NAG C Library Function Document

nag_1d_quad_inf_1 (d01smc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{f}$ – function, supplied by the userExternal 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:    $\mathbf{x}$ – doubleInput

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:

p – Pointer 

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 nag_1d_quad_inf_1 (d01smc). If your code inadvertently does return any NaNs or infinities, nag_1d_quad_inf_1 (d01smc) is likely to produce unexpected results.

2:    $\mathbf{boundinf}$ – Nag_BoundIntervalInput

On entry: indicates the kind of integration interval.

$\mathbf{boundinf}=\mathrm{Nag\_UpperSemiInfinite}$

The interval is $\left[\mathbf{bound},+\infty \right)$.

$\mathbf{boundinf}=\mathrm{Nag\_LowerSemiInfinite}$

The interval is $\left(-\infty ,\mathbf{bound}\right]$.

$\mathbf{boundinf}=\mathrm{Nag\_Infinite}$

The interval is $\left(-\infty ,+\infty \right)$.

Constraint: $\mathbf{boundinf}=\mathrm{Nag\_UpperSemiInfinite}$, $\mathrm{Nag\_LowerSemiInfinite}$ or $\mathrm{Nag\_Infinite}$.

3:    $\mathbf{bound}$ – doubleInput

On entry: the finite limit of the integration interval (if present). bound is not used if $\mathbf{boundinf}=\mathrm{Nag\_Infinite}$.

4:    $\mathbf{epsabs}$ – doubleInput

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

5:    $\mathbf{epsrel}$ – doubleInput

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

6:    $\mathbf{max\_num\_subint}$ – 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:    $\mathbf{result}$ – 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 $\left|I-\mathbf{result}\right|$.

9:    $\mathbf{qp}$ – Nag_QuadProgress *

Pointer to structure of type Nag_QuadProgress with the following members:

num_subint – IntegerOutput

On exit: the actual number of sub-intervals used.

fun_count – IntegerOutput

On exit: the number of function evaluations performed by nag_1d_quad_inf_1 (d01smc).

sub_int_beg_pts – double *Output

sub_int_end_pts – double *Output

sub_int_result – double *Output

sub_int_error – double *Output

On exit: these pointers are allocated memory internally with max_num_subint elements. If an error exit other than NE_INT_ARG_LT, NE_BAD_PARAM or NE_ALLOC_FAIL occurs, these arrays will contain information which may be useful. For details, see Section 9.

Before a subsequent call to nag_1d_quad_inf_1 (d01smc) 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:

p – Pointer 

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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument boundinf had an illegal value.

NE_INT_ARG_LT

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

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_INTS

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

NE_QUAD_MAX_SUBDIV

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.

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_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

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.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (d01smce.c)

10.2

Program Data

10.3

Program Results

Program Results (d01smce.r)