NAG Library Function Document

nag_1d_quad_inf_wt_trig (d01asc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     g – function, supplied by the userExternal Function

g must return the value of the function $g$ at a given point.

The specification of g is:

double  g (double x)

1:     x – doubleInput

On entry: the point at which the function $g$ must be evaluated.

2:     a – doubleInput

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

3:     omega – doubleInput

On entry: the argument $\omega$ in the weight function of the transform.

4:     wt_func – Nag_TrigTransformInput

On entry: indicates which integral is to be computed:

• if $\mathbf{wt\_func}=\mathrm{Nag\_Cosine}$, $w\left(x\right)=\cos \left(\omega x\right)$;
• if $\mathbf{wt\_func}=\mathrm{Nag\_Sine}$, $w\left(x\right)=\sin \left(\omega x\right)$.

Constraint: $\mathbf{wt\_func}=\mathrm{Nag\_Cosine}$ or $\mathrm{Nag\_Sine}$.

5:     maxintervals – IntegerInput

On entry: an upper bound on the number of intervals $C_k$ needed for the integration.

Suggested value: $\mathbf{maxintervals}=50$ is adequate for most problems.

Constraint: $\mathbf{maxintervals}\ge 3$.

6:     max_num_subint – IntegerInput

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:     epsabs – doubleInput

On entry: the absolute accuracy required. If epsabs is negative, the absolute value is used. See Section 7.

8:     result – double *Output

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

9:     abserr – double *Output

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

10:   qpsub – Nag_QuadSubProgress *

Pointer to structure of type Nag_QuadSubProgress with the following members:

intervals – IntegerOutput

On exit: the number of intervals $C_k$ actually used for the integration.

fun_count – IntegerOutput

On exit: the number of function evaluations performed by nag_1d_quad_inf_wt_trig (d01asc).

subints_per_interval – IntegerOutput

On exit: the maximum number of sub-intervals actually used for integrating over any of the intervals $C_k$.

interval_error – double *Output

On exit: the error estimate corresponding to the integral contribution over the interval $C_k$, for $\mathit{k}=1,2,\dots ,\mathbf{intervals}$.

interval_result – double *Output

On exit: the corresponding integral contribution over the interval $C_{\mathit{k}}$, for $\mathit{k}=1,2,\dots ,\mathbf{intervals}$.

interval_flag – Integer *Output

On exit: the error flag corresponding to $\mathbf{interval\_result}$, for $k=1,2,\dots ,\mathbf{intervals}$. See also Section 6.

When the information available in the arrays $\mathbf{interval\_error}$, $\mathbf{interval\_result}$ and $\mathbf{interval\_flag}$ is no longer useful, or before a subsequent call to nag_1d_quad_inf_wt_trig (d01asc) with the same argument qpsub is made, you should free the storage contained in this pointer using the NAG macro NAG_FREE. Note that these arrays do not need to be freed if one of the error exits NE_INT_ARG_LT, NE_BAD_PARAM or NE_ALLOC_FAIL occurred.

11:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

In the cases where $\mathbf{fail}\mathbf{.}\mathbf{code}=\mathbf{NE\_QUAD\_BAD\_SUBDIV\_INT}\text{, }\mathbf{NE\_QUAD\_MAX\_INT}\text{ or }\mathbf{NE\_QUAD\_EXTRAPL\_INT}$, additional information about the cause of the error can be obtained from the array $\mathbf{qpsub}\mathbf{\to }\mathbf{interval\_flag}$, as follows:

• $\mathbf{qpsub}\mathbf{\to }\mathbf{interval\_flag}\left[k-1\right]=1$ 
  • The maximum number of subdivisions $\left(=\mathbf{max\_num\_subint}\right)$ has been achieved on the $k$th interval.
• $\mathbf{qpsub}\mathbf{\to }\mathbf{interval\_flag}\left[k-1\right]=2$ 
  • Occurrence of round-off error is detected and prevents the tolerance imposed on the $k$th interval from being achieved.
• $\mathbf{qpsub}\mathbf{\to }\mathbf{interval\_flag}\left[k-1\right]=3$ 
  • Extremely bad integrand behaviour occurs at some points of the $k$th interval.
• $\mathbf{qpsub}\mathbf{\to }\mathbf{interval\_flag}\left[k-1\right]=4$ 
  • The integration procedure over the $k$th interval does not converge (to within the required accuracy) due to round-off in the extrapolation procedure invoked on this interval. It is assumed that the result on this interval is the best which can be obtained.
• $\mathbf{qpsub}\mathbf{\to }\mathbf{interval\_flag}\left[k-1\right]=5$ 
  • The integral over the $k$th interval is probably divergent or slowly convergent. It must be noted that divergence can occur with any other value of $\mathbf{qpsub}\mathbf{\to }\mathbf{interval\_flag}\left[k-1\right]$.

If users declare and initialize fail and set $\mathbf{fail}\mathbf{.}\mathbf{print}=\mathrm{Nag\_TRUE}$ as recommended then

may be produced supplemented by messages indicating more precisely where problems were encountered by the function. However, if the default error handling, NAGERR_DEFAULT, is used then one of the following errors may occur. Please note the program will terminate when the first of such errors is detected.

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument wt_func had an illegal value.

NE_INT_ARG_LT

On entry, $\mathbf{maxintervals}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{maxintervals}\ge 3$.

On entry, max_num_subint must not be less than 1: $\mathbf{max\_num\_subint}=〈\mathit{\text{value}}〉$.

NE_QUAD_BAD_SPEC_INT

Bad integrand behaviour occurs at some points of the $〈\mathit{\text{value}}〉$ interval.
$\mathbf{qpsub}\mathbf{\to }\mathbf{interval\_flag}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$ over sub-interval $\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$.

NE_QUAD_BAD_SUBDIV

Extremely bad integrand behaviour occurs around the sub-interval $\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$.
The same advice applies as in the case of NE_QUAD_MAX_SUBDIV.

NE_QUAD_BAD_SUBDIV_INT

Bad integration behaviour has occurred within one or more intervals.

NE_QUAD_DIVERGENCE_SPEC_INT

The integral is probably divergent on the $〈\mathit{\text{value}}〉$ interval.
$\mathbf{qpsub}\mathbf{\to }\mathbf{interval\_flag}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$ over sub-interval $\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$.

NE_QUAD_EXTRAPL_INT

The extrapolation table constructed for convergence acceleration of the series formed by the integral contribution over the integral does not converge.

NE_QUAD_MAX_INT

Maximum number of intervals allowed has been achieved. Increase the value of maxintervals.

NE_QUAD_MAX_SUBDIV

The maximum number of subdivisions has been reached: $\mathbf{max\_num\_subint}=〈\mathit{\text{value}}〉$.

The maximum number of subdivisions within an interval 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 this function on the infinite subrange and an appropriate integrator on the finite subrange. Alternatively, consider relaxing the accuracy requirements specified by epsabs or increasing the value of max_num_subint.

NE_QUAD_MAX_SUBDIV_SPEC_INT

The maximum number of subdivisions has been reached,
$\mathbf{max\_num\_subint}=〈\mathit{\text{value}}〉$ on the $〈\mathit{\text{value}}〉$ interval.
$\mathbf{qpsub}\mathbf{\to }\mathbf{interval\_flag}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$ over sub-interval $\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$.

NE_QUAD_NO_CONV

The integral is probably divergent or slowly convergent.
Please note that divergence can also occur with any error exit other than NE_INT_ARG_LT, NE_BAD_PARAM or NE_ALLOC_FAIL.

NE_QUAD_NO_CONV_SPEC_INT

The integral has failed to converge on the $〈\mathit{\text{value}}〉$ interval.
$\mathbf{qpsub}\mathbf{\to }\mathbf{interval\_flag}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$ over sub-interval $\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$.

NE_QUAD_ROUNDOFF_ABS_TOL

Round-off error prevents the requested tolerance from being achieved: $\mathbf{epsabs}=〈\mathit{\text{value}}〉$.
The error may be underestimated. Consider relaxing the accuracy requirements specified by epsabs.

NE_QUAD_ROUNDOFF_EXTRAPL

Round-off error is detected during extrapolation.
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 NE_QUAD_MAX_SUBDIV.

NE_QUAD_ROUNDOFF_TOL_SPEC_INT

Round-off error prevents the requested tolerance from being achieved on the $〈\mathit{\text{value}}〉$ interval.
$\mathbf{qpsub}\mathbf{\to }\mathbf{interval\_flag}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$ over sub-interval $\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (d01asce.c)

9.2  Program Data

9.3  Program Results

Program Results (d01asce.r)