NAG CL Interface
e01abc (dim1_​everett)

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1 Purpose

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>
void  e01abc (Integer n, double p, double a[], double g[], NagError *fail)
The function may be called by the names: e01abc, nag_interp_dim1_everett or nag_1d_everett_interp.

3 Description

e01abc interpolates a function of one variable at a given point
where -1p1 and h is the interval of differencing, from a table of values xm=x0+mh and ym where m=-(n-1),-(n-2),,-1,0,1,,n. The formula used is that of Fröberg (1970), neglecting the remainder term:
yp=r=0 n-1 (1-p+r 2r+1 ) δ2ry0+r=0 n-1 (p+r 2r+1 ) δ2ry1.  
The values of δ2ry0 and δ2ry1 are stored on exit from the function in addition to the interpolated function value yp.

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 double Input
On entry: the point p at which the interpolated function value is required, i.e., p=(x-x0)/h with -1.0<p<1.0.
Constraint: -1.0p1.0.
3: a[2×n] double Input/Output
On entry: a[i-1] must be set to the function value yi-n, for i=1,2,,2n.
On exit: the contents of a are unspecified.
4: g[2×n+1] double Output
On exit: the array contains
δ2ry0 in g[0]
δ2ry1 in g[1]
δ2ry0 in g[2r]
δ2ry1 in g[2r+1], for r=1,2,,n-1.
The interpolated function value yp is stored in g[2n].
5: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, p=value.
Constraint: p1.0.
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., 0.0p<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×(|g[2n-2]|+|g[2n-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.
Note that if p=1, y1 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.
e01abc 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
( xi -1.00 -0.50 0.00 0.50 1.00 1.50 yi 0.00 -0.53 -1.00 -0.46 2.00 11.09 ) .  
We take n=3 and p=0.56.

10.1 Program Text

Program Text (e01abce.c)

10.2 Program Data

Program Data (e01abce.d)

10.3 Program Results

Program Results (e01abce.r)