NAG Library Function Document

nag_1d_quad_vals (d01gac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_1d_quad_vals (d01gac) integrates a function which is specified numerically at four or more points, over the whole of its specified range, using third-order finite difference formulae with error estimates, according to a method due to Gill and Miller (1972).

2 Specification

#include <nag.h>

#include <nagd01.h>

void	nag_1d_quad_vals (Integer n, const double x[], const double y[], double ans, double er, NagError *fail)

3 Description

nag_1d_quad_vals (d01gac) evaluates the definite integral

I = \int_{x_{1}}^{x_{n}} y (x) d x,

where the function

y

is specified at the

n

-points

x_{1}, x_{2}, \dots, x_{n}

, which should be all distinct, and in either ascending or descending order. The integral between successive points is calculated by a four-point finite difference formula centred on the interval concerned, except in the case of the first and last intervals, where four-point forward and backward difference formulae respectively are employed. If

n

is less than

4

, the function fails. An approximation to the truncation error is integrated and added to the result. It is also returned separately to give an estimate of the uncertainty in the result. The method is due to Gill and Miller (1972).

4 References

Gill P E and Miller G F (1972) An algorithm for the integration of unequally spaced data Comput. J. 15 80–83

5 Arguments

1: n – IntegerInput: On entry: $n$ , the number of points.
Constraint: $n \geq 4$ .
2: x[n] – const doubleInput: On entry: the values of the independent variable, i.e., the $x_{1}, x_{2}, \dots, x_{n}$ .
Constraint: either $x [0] < x [1] < \dots < x [n - 1]$ or $x [0] > x [1] > \dots > x [n - 1]$ .
3: y[n] – const doubleInput: On entry: the values of the dependent variable $y_{i}$ at the points $x_{i}$ , for $i = 1, 2, \dots, n$ .
4: ans – double *Output: On exit: the estimated value of the integral.
5: er – double *Output: On exit: an estimate of the uncertainty in ans.
6: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 4$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_NOT_STRICTLY_DECREASING: The sequence x is not strictly decreasing: $x [⟨value⟩] = ⟨value⟩$ and $x [⟨value⟩] = ⟨value⟩$ .
NE_NOT_STRICTLY_INCREASING: The sequence x is not strictly increasing: $x [⟨value⟩] = ⟨value⟩$ , $x [⟨value⟩] = ⟨value⟩$ .
NE_QUAD_FIRST_TWO_PTS_EQL: The sequence x has first two points equal: $x [0] = ⟨value⟩$ and $x [1] = ⟨value⟩$ .

7 Accuracy

No accuracy level is specified by you before calling nag_1d_quad_vals (d01gac) but on return the absolute value of er is an approximation to, but not necessarily a bound for,

|I - ans|

. If on exit

fail . code =

NE_INT, NE_NOT_STRICTLY_DECREASING, NE_NOT_STRICTLY_INCREASING or NE_QUAD_FIRST_TWO_PTS_EQL, both ans and er are returned as zero.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_1d_quad_vals (d01gac) depends on the number of points supplied,

n

In their paper, Gill and Miller (1972) do not add the quantity er to ans before return. However, extensive tests have shown that a dramatic reduction in the error often results from such addition. In other cases, it does not make an improvement, but these tend to be cases of low accuracy in which the modified answer is not significantly inferior to the unmodified one. You have the option of recovering the Gill–Miller answer by subtracting er from ans on return from the function.

10 Example

This example evaluates the integral

\int_{0}^{1} \frac{4}{1 + x^{2}} d x = π

reading in the function values at

21

unequally spaced points.

NAG Library Function Documentnag_1d_quad_vals (d01gac)