naginterfaces.library.interp.dim1_​everett

naginterfaces.library.interp.dim1_everett(p, a)

dim1_everett 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.

For full information please refer to the NAG Library document for e01ab

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/e01/e01abf.html

Parameters

pfloat

The point $p$ at which the interpolated function value is required, i.e., $p=(x-x_0)/h$ with $-1.0<p<1.0$.

afloat, array-like, shape $(2\times n)$

$a[\text{i}-1]$ must be set to the function value $y_{\text{i}-n}$, for $\text{i}=1,2,\dots ,2n$.

Returns

afloat, ndarray, shape $(2\times n)$

The contents of $a$ are unspecified.

gfloat, ndarray, shape $(2\times n+1)$

The array contains

$y_0$	in $g\left[0\right]$
$y_1$	in $g\left[1\right]$
${\delta }^{2r}{y}_0$	in $g\left[2r\right]$
${\delta }^{2r}{y}_1$	in $g[2\text{r}+1]$, for $\text{r}=1,2,\dots ,n-1$.

The interpolated function value $y_p$ is stored in $g\left[2n\right]$.

Raises

NagValueError

(errno $1$)

On entry, $p=⟨value⟩$.

Constraint: $p\le 1.0$.

(errno $1$)

On entry, $p=⟨value⟩$.

Constraint: $p\ge -1.0$.

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n>0$.

Notes

dim1_everett interpolates a function of one variable at a given point

$$x=x_0+ph\text{,}$$

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 $m=-(n-1),-(n-2),\dots ,-1,0,1,\dots ,n$. The formula used is that of Fröberg (1970), neglecting the remainder term:

$$y_p=\sum _{r=0}^{n-1}\left(\frac{1-p+r}{2r+1}\right){\delta }^{2r}{y}_0+\sum _{r=0}^{n-1}\left(\frac{p+r}{2r+1}\right){\delta }^{2r}{y}_1\text{.}$$

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 $y_p$.

References

Fröberg, C E, 1970, Introduction to Numerical Analysis, Addison–Wesley