nag_1d_aitken_interp (e01aac) : NAG Library, Mark 24

nag_1d_aitken_interp (e01aac) interpolates a function of one variable at a given point

x

from a table of function values

y_{i}

evaluated at equidistant or non-equidistant points

x_{i}

, for

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

, using Aitken's technique of successive linear interpolations.

nag_1d_aitken_interp (e01aac) 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.

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

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

On entry:

a [i - 1]

must contain the

x

-component of the

i

th data point,

x_{i}

, for

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

On exit:

a [i - 1]

contains the value

x_{i} - x

, for

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

On entry:

b [i - 1]

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.

On exit:

$c [0], \dots, c [n - 1]$ contain the first set of linear interpolations,
$c [n], \dots, c [2 \times n - 2]$ contain the second set of linear interpolations,
$c [2 n - 1], \dots, c [3 \times n - 4]$ contain the third set of linear interpolations,
$⋮$
$c [n \times (n + 1) / 2 - 1]$ contains the interpolated function value at the point $x$ .

On entry: the point

x

at which the interpolation is required.

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

On entry, argument

⟨value⟩

had an illegal value.

On entry,

n = ⟨value⟩

.
Constraint:

n > 0

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.

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.

The computation time for interpolation at any point

x

is proportional to

n \times (n + 1) / 2

This example interpolates at

x = 0.28

the function value of a curve defined by the points

(\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}) .

NAG Library Function Document

nag_1d_aitken_interp (e01aac)

+− 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 Library Function Documentnag_1d_aitken_interp (e01aac)

+− 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 Library Function Document

nag_1d_aitken_interp (e01aac)