# NAG FL Interfacee01bhf (dim1_​monotonic_​intg)

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## 1Purpose

e01bhf evaluates the definite integral of a piecewise cubic Hermite interpolant over the interval $\left[a,b\right]$.

## 2Specification

Fortran Interface
 Subroutine e01bhf ( n, x, f, d, a, b, pint,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(n), f(n), d(n), a, b Real (Kind=nag_wp), Intent (Out) :: pint
#include <nag.h>
 void e01bhf_ (const Integer *n, const double x[], const double f[], const double d[], const double *a, const double *b, double *pint, Integer *ifail)
The routine may be called by the names e01bhf or nagf_interp_dim1_monotonic_intg.

## 3Description

e01bhf evaluates the definite integral of a piecewise cubic Hermite interpolant, as computed by e01bef, over the interval $\left[a,b\right]$.
If either $a$ or $b$ lies outside the interval from ${\mathbf{x}}\left(1\right)$ to ${\mathbf{x}}\left({\mathbf{n}}\right)$ computation of the integral involves extrapolation and a warning is returned.
The routine is derived from routine PCHIA in Fritsch (1982).

## 4References

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

## 5Arguments

1: $\mathbf{n}$Integer Input
2: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
3: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
4: $\mathbf{d}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: n, x, f and d must be unchanged from the previous call of e01bef.
5: $\mathbf{a}$Real (Kind=nag_wp) Input
6: $\mathbf{b}$Real (Kind=nag_wp) Input
On entry: the interval $\left[a,b\right]$ over which integration is to be performed.
7: $\mathbf{pint}$Real (Kind=nag_wp) Output
On exit: the value of the definite integral of the interpolant over the interval $\left[a,b\right]$.
8: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 2$.
${\mathbf{ifail}}=2$
On entry, $r=⟨\mathit{\text{value}}⟩$, ${\mathbf{x}}\left(r-1\right)=⟨\mathit{\text{value}}⟩$ and ${\mathbf{x}}\left(r\right)=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}\left(r-1\right)<{\mathbf{x}}\left(r\right)$ for all $r$.
${\mathbf{ifail}}=3$
Warning – either a or b is outside the range ${\mathbf{x}}\left(1\right)\cdots {\mathbf{x}}\left({\mathbf{n}}\right)$. The result has been computed by extrapolation and is unreliable. ${\mathbf{a}}=⟨\mathit{\text{value}}⟩$ ${\mathbf{b}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

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

## 8Parallelism and Performance

e01bhf is not threaded in any implementation.

The time taken by e01bhf is approximately proportional to the number of data points included within the interval $\left[a,b\right]$.

## 10Example

This example 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 $\left[{\mathbf{a}},{\mathbf{b}}\right]$ until end-of-file is reached.

### 10.1Program Text

Program Text (e01bhfe.f90)

### 10.2Program Data

Program Data (e01bhfe.d)

### 10.3Program Results

Program Results (e01bhfe.r)