NAG Library Function Document

1Purpose

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.

2Specification

 #include #include
 void nag_1d_everett_interp (Integer n, double p, double a[], double g[], NagError *fail)

3Description

nag_1d_everett_interp (e01abc) interpolates a function of one variable at a given point
 $x=x0+ph,$
where $-1\le p\le 1$ and $h$ is the interval of differencing, from a table of values ${x}_{m}={x}_{0}+mh$ and ${y}_{m}$ where $m=-\left(n-1\right),-\left(n-2\right),\dots ,-1,0,1,\dots ,n$. 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 ${\delta }^{2r}{y}_{0}$ and ${\delta }^{2r}{y}_{1}$ are stored on exit from the function in addition to the interpolated function value ${y}_{p}$.
Fröberg C E (1970) Introduction to Numerical Analysis Addison–Wesley

5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, half the number of points to be used in the interpolation.
Constraint: ${\mathbf{n}}>0$.
2:    $\mathbf{p}$doubleInput
On entry: the point $p$ at which the interpolated function value is required, i.e., $p=\left(x-{x}_{0}\right)/h$ with $-1.0.
Constraint: $-1.0\le {\mathbf{p}}\le 1.0$.
3:    $\mathbf{a}\left[2×n\right]$doubleInput/Output
On entry: ${\mathbf{a}}\left[\mathit{i}-1\right]$ must be set to the function value ${y}_{\mathit{i}-n}$, for $\mathit{i}=1,2,\dots ,2n$.
On exit: the contents of a are unspecified.
4:    $\mathbf{g}\left[2×n+1\right]$doubleOutput
On exit: the array contains
 $\phantom{{\delta }^{2r}}{y}_{0}$ in ${\mathbf{g}}\left[0\right]$ $\phantom{{\delta }^{2r}}{y}_{1}$ in ${\mathbf{g}}\left[1\right]$ ${\delta }^{2r}{y}_{0}$ in ${\mathbf{g}}\left[2r\right]$ ${\delta }^{2r}{y}_{1}$ in ${\mathbf{g}}\left[2\mathit{r}+1\right]$, for $\mathit{r}=1,2,\dots ,n-1$.
The interpolated function value ${y}_{p}$ is stored in ${\mathbf{g}}\left[2n\right]$.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}\le 1.0$.
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}\ge -1.0$.

7Accuracy

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., $0.0\le p<1.0$), 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 $a×\left(\left|{\mathbf{g}}\left[2n-2\right]\right|+\left|{\mathbf{g}}\left[2n-1\right]\right|\right)$, 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 ${\mathbf{p}}=1$, ${y}_{1}$ is returned. If ${\mathbf{p}}=-1$ and ${\mathbf{n}}>1$, ${y}_{-1}$ is returned. In these cases, no interpolation is necessary and there is no loss of accuracy.

8Parallelism and Performance

nag_1d_everett_interp (e01abc) is not threaded in any implementation.

The computation time increases as the order of $n$ increases.

10Example

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 $p=0.56$.

10.1Program Text

Program Text (e01abce.c)

10.2Program Data

Program Data (e01abce.d)

10.3Program Results

Program Results (e01abce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017