nag_1d_everett_interp (e01abc) (PDF version)
e01 Chapter Contents
e01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_1d_everett_interp (e01abc)

+ Contents

    1  Purpose
    7  Accuracy

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=x0+ph,
where -1<p<1 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:     nIntegerInput
On entry: n, half the number of points to be used in the interpolation.
Constraint: n>0.
2:     pdoubleInput
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: -1.0<p<1.0.
3:     a[2×n]doubleInput/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]doubleOutput
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:     failNagError *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., 0.0p<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.

8  Further Comments

The computation time increases as the order of n increases.

9  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.

9.1  Program Text

Program Text (e01abce.c)

9.2  Program Data

Program Data (e01abce.d)

9.3  Program Results

Program Results (e01abce.r)


nag_1d_everett_interp (e01abc) (PDF version)
e01 Chapter Contents
e01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012