naginterfaces.library.fit.dim1_​spline_​eval

naginterfaces.library.fit.dim1_spline_eval(lamda, c, x)

dim1_spline_eval evaluates a cubic spline from its B-spline representation.

For full information please refer to the NAG Library document for e02bb

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/e02/e02bbf.html

Parameters

lamdafloat, array-like, shape $\left(\text{ncap7}\right)$

$lamda[\text{j}-1]$ must be set to the value of the $\text{j}$th member of the complete set of knots, ${\lambda }_{\text{j}}$, for $\text{j}=1,2,\dots ,\overline{n}+7$.

cfloat, array-like, shape $\left(\text{ncap7}\right)$

The coefficient $c_{\text{i}}$ of the B-spline $N_{\text{i}}\left(x\right)$, for $\text{i}=1,2,\dots ,\overline{n}+3$. The remaining elements of the array are not referenced.

xfloat

The argument $x$ at which the cubic spline is to be evaluated.

Returns

sfloat

The value of the spline, $s\left(x\right)$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$, $\text{ncap7}=⟨value⟩$ and $lamda[\text{ncap7}-4]=⟨value⟩$.

Constraint: $x\le lamda[\text{ncap7}-4]$.

(errno $1$)

On entry, $x=⟨value⟩$ and $lamda\left[3\right]=⟨value⟩$.

Constraint: $x\ge lamda\left[3\right]$.

(errno $2$)

On entry, $\text{ncap7}=⟨value⟩$.

Constraint: $\text{ncap7}\ge 8$.

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_eval evaluates the cubic spline $s\left(x\right)$ at a prescribed argument $x$ from its augmented knot set ${\lambda }_{\text{i}}$, for $\text{i}=1,2,\dots ,n+7$, (see dim1_spline_knots()) and from the coefficients $c_{\text{i}}$, for $\text{i}=1,2,\dots ,q$ in its B-spline representation

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

Here $q=\overline{n}+3$, where $\overline{n}$ is the number of intervals of the spline, and $N_i\left(x\right)$ denotes the normalized B-spline of degree $3$ defined upon the knots ${\lambda }_i,{\lambda }_{i+1},\dots ,{\lambda }_{i+4}$. The prescribed argument $x$ must satisfy ${\lambda }_4\le x\le {\lambda }_{\overline{n}+4}$.

It is assumed that ${\lambda }_{\text{j}}\ge {\lambda }_{\text{j}-1}$, for $\text{j}=2,3,\dots ,\overline{n}+7$, and ${\lambda }_{\overline{n}+4}>{\lambda }_4$.

If $x$ is a point at which $4$ knots coincide, $s\left(x\right)$ is discontinuous at $x$; in this case, $s$ contains the value defined as $x$ is approached from the right.

The method employed is that of evaluation by taking convex combinations due to de Boor (1972). For further details of the algorithm and its use see Cox (1972) and Cox and Hayes (1973).

It is expected that a common use of dim1_spline_eval will be the evaluation of the cubic spline approximations produced by dim1_spline_knots(). A generalization of dim1_spline_eval which also forms the derivative of $s\left(x\right)$ is dim1_spline_deriv(). dim1_spline_deriv() takes about $50\%$ longer than dim1_spline_eval.

References

Cox, M G, 1972, The numerical evaluation of B-splines, J. Inst. Math. Appl. (10), 134–149

Cox, M G, 1978, The numerical evaluation of a spline from its B-spline representation, J. Inst. Math. Appl. (21), 135–143

Cox, M G and Hayes, J G, 1973, Curve fitting: a guide and suite of algorithms for the non-specialist user, NPL Report NAC26, National Physical Laboratory

de Boor, C, 1972, On calculating with B-splines, J. Approx. Theory (6), 50–62