NAG Library Function Document

nag_1d_everett_interp (e01abc)

+− 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

1 Purpose

nag_1d_everett_interp (e01abc) interpolates a function of one variable at a given point

x

from a table of function values evaluated at equidistant points, using Everett's formula.

2 Specification

#include <nag.h>

#include <nage01.h>

void	nag_1d_everett_interp (Integer n, double p, double a[], double g[], NagError *fail)

3 Description

nag_1d_everett_interp (e01abc) interpolates a function of one variable at a given point

x = x_{0} + p h,

where

- 1 < p < 1

and

h

is the interval of differencing, from a table of values

x_{m} = x_{0} + m h

and

y_{m}

where

m = - (n - 1), - (n - 2), \dots, - 1, 0, 1, \dots, n

. The formula used is that of Fröberg (1970), neglecting the remainder term:

y_{p} = \sum_{r = 0}^{n - 1} (\frac{1 - p + r}{2 r + 1}) δ^{2 r} y_{0} + \sum_{r = 0}^{n - 1} (\frac{p + r}{2 r + 1}) δ^{2 r} y_{1} .

The values of

δ^{2 r} y_{0}

and

δ^{2 r} y_{1}

are stored on exit from the function in addition to the interpolated function value

y_{p}

4 References

Fröberg C E (1970) Introduction to Numerical Analysis Addison–Wesley

5 Arguments

1: n – IntegerInput

On entry:

n

, half the number of points to be used in the interpolation.

Constraint:

n > 0

2: p – doubleInput

On entry: 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[ $2 \times n$ ] – doubleInput/Output

On entry:

a [i - 1]

must be set to the function value

y_{i - n}

, for

i = 1, 2, \dots, 2 n

On exit: the contents of a are unspecified.

4: g[ $2 \times n + 1$ ] – doubleOutput

On exit: the array contains

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

The interpolated function value

y_{p}

is stored in

g [2 n]

5: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n > 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL: On entry, $p = ⟨value⟩$ .
Constraint: $p < 1.0$ .
On entry, $p = ⟨value⟩$ .
Constraint: $p > - 1.0$ .

7 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 - 2]| + |g [2 n - 1]|)

, 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.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The computation time increases as the order of

n

increases.

10 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

NAG Library Function Documentnag_1d_everett_interp (e01abc)