NAG Library Function Document

nag_monotonic_intg (e01bhc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_monotonic_intg (e01bhc) evaluates the definite integral of a piecewise cubic Hermite interpolant over the interval a,b .

2
Specification

#include <nag.h>
#include <nage01.h>
void  nag_monotonic_intg (Integer n, const double x[], const double f[], const double d[], double a, double b, double *integral, NagError *fail)

3
Description

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  or b  lies outside the interval from x[0]  to 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 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_INT_ARG_LT
On entry, n=value.
Constraint: n2.
NE_NOT_MONOTONIC
On entry, x[r-1] x[r]  for r=value  : x[r-1] = value, x[r] = value.
The values of x[r] , for r=0,1,,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

nag_monotonic_intg (e01bhc) is not threaded in any implementation.

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.

10.1
Program Text

Program Text (e01bhce.c)

10.2
Program Data

Program Data (e01bhce.d)

10.3
Program Results

Program Results (e01bhce.r)

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