void	nag_monotonic_deriv (Integer n, const double x[], const double f[], const double d[], Integer m, const double px[], double pf[], double pd[], NagError *fail)

3 Description

nag_monotonic_deriv (e01bgc) evaluates a piecewise cubic Hermite interpolant, as computed by the NAG function nag_monotonic_interpolant (e01bec), at the points

px [i]

, for

i = 0, 1, \dots, m - 1

. The first derivatives at the points are also computed. If any point lies outside the interval from

x [0]

x [n - 1]

, values of the interpolant and its derivative are extrapolated from the nearest extreme cubic, and a warning is returned.

If values of the interpolant only, and not of its derivative, are required, nag_monotonic_evaluate (e01bfc) should be used.

The function is derived from routine PCHFD 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: $m$ – IntegerInput: On entry: $m$ , the number of points at which the interpolant is to be evaluated.

Constraint: $m \geq 1$ .
6: $px [m]$ – const doubleInput: On entry: the $m$ values of $x$ at which the interpolant is to be evaluated.
7: $pf [m]$ – doubleOutput: On exit: $pf [i]$ contains the value of the interpolant evaluated at the point $px [i]$ , for $i = 0, 1, \dots, m - 1$ .
8: $pd [m]$ – doubleOutput: On exit: $pd [i]$ contains the first derivative of the interpolant evaluated at the point $px [i]$ , for $i = 0, 1, \dots, m - 1$ .
9: $fail$ – NagError *Input/Output: The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_INT_ARG_LT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

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_EXTRAPOLATE: Warning – some points in array px lie outside the range $x [0] \dots x [n - 1]$ . Values at these points are unreliable as they have been computed by extrapolation.

7 Accuracy

The computational errors in the arrays pf and pd should be negligible in most practical situations.

8 Parallelism and Performance

nag_monotonic_deriv (e01bgc) is not threaded in any implementation.

9 Further Comments

The time taken by nag_monotonic_deriv (e01bgc) is approximately proportional to the number of evaluation points,

m

. The evaluation will be most efficient if the elements of px are in nondecreasing order (or, more generally, if they are grouped in increasing order of the intervals

[x [r - 1], x [r]]

). A single call of nag_monotonic_deriv (e01bgc) with

m > 1

is more efficient than several calls with

m = 1

10 Example

This example program reads in values of n, x, f and d and calls nag_monotonic_deriv (e01bgc) to compute the values of the interpolant and its derivative at equally spaced points.

NAG Library Function Documentnag_monotonic_deriv (e01bgc)

▸▿ 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

NAG Library Function Document

nag_monotonic_deriv (e01bgc)