NAG Library Function Document

nag_monotonic_intg (e01bhc)

nag_monotonic_intg (e01bhc) evaluates the definite integral of a piecewise cubic Hermite interpolant, as computed by nag_monotonic_interpolant (e01bec), over the interval

[a, b]

If either

a

b

lies outside the interval from

x [0]

x [n - 1]

, computation of the integral involves extrapolation and a warning is returned.

The function is derived from routine PCHIA in Fritsch (1982).

4 References

Fritsch F N (1982) PCHIP final specifications Report UCID-30194 Lawrence Livermore National Laboratory

5 Arguments

1: n – IntegerInput: On entry: n must be unchanged from the previous call of nag_monotonic_interpolant (e01bec).
2: x[n] – const doubleInput
3: f[n] – const doubleInput
4: d[n] – const doubleInput: On entry: x, f and d must be unchanged from the previous call of nag_monotonic_interpolant (e01bec).
5: a – doubleInput
6: b – doubleInput: On entry: the interval $[a, b]$ over which integration is to be performed.
7: integral – double *Output: On exit: the value of the definite integral of the interpolant over the interval $[a, b]$ .
8: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_INT_ARG_LT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 2$ .
NE_NOT_MONOTONIC: On entry, $x [r - 1] \geq x [r]$ for $r = ⟨value⟩$ : $x [r - 1] = ⟨value⟩$ , $x [r] = ⟨value⟩$ .
The values of $x [r]$ , for $r = 0, 1, \dots, n - 1$ , are not in strictly increasing order.
NW_INTERVAL_EXTRAPOLATE: On entry, limits a, b must not be outside interval $[x [0], x [n - 1]]$ , $a = ⟨value⟩$ , $b = ⟨value⟩$ , $x [0] = ⟨value⟩$ , $x [⟨value⟩] = ⟨value⟩$ . Extrapolation was performed to compute the integral. The value returned is therefore unreliable.

7 Accuracy

The computational error in the value returned for integral should be negligible in most practical situations.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_monotonic_intg (e01bhc) is approximately proportional to the number of data points included within the interval

[a, b]

10 Example

This example program reads in values of n, x, f and d. It then reads in pairs of values for a and b, and evaluates the definite integral of the interpolant over the interval

(a, b)

until end-of-file is reached.

NAG Library Function Documentnag_monotonic_intg (e01bhc)

+− Contents

1 Purpose

2 Specification

3 Description