naginterfaces.library.fit.dim1_​spline_​deriv

naginterfaces.library.fit.dim1_spline_deriv(lamda, c, x, left)

dim1_spline_deriv evaluates a cubic spline and its first three derivatives from its B-spline representation.

For full information please refer to the NAG Library document for e02bc

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/e02/e02bcf.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 and its derivatives are to be evaluated.

leftint

Specifies whether left- or right-hand values of the spline and its derivatives are to be computed (see Notes). Left - or right-hand values are formed according to whether $left$ is equal or not equal to $1$.

If $x$ does not coincide with a knot, the value of $left$ is immaterial.

If $x=lamda\left[3\right]$, right-hand values are computed.

If $x=lamda[\text{ncap7}-4]$, left-hand values are formed, regardless of the value of $left$.

Returns

sfloat, ndarray, shape $\left(4\right)$

$s[\text{j}-1]$ contains the value of the $(\text{j}-1)$th derivative of the spline at the argument $x$, for $\text{j}=1,2,\dots ,4$. Note that $s\left[0\right]$ contains the value of the spline.

Raises

NagValueError

(errno $1$)

On entry, $\text{ncap7}=⟨value⟩$.

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

(errno $2$)

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

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

(errno $2$)

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

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

(errno $2$)

On entry, $lamda\left[3\right]=⟨value⟩$, $\text{ncap7}=⟨value⟩$ and $lamda[\text{ncap7}-4]=⟨value⟩$.

Constraint: $lamda\left[3\right]<lamda[\text{ncap7}-4]$.

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_deriv evaluates the cubic spline $s\left(x\right)$ and its first three derivatives at a prescribed argument $x$. It is assumed that $s\left(x\right)$ is represented in terms of its B-spline coefficients $c_{\text{i}}$, for $\text{i}=1,2,\dots ,\overline{n}+3$ and (augmented) ordered knot set ${\lambda }_{\text{i}}$, for $\text{i}=1,2,\dots ,\overline{n}+7$, (see dim1_spline_knots()), i.e.,

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

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

$${\lambda }_4\le x\le {\lambda }_{\overline{n}+4}\text{.}$$

At a simple knot ${\lambda }_i$ (i.e., one satisfying ${\lambda }_{i-1}<{\lambda }_i<{\lambda }_{i+1}$), the third derivative of the spline is in general discontinuous. At a multiple knot (i.e., two or more knots with the same value), lower derivatives, and even the spline itself, may be discontinuous. Specifically, at a point $x=u$ where (exactly) $r$ knots coincide (such a point is termed a knot of multiplicity $r$), the values of the derivatives of order $4-\text{j}$, for $\text{j}=1,2,\dots ,r$, are in general discontinuous. (Here $1\le r\le 4$; $r>4$ is not meaningful.) You must specify whether the value at such a point is required to be the left- or right-hand derivative.

The method employed is based upon:

1. carrying out a binary search for the knot interval containing the argument $x$ (see Cox (1978)),

2. evaluating the nonzero B-splines of orders $1$, $2$, $3$ and $4$ by recurrence (see Cox (1972) and Cox (1978)),

3. computing all derivatives of the B-splines of order $4$ by applying a second recurrence to these computed B-spline values (see de Boor (1972)),

4. multiplying the fourth-order B-spline values and their derivative by the appropriate B-spline coefficients, and summing, to yield the values of $s\left(x\right)$ and its derivatives.

dim1_spline_deriv can be used to compute the values and derivatives of cubic spline fits and interpolants produced by dim1_spline_knots().

If only values and not derivatives are required, dim1_spline_eval() may be used instead of dim1_spline_deriv, which 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

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