Integer, Intent (In)	::	n, n1, n2
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	p
Real (Kind=nag_wp), Intent (Inout)	::	a(2*n)
Real (Kind=nag_wp), Intent (Out)	::	g(2*n+1)

C Header Interface

#include <nag.h>

void	e01abf_ (const Integer n, const double p, double a[], double g[], const Integer n1, const Integer n2, Integer *ifail)

The routine may be called by the names e01abf or nagf_interp_dim1_everett.

3 Description

e01abf interpolates a function of one variable at a given point

x = x_{0} + p h,

where

- 1 \leq p \leq 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 routine in addition to the interpolated function value

y_{p}

4 References

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

5 Arguments

1: $n$ – Integer Input

On entry:

n

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

Constraint:

n > 0

2: $p$ – Real (Kind=nag_wp) Input

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 \leq p \leq 1.0

3: $a (2 \times n)$ – Real (Kind=nag_wp) array Input/Output

On entry:

a (i)

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: $g (2 \times n + 1)$ – Real (Kind=nag_wp) array Output

On exit: 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)

5: $n1$ – Integer Input

6: $n2$ – Integer Input

On entry: these arguments are no longer referenced, but is included for backwards compatability.

7: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

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

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $p = ⟨ value ⟩$ .
Constraint: $p \leq 1.0$ .

On entry, $p = ⟨ value ⟩$ .
Constraint: $p \geq - 1.0$ .

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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 - 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.

Note that if

p = 1

y_{1}

is returned. If

p = −1

and

n > 1

y_{−1}

is returned. In these cases, no interpolation is necessary and there is no loss of accuracy.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

e01abf is not threaded in any implementation.

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

e01ab: FL CL CPP AD PY MB

NAG FL Interfacee01abf (dim1_​everett)

▸▿ 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 FL Interface
e01abf (dim1_everett)