1: $n$ – int64int32nag_int scalar: $n$ , half the number of points to be used in the interpolation.

Constraint: $n > 0$ .
2: $p$ – double scalar: The point $p$ at which the interpolated function value is required, i.e., $p = (x - x_{0}) / h$ with $- 1.0 < p < 1.0$ .

Constraint: $- 1.0 < p < 1.0$ .
3: $a (n1)$ – double array: $a (i)$ must be set to the function value $y_{i - n}$ , for $i = 1, 2, \dots, 2 n$ .

Optional Input Parameters

None.

Output Parameters

1: $a (n1)$ – double array

n1 = 2 \times n

The contents of a are unspecified.

2: $g (n2)$ – double array

n2 = 2 \times n + 1

The array contains

$y_{0}$	in $g (1)$
$y_{1}$	in $g (2)$
$δ^{2 r} y_{0}$	in $g (2 r + 1)$
$δ^{2 r} y_{1}$	in $g (2 r + 2)$ , for $r = 1, 2, \dots, n - 1$ .

The interpolated function value

y_{p}

is stored in

g (2 n + 1)

3: $ifail$ – int64int32nag_int scalar

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$p \leq - 1.0$ ,
or	$p \geq 1.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

In general, increasing

n

improves the accuracy of the result until full attainable accuracy is reached, after which it might deteriorate. If

x

lies in the central interval of the data (i.e.,

0.0 \leq p < 1.0

), as is desirable, an upper bound on the contribution of the highest order differences (which is usually an upper bound on the error of the result) is given approximately in terms of the elements of the array g by

a \times (|g (2 n - 1)| + |g (2 n)|)

, where

a = 0.1

0.02

0.005

0.001

0.0002

for

n = 1, 2, 3, 4, 5

respectively, thereafter decreasing roughly by a factor of

4

each time.

Further Comments

The computation time increases as the order of

n

increases.

Example

This example interpolates at the point

x = 0.28

from the function values

(\begin{array}{r} x_{i} & - 1.00 & - 0.50 & 0.00 & 0.50 & 1.00 & 1.50 \\ y_{i} & 0.00 & - 0.53 & - 1.00 & - 0.46 & 2.00 & 11.09 \end{array}) .

We take

n = 3

and

p = 0.56

Open in the MATLAB editor: e01ab_example

function e01ab_example


fprintf('e01ab example results\n\n');

a = [-1  -0.50   0    0.50    1    1.50];
b = [ 0  -0.53  -1   -0.46    2   11.09];
n = int64(size(a,2)/2);

x = 0.28;
% We get p from x = a(n) + p*h, where h = 0.5
p = (x-a(n))/0.5;

[bx, c, ifail] = e01ab(n, p, b);

for k = 0:n-1
  fprintf('Central differences order %4d of y_0 = %12.5f\n', k, c(2*k+1));
  fprintf('%37s = %12.5f\n', 'y_1',c(2*k+2));
end
fprintf('\nFunction value at interpolation point = %12.5f\n', c(end));

e01ab example results

Central differences order    0 of y_0 =     -1.00000
                                  y_1 =     -0.46000
Central differences order    1 of y_0 =      1.01000
                                  y_1 =      1.92000
Central differences order    2 of y_0 =     -0.04000
                                  y_1 =      3.80000

Function value at interpolation point =     -0.83591

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox