naginterfaces.library.interp.dim1_​spline

naginterfaces.library.interp.dim1_spline(x, y)

dim1_spline determines a cubic spline interpolant to a given set of data.

For full information please refer to the NAG Library document for e01ba

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/e01/e01baf.html

Parameters

xfloat, array-like, shape $\left(m\right)$

$x[\text{i}-1]$ must be set to $x_{\text{i}}$, the $\text{i}$th data value of the independent variable $x$, for $\text{i}=1,2,\dots ,m$.

yfloat, array-like, shape $\left(m\right)$

$y[\text{i}-1]$ must be set to $y_{\text{i}}$, the $\text{i}$th data value of the dependent variable $y$, for $\text{i}=1,2,\dots ,m$.

Returns

lamdafloat, ndarray, shape $(m+4)$

The value of ${\lambda }_{\text{i}}$, the $\text{i}$th knot, for $\text{i}=1,2,\dots ,m+4$.

cfloat, ndarray, shape $\left(m\right)$

The coefficient $c_{\text{i}}$ of the B-spline $N_{\text{i}}\left(x\right)$, for $\text{i}=1,2,\dots ,m$. The remaining elements of the array are not used.

Raises

NagValueError

(errno $1$)

On entry, $\text{lck}=⟨value⟩$ and $m=⟨value⟩$.

Constraint: $\text{lck}\ge m+4$.

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 4$.

(errno $2$)

On entry, $\text{I}=⟨value⟩$, $x[\text{I}-1]=⟨value⟩$ and $x[\text{I}-2]=⟨value⟩$.

Constraint: $x[\text{I}-1]>x[\text{I}-2]$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

dim1_spline determines a cubic spline $s\left(x\right)$, defined in the range $x_1\le x\le x_m$, which interpolates (passes exactly through) the set of data points $(x_{\text{i}},y_{\text{i}})$, for $\text{i}=1,2,\dots ,m$, where $m\ge 4$ and $x_1<x_2<\cdots <x_m$. Unlike some other spline interpolation algorithms, derivative end conditions are not imposed. The spline interpolant chosen has $m-4$ interior knots ${\lambda }_5,{\lambda }_6,\dots ,{\lambda }_m$, which are set to the values of $x_3,x_4,\dots ,x_{m-2}$ respectively. This spline is represented in its B-spline form (see Cox (1975)):

$$s\left(x\right)=\sum _{i=1}^m{c}_i{N}_i\left(x\right)\text{,}$$

where $N_i\left(x\right)$ denotes the normalized B-spline of degree $3$, defined upon the knots ${\lambda }_i,{\lambda }_{i+1},\dots ,{\lambda }_{i+4}$, and $c_i$ denotes its coefficient, whose value is to be determined by the function.

The use of B-splines requires eight additional knots ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, ${\lambda }_4$, ${\lambda }_{m+1}$, ${\lambda }_{m+2}$, ${\lambda }_{m+3}$ and ${\lambda }_{m+4}$ to be specified; dim1_spline sets the first four of these to $x_1$ and the last four to $x_m$.

The algorithm for determining the coefficients is as described in Cox (1975) except that $QR$ factorization is used instead of $LU$ decomposition. The implementation of the algorithm involves setting up appropriate information for the related function fit.dim1_spline_knots followed by a call of that function. (See fit.dim1_spline_knots for further details.)

Values of the spline interpolant, or of its derivatives or definite integral, can subsequently be computed as detailed in Further Comments.

References

Cox, M G, 1975, An algorithm for spline interpolation, J. Inst. Math. Appl. (15), 95–108

Cox, M G, 1977, A survey of numerical methods for data and function approximation, The State of the Art in Numerical Analysis, (ed D A H Jacobs), 627–668, Academic Press