NAG Library Function Document

1Purpose

nag_1d_quad_wt_cauchy_1 (d01sqc) calculates an approximation to the Hilbert transform of a function $g\left(x\right)$ over $\left[a,b\right]$:
 $I = ∫ a b g x x-c dx$
for user-specified values of $a$, $b$ and $c$.

2Specification

 #include #include
 double (*g)(double x, Nag_User *comm),
double a, double b, double c, double epsabs, double epsrel, Integer max_num_subint, double *result, double *abserr, Nag_QuadProgress *qp, Nag_User *comm, NagError *fail)

3Description

nag_1d_quad_wt_cauchy_1 (d01sqc) is based upon the QUADPACK routine QAWC (Piessens et al. (1983)) and integrates a function of the form $g\left(x\right)w\left(x\right)$, where the weight function
 $w x = 1 x-c$
is that of the Hilbert transform. (If $a the integral has to be interpreted in the sense of a Cauchy principal value.) It is an adaptive function which employs a ‘global’ acceptance criterion (as defined by Malcolm and Simpson (1976)). Special care is taken to ensure that $c$ is never the end-point of a sub-interval (Piessens et al. (1976)). On each sub-interval $\left({c}_{1},{c}_{2}\right)$ modified Clenshaw–Curtis integration of orders 12 and 24 is performed if ${c}_{1}-d\le c\le {c}_{2}+d$ where $d=\left({c}_{2}-{c}_{1}\right)/20$. Otherwise the Gauss 7-point and Kronrod 15-point rules are used. The local error estimation is described by Piessens et al. (1983).
Malcolm M A and Simpson R B (1976) Local versus global strategies for adaptive quadrature ACM Trans. Math. Software 1 129–146
Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag
Piessens R, van Roy–Branders M and Mertens I (1976) The automatic evaluation of Cauchy principal value integrals Angew. Inf. 18 31–35

5Arguments

1:    $\mathbf{g}$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, Nag_User *comm)
1:    $\mathbf{x}$doubleInput
On entry: the point at which the function $g$ 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: g should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_1d_quad_wt_cauchy_1 (d01sqc). If your code inadvertently does return any NaNs or infinities, nag_1d_quad_wt_cauchy_1 (d01sqc) is likely to produce unexpected results.
2:    $\mathbf{a}$doubleInput
On entry: the lower limit of integration, $a$.
3:    $\mathbf{b}$doubleInput
On entry: the upper limit of integration, $b$. It is not necessary that $a.
4:    $\mathbf{c}$doubleInput
On entry: the argument $c$ in the weight function.
Constraint: ${\mathbf{c}}\ne {\mathbf{a}}\text{​ or ​}{\mathbf{b}}$.
5:    $\mathbf{epsabs}$doubleInput
On entry: the absolute accuracy required. If epsabs is negative, the absolute value is used. See Section 7.
6:    $\mathbf{epsrel}$doubleInput
On entry: the relative accuracy required. If epsrel is negative, the absolute value is used. See Section 7.
7:    $\mathbf{max_num_subint}$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: ${\mathbf{max_num_subint}}\ge 1$.
8:    $\mathbf{result}$double *Output
On exit: the approximation to the integral $I$.
9:    $\mathbf{abserr}$double *Output
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $\left|I-{\mathbf{result}}\right|$.
10:  $\mathbf{qp}$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 nag_1d_quad_wt_cauchy_1 (d01sqc).
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, NE_2_REAL_ARG_EQ 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_wt_cauchy_1 (d01sqc) 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.
11:  $\mathbf{comm}$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 g(). 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 *.
12:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_2_REAL_ARG_EQ
On entry, ${\mathbf{c}}=〈\mathit{\text{value}}〉$ while ${\mathbf{a}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{c}}\ne {\mathbf{a}}$.
On entry, ${\mathbf{c}}=〈\mathit{\text{value}}〉$ while ${\mathbf{b}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{c}}\ne {\mathbf{b}}$.
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. 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

nag_1d_quad_wt_cauchy_1 (d01sqc) 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

The time taken by nag_1d_quad_wt_cauchy_1 (d01sqc) depends on the integrand and the accuracy required.
If the function fails with an error exit other than NE_INT_ARG_LT, NE_2_REAL_ARG_EQ 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 nag_1d_quad_wt_cauchy_1 (d01sqc) along with the integral contributions and error estimates over the sub-intervals.
Specifically, for $i=1,2,\dots ,n$, 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, ${\int }_{{a}_{i}}^{{b}_{i}}g\left(x\right)w\left(x\right)dx\simeq {r}_{i}$ 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
 $∫ -1 1 dx x 2 + 0.01 2 x - 1 2 .$

10.1Program Text

Program Text (d01sqce.c)

None.

10.3Program Results

Program Results (d01sqce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017