, which determine a piecewise cubic Hermite interpolant to the data, that preserves monotonicity over ranges where the data points are monotonic. If the data points are only piecewise monotonic, the interpolant will have an extremum at each point where monotonicity switches direction. The estimates of the derivatives are computed by a formula due to Brodlie, which is described in Fritsch and Butland (1984), with suitable changes at the boundary points.

The algorithm is derived from routine PCHIM in Fritsch (1982).

Values of the computed interpolant can subsequently be computed by calling e01bfc.

4 References

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

Fritsch F N and Butland J (1984) A method for constructing local monotone piecewise cubic interpolants SIAM J. Sci. Statist. Comput. 5 300–304

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , the number of data points.

Constraint: $n \geq 2$ .
2: $x [n]$ – const double Input: On entry: $x [r]$ must be set to $x_{r}$ , the $r$ th value of the independent variable (abscissa), for $r = 0, 1, \dots, n - 1$ .

Constraint: $x [r] < x [r + 1]$ .
3: $f [n]$ – const double Input: On entry: $f [r]$ must be set to $f_{r}$ , the $r$ th value of the dependent variable (ordinate), for $r = 0, 1, \dots, n - 1$ .
4: $d [n]$ – double Output: On exit: estimates of derivatives at the data points. $d [r]$ contains the derivative at $x [r]$ .
5: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

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 $r = ⟨ value ⟩$ : $x [r - 1]$ , $x [r] = ⟨ values ⟩$ .
The values of $x [r]$ , for $r = 0, 1, \dots, n - 1$ , are not in strictly increasing order.

7 Accuracy

The computational errors in the array d should be negligible in most practical situations.

8 Parallelism and Performance

e01bec is not threaded in any implementation.

9 Further Comments

The time taken by e01bec is approximately proportional to

n

The values of the computed interpolant at the points

[i]

, for

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

, may be obtained in the real array pf, of length at least

m

, by the call:

e01bfc (n, x, f, d, m, px, pf, &fail)

where n, x and f are the input arguments to e01bec and d is the output argument from e01bec.

10 Example

This example program reads in a set of data points, calls e01bec to compute a piecewise monotonic interpolant, and then calls e01bfc to evaluate the interpolant at equally spaced points.

e01be: FL CL CPP AD

NAG CL Interfacee01bec (dim1_​monotonic)

▸▿ 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 CL Interface
e01bec (dim1_monotonic)