nag_1d_quad_inf_wt_trig_1 (d01ssc) calculates an approximation to the sine or the cosine transform of a function
over
:
(for a user-specified value of
).
nag_1d_quad_inf_wt_trig_1 (d01ssc) is based upon the QUADPACK routine QAWFE (
Piessens et al. (1983)). It is an adaptive function, designed to integrate a function of the form
over a semi-infinite interval, where
is either
or
. Over successive intervals
integration is performed by the same algorithm as is used by
nag_1d_quad_wt_trig_1 (d01snc). The intervals
are of constant length
where
represents the largest integer less than or equal to
. Since
equals an odd number of half periods, the integral contributions over succeeding intervals will alternate in sign when the function
is positive and monotonically decreasing over
. The algorithm, described by
Piessens et al. (1983), incorporates a global acceptance criterion (as defined by
Malcolm and Simpson (1976)) together with the
-algorithm (
Wynn (1956)) to perform extrapolation. The local error estimation is described by
Piessens et al. (1983).
If
and
, the function uses the same algorithm as
nag_1d_quad_inf_1 (d01smc) (with
).
In contrast to most other functions in
Chapter d01, nag_1d_quad_inf_wt_trig_1 (d01ssc) works only with a user-specified absolute error tolerance (
epsabs). Over the interval
it attempts to satisfy the absolute accuracy requirement
where
, for
and
.
However, when difficulties occur during the integration over the
th interval
such that the error flag
is nonzero, the accuracy requirement over subsequent intervals is relaxed. See
Piessens et al. (1983) for more details.
Malcolm M A and Simpson R B (1976) Local versus global strategies for adaptive quadrature ACM Trans. Math. Software 1 129–146
- 1:
g – function, supplied by the userExternal Function
-
g must return the value of the function
at a given point.
The specification of
g is:
double |
g (double x,
Nag_User *comm)
|
|
- 1:
x – doubleInput
-
On entry: the point at which the function must be evaluated.
- 2:
comm – Nag_User *
-
Pointer to a structure of type Nag_User with the following member:
- p – Pointer
-
On entry/exit: the pointer
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.)
- 2:
a – doubleInput
-
On entry: the lower limit of integration, .
- 3:
omega – doubleInput
-
On entry: the argument in the weight function of the transform.
- 4:
wt_func – Nag_TrigTransformInput
On entry: indicates which integral is to be computed:
- if , ;
- if , .
Constraint:
or .
- 5:
maxintervals – IntegerInput
-
On entry: an upper bound on the number of intervals needed for the integration.
Suggested value:
is adequate for most problems.
Constraint:
.
- 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:
.
- 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 .
- 9:
abserr – double *Output
-
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for .
- 10:
qpsub – Nag_QuadSubProgress *
-
Pointer to structure of type Nag_QuadSubProgress with the following members:
- intervals – IntegerOutput
-
On exit: the number of intervals actually used for the integration.
- fun_count – IntegerOutput
-
On exit: the number of function evaluations performed by nag_1d_quad_inf_wt_trig_1 (d01ssc).
- subints_per_interval – Integer *Output
-
On exit: the maximum number of sub-intervals actually used for integrating over any of the intervals .
- interval_error – double *Output
-
On exit: the error estimate corresponding to the integral contribution over the interval , for .
- interval_result – double *Output
-
On exit: the corresponding integral contribution over the interval , for .
- interval_flag – Integer *Output
-
On exit: the error flag corresponding to
, for
. See also
Section 6.
When the information available in the arrays
,
and
is no longer useful, or before a subsequent call to nag_1d_quad_inf_wt_trig_1 (d01ssc) 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:
comm – Nag_User *
-
Pointer to a structure of type Nag_User with the following member:
- p – Pointer
-
On entry/exit: the pointer
, of type Pointer, allows you to communicate information to and from
g(). An object of the required type should be declared, e.g., a structure, and its address assigned to the pointer
by means of a cast to Pointer in the calling program, e.g.,
comm.p = (Pointer)&s. The type Pointer is
void *.
- 12:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
- In the cases where , NE_QUAD_MAX_INT or NE_QUAD_EXTRAPL_INT, additional information about the cause of the error can be obtained from the array , as follows:
-
- The maximum number of subdivisions has been achieved on the th interval.
-
- Occurrence of round-off error is detected and prevents the tolerance imposed on the th interval from being achieved.
-
- Extremely bad integrand behaviour occurs at some points of the th interval.
-
- The integration procedure over the 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.
-
- The integral over the th interval is probably divergent or slowly convergent. It must be noted that divergence can occur with any other value of .
- If you declare and initialize fail and set as recommended then NE_QUAD_NO_CONV 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 NE_QUAD_MAX_SUBDIV_SPEC_INT, NE_QUAD_ROUNDOFF_TOL_SPEC_INT, NE_QUAD_BAD_SPEC_INT, NE_QUAD_NO_CONV_SPEC_INT and NE_QUAD_DIVERGENCE_SPEC_INT 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, .
Constraint: .
On entry,
max_num_subint must not be less than 1:
.
- NE_QUAD_BAD_SPEC_INT
-
Bad integrand behaviour occurs at some points of the interval.
over sub-interval .
- NE_QUAD_BAD_SUBDIV
-
Extremely bad integrand behaviour occurs around the sub-interval
.
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 interval.
over sub-interval .
-
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: .
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,
on the interval.
over sub-interval .
- 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 interval.
over sub-interval .
- NE_QUAD_ROUNDOFF_ABS_TOL
-
Round-off error prevents the requested tolerance from being achieved:
.
The error may be underestimated. Consider relaxing the accuracy requirements specified by
epsabs.
-
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 interval.
over sub-interval .
nag_1d_quad_inf_wt_trig_1 (d01ssc) cannot guarantee, but in practice usually achieves, the following accuracy:
where
epsabs is the user-specified absolute error tolerance. Moreover it returns the quantity
abserr which, in normal circumstances, satisfies
Not applicable.
The time taken by nag_1d_quad_inf_wt_trig_1 (d01ssc) depends on the integrand and on the accuracy required.
None.