naginterfaces.library.interp.dim1_​aitken

naginterfaces.library.interp.dim1_aitken(a, b, x)

dim1_aitken 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 $\text{i}=1,2,\dots ,n+1$, using Aitken’s technique of successive linear interpolations.

For full information please refer to the NAG Library document for e01aa

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/e01/e01aaf.html

Parameters

afloat, array-like, shape $(n+1)$

$a[\text{i}-1]$ must contain the $x$-component of the $\text{i}$th data point, $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n+1$.

bfloat, array-like, shape $(n+1)$

$b[\text{i}-1]$ must contain the $y$-component (function value) of the $\text{i}$th data point, $y_{\text{i}}$, for $\text{i}=1,2,\dots ,n+1$.

xfloat

The point $x$ at which the interpolation is required. Note that $x$ may lie outside the interval defined by the minimum and maximum values in $a$, in which case an extrapolated value will be computed; extrapolated results should be treated with considerable caution since there is no information on the behaviour of the function outside the defined interval.

Returns

afloat, ndarray, shape $(n+1)$

$a[\text{i}-1]$ contains the value $x_{\text{i}}-x$, for $\text{i}=1,2,\dots ,n+1$.

bfloat, ndarray, shape $(n+1)$

The contents of $b$ are unspecified.

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

$c\left[0\right],\dots ,c[n-1]$ contain the first set of linear interpolations,

$c\left[n\right],\dots ,c[2\times n-2]$ contain the second set of linear interpolations,

$c[2n-1],\dots ,c[3\times n-4]$ contain the third set of linear interpolations,

$⋮$

$c[n\times (n+1)/2-1]$ contains the interpolated function value at the point $x$.

Raises

NagValueError

(errno $6$)

On entry, error in parameter $\text{n}$.

Constraint: $n>0$.

Notes

dim1_aitken 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.

References

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