NAG CL Interfaced01skc (dim1_​osc)

Note: this function is deprecated and will be withdrawn at Mark 31.3. Replaced by d01rkc.
d01rkc requires the user-supplied function f to calculate a vector of abscissae at once for greater efficiency and returns additional information on the computation (in the arrays rinfo and iinfo rather than $qp$ previously).
Callbacks

Old Code

`double (*f)(double x)`

New Code

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

Old Code

```nag_quad_dim1_osc(f, a, b, epsabs, epsrel, max_num_subint, &result, &abserr,
&qp, &comm, &fail);```

New Code

```key = 6;
nag_quad_dim1_fin_osc_fn(f, a, b, key, epsabs, epsrel, maxsub, &result, &abserr,
rinfo, iinfo, &comm, &fail);```

Settings help

CL Name Style:

1Purpose

d01skc is an adaptive integrator, especially suited to oscillating, nonsingular integrands, which calculates an approximation to the integral of a function $f\left(x\right)$ over a finite interval $\left[a,b\right]$:
 $I = ∫ a b f (x) dx .$

2Specification

 #include
void  d01skc (
 double (*f)(double x, Nag_User *comm),
double a, double b, double epsabs, double epsrel, Integer max_num_subint, double *result, double *abserr, Nag_QuadProgress *qp, Nag_User *comm, NagError *fail)
The function may be called by the names: d01skc, nag_quad_dim1_osc or nag_1d_quad_osc_1.

3Description

d01skc is based upon the QUADPACK routine QAG (Piessens et al. (1983)). It is an adaptive function, using the Gauss 30-point and Kronrod 61-point rules. A ‘global’ acceptance criterion (as defined by Malcolm and Simpson (1976)) is used. The local error estimation is described by Piessens et al. (1983).
As this function is based on integration rules of high order, it is especially suitable for nonsingular oscillating integrands.
This function requires you to supply a function to evaluate the integrand at a single point.

4References

Malcolm M A and Simpson R B (1976) Local versus global strategies for adaptive quadrature ACM Trans. Math. Software 1 129–146
Piessens R (1973) An algorithm for automatic integration Angew. Inf. 15 399–401
Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag

5Arguments

1: $\mathbf{f}$function, supplied by the user External Function
f must return the value of the integrand $f$ at a given point.
The specification of f is:
 double f (double x, Nag_User *comm)
1: $\mathbf{x}$double Input
On entry: the point at which the integrand $f$ must be evaluated.
2: $\mathbf{comm}$Nag_User *
Pointer to a structure of type Nag_User with the following member:
pPointer
On entry/exit: the pointer $\mathbf{comm}\mathbf{\to }\mathbf{p}$ should be cast to the required type, e.g., struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type. (See the argument comm below.)
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01skc. If your code inadvertently does return any NaNs or infinities, d01skc is likely to produce unexpected results.
2: $\mathbf{a}$double Input
On entry: the lower limit of integration, $a$.
3: $\mathbf{b}$double Input
On entry: the upper limit of integration, $b$. It is not necessary that $a.
4: $\mathbf{epsabs}$double Input
On entry: the absolute accuracy required. If epsabs is negative, the absolute value is used. See Section 7.
5: $\mathbf{epsrel}$double Input
On entry: the relative accuracy required. If epsrel is negative, the absolute value is used. See Section 7.
6: $\mathbf{max_num_subint}$Integer Input
On entry: the upper bound on the number of sub-intervals into which the interval of integration may be divided by the function. The more difficult the integrand, the larger max_num_subint should be.
Constraint: ${\mathbf{max_num_subint}}\ge 1$.
7: $\mathbf{result}$double * Output
On exit: the approximation to the integral $I$.
8: $\mathbf{abserr}$double * Output
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $|I-{\mathbf{result}}|$.
9: $\mathbf{qp}$Nag_QuadProgress *
Pointer to structure of type Nag_QuadProgress with the following members:
num_subintIntegerOutput
On exit: the actual number of sub-intervals used.
fun_countIntegerOutput
On exit: the number of function evaluations performed by d01skc.
sub_int_beg_ptsdouble *Output
sub_int_end_ptsdouble *Output
sub_int_resultdouble *Output
sub_int_errordouble *Output
On exit: these pointers are allocated memory internally with max_num_subint elements. If an error exit other than NE_INT_ARG_LT or NE_ALLOC_FAIL occurs, these arrays will contain information which may be useful. For details, see Section 9.
Before a subsequent call to d01skc is made, or when the information contained in these arrays is no longer useful, you should free the storage allocated by these pointers using the NAG macro NAG_FREE.
10: $\mathbf{comm}$Nag_User *
Pointer to a structure of type Nag_User with the following member:
pPointer
On entry/exit: the pointer $\mathbf{comm}\mathbf{\to }\mathbf{p}$, of type Pointer, allows you to communicate information to and from f(). An object of the required type should be declared, e.g., a structure, and its address assigned to the pointer $\mathbf{comm}\mathbf{\to }\mathbf{p}$ by means of a cast to Pointer in the calling program, e.g., comm.p = (Pointer)&s. The type Pointer is void *.
11: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_INT_ARG_LT
On entry, max_num_subint must not be less than 1: ${\mathbf{max_num_subint}}=⟨\mathit{\text{value}}⟩$.
Extremely bad integrand behaviour occurs around the sub-interval $\left(⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right)$.
The maximum number of subdivisions has been reached: ${\mathbf{max_num_subint}}=⟨\mathit{\text{value}}⟩$.
The maximum number of subdivisions 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.) you will probably gain from splitting up the interval at this point and calling the integrator on the sub-intervals. 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 value of max_num_subint.
Round-off error prevents the requested tolerance from being achieved: ${\mathbf{epsabs}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{epsrel}}=⟨\mathit{\text{value}}⟩$.
The error may be underestimated. Consider relaxing the accuracy requirements specified by epsabs and epsrel.

7Accuracy

d01skc cannot guarantee, but in practice usually achieves, the following accuracy:
 $|I-result| ≤ tol$
where
 $tol = max{|epsabs|, |epsrel| × |I| }$
and epsabs and epsrel are user-specified absolute and relative error tolerances. Moreover it returns the quantity abserr which, in normal circumstances, satisfies
 $|I-result| ≤ abserr ≤ tol .$

8Parallelism and Performance

d01skc is not threaded in any implementation.

The time taken by d01skc depends on the integrand and the accuracy required.
If the function fails with an error exit other than NE_INT_ARG_LT or NE_ALLOC_FAIL, then you may wish to examine the contents of the structure qp. These contain the end-points of the sub-intervals used by d01skc along with the integral contributions and error estimates over these sub-intervals.
Specifically, $i=1,2,\dots ,n$, let ${r}_{i}$ denote the approximation to the value of the integral over the sub-interval $\left[{a}_{i},{b}_{i}\right]$ in the partition of $\left[a,b\right]$ and ${e}_{i}$ be the corresponding absolute error estimate.
Then, ${\int }_{{a}_{i}}^{{b}_{i}}f\left(x\right)dx\simeq {r}_{i}$ and ${\mathbf{result}}={\sum }_{i=1}^{n}{r}_{i}$. The value of $n$ is returned in $\mathbf{qp}\mathbf{\to }\mathbf{num_subint}$, and the values ${a}_{i}$, ${b}_{i}$, ${r}_{i}$ and ${e}_{i}$ are stored in the structure qp as
• ${a}_{i}=\mathbf{qp}\mathbf{\to }\mathbf{sub_int_beg_pts}\left[i-1\right]$,
• ${b}_{i}=\mathbf{qp}\mathbf{\to }\mathbf{sub_int_end_pts}\left[i-1\right]$,
• ${r}_{i}=\mathbf{qp}\mathbf{\to }\mathbf{sub_int_result}\left[i-1\right]$ and
• ${e}_{i}=\mathbf{qp}\mathbf{\to }\mathbf{sub_int_error}\left[i-1\right]$.

10Example

This example computes
 $∫ 0 2π sin(30x) cos⁡x dx .$

10.1Program Text

Program Text (d01skce.c)

None.

10.3Program Results

Program Results (d01skce.r)