E01AAF : NAG Library, Mark 24

E01AAF interpolates a function of one variable at a given point

x

from a table of values

x_{i}

and

y_{i}

, for

i = 1, 2, \dots, n + 1

using Aitken's method (see Fröberg (1970)). The intermediate values of linear interpolations are stored to enable an estimate of the accuracy of the results to be made.

5 Parameters

1: A(N1) – REAL (KIND=nag_wp) arrayInput/Output

On entry:

A (i)

must contain the

x

-component of the

i

th data point,

x_{i}

, for

i = 1, 2, \dots, n + 1

On exit:

A (i)

contains the value

x_{i} - x

, for

i = 1, 2, \dots, n + 1

2: B(N1) – REAL (KIND=nag_wp) arrayInput/Output

On entry:

B (i)

must contain the

y

-component (function value) of the

i

th data point,

y_{i}

, for

i = 1, 2, \dots, n + 1

On exit: the contents of B are unspecified.

3: C(N2) – REAL (KIND=nag_wp) arrayOutput

On exit:

$C (1), \dots, C (n)$ contain the first set of linear interpolations,
$C (n + 1), \dots, C (2 \times n - 1)$ contain the second set of linear interpolations,
$C (2 n), \dots, C (3 \times n - 3)$ contain the third set of linear interpolations,
$⋮$
$C (n \times (n + 1) / 2)$ contains the interpolated function value at the point $x$ .

4: N1 – INTEGERInput

On entry: the value

n + 1

where

n

is the number of intervals; that is, N1 is the number of data points.

5: N2 – INTEGERInput

On entry: the value

n \times (n + 1) / 2

where

n

is the number of intervals.

6: N – INTEGERInput

On entry: the number of intervals which are to be used in interpolating the value at

x

; that is, there are

n + 1

data points

(x_{i}, y_{i})

Constraint:

N > 0

7: X – REAL (KIND=nag_wp)Input

On entry: the point

x

at which the interpolation is required.

7 Accuracy

An estimate of the accuracy of the result can be made from a comparison of the final result and the previous interpolates, given in the array C. In particular, the first interpolate in the

i

th set, for

i = 1, 2, \dots, n

, is the value at

x

of the polynomial interpolating the first

(i + 1)

data points. It is given in position

(i - 1) (2 n - i + 2) / 2

of the array C. Ideally, providing

n

is large enough, this set of

n

interpolates should exhibit convergence to the final value, the difference between one interpolate and the next settling down to a roughly constant magnitude (but with varying sign). This magnitude indicates the size of the error (any subsequent increase meaning that the value of

n

is too high). Better convergence will be obtained if the data points are supplied, not in their natural order, but ordered so that the first

i

data points give good coverage of the neighbourhood of

x

, for all

i

. To this end, the following ordering is recommended as widely suitable: first the point nearest to

x

, then the nearest point on the opposite side of

x

, followed by the remaining points in increasing order of their distance from

x

, that is of

|x_{r} - x|

. With this modification the Aitken method will generally perform better than the related method of Neville, which is often given in the literature as superior to that of Aitken.

NAG Library Routine Document

E01AAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentE01AAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

E01AAF