{abserr}$ – double * Output

On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $|I-\mathbf{result}|$.

11: $\mathbf{rinfo}\left[\mathit{dim}\right]$ – double Output

Note: the dimension, dim, of the array rinfo must be at least $4\times \max (\mathbf{maxsub},\mathbf{npts})+\mathbf{npts}+6$.

On exit: details of the computation. See Section 9 for more information.

12: $\mathbf{iinfo}\left[\mathit{dim}\right]$ – Integer Output

Note: the dimension, dim, of the array iinfo must be at least $2\times \max (\mathbf{maxsub},\mathbf{npts})+\mathbf{npts}+4$.

On exit: details of the computation. See Section 9 for more information.

13: $\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

14: $\mathbf{fail}$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.

NE_BAD_PARAM

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.

NE_INT

On entry, $\mathbf{maxsub}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{maxsub}\ge 1$.

On entry, $\mathbf{npts}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{npts}\ge 0$.

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 7.5 in the Introduction to the NAG Library CL Interface for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

NE_QUAD_BAD_SUBDIV

Extremely bad integrand behaviour occurs around the sub-interval $(⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩)$. The same advice applies as in the case of $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_QUAD_MAX_SUBDIV.

NE_QUAD_BRKPTS_INVAL

On entry, break-points are specified outside $(\mathbf{a},\mathbf{b})$: $\mathbf{a}=⟨\mathit{\text{value}}⟩$ and $\mathbf{b}=⟨\mathit{\text{value}}⟩$.

NE_QUAD_MAX_SUBDIV

The maximum number of subdivisions allowed with the given workspace has been reached without the accuracy requirements being achieved. Look at the integrand in order to determine the integration difficulties. If the position of a local difficulty within the interval can be determined (e.g., a singularity of the integrand or its derivative, a peak, a discontinuity, etc.) it should be supplied to the function as an element of the vector points. If necessary, another integrator, which is designed for handling the type of difficulty involved, must be used. Alternatively, consider relaxing the accuracy requirements specified by epsabs and epsrel, or increasing the amount of workspace.

NE_QUAD_NO_CONV

The integral is probably divergent or slowly convergent.

NE_QUAD_ROUNDOFF_EXTRAPL

Round-off error is detected in the extrapolation table. The requested tolerance cannot be achieved because the extrapolation does not increase the accuracy satisfactorily; the returned result is the best that can be obtained. The same advice applies as in the case of $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_QUAD_MAX_SUBDIV.

NE_QUAD_ROUNDOFF_TOL

Round-off error prevents the requested tolerance from being achieved: $\mathbf{epsabs}=⟨\mathit{\text{value}}⟩$ and $\mathbf{epsrel}=⟨\mathit{\text{value}}⟩$.

NE_USER_STOP

Exit from f with $\mathbf{iflag}<0$.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results