naginterfaces.library.fit.dim1_​spline_​integ

naginterfaces.library.fit.dim1_spline_integ(lamda, c)

dim1_spline_integ computes the definite integral of a cubic spline from its B-spline representation.

For full information please refer to the NAG Library document for e02bd

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/e02/e02bdf.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.

Returns

dintfloat

The value of the definite integral of $s\left(x\right)$ between the limits $x=a$ and $x=b$, where $a={\lambda }_4$ and $b={\lambda }_{\overline{n}+4}$.

Raises

NagValueError

(errno $1$)

On entry, $\text{ncap7}=⟨value⟩$.

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

(errno $2$)

On entry, $J=⟨value⟩$, $lamda[J-1]=⟨value⟩$ and $lamda[J-2]=⟨value⟩$.

Constraint: $lamda[J-1]\ge lamda[J-2]$.

(errno $2$)

On entry, $\text{ncap7}=⟨value⟩$, $lamda[\text{ncap7}-2]=⟨value⟩$ and $lamda[\text{ncap7}-1]=⟨value⟩$.

Constraint: $lamda[\text{ncap7}-2]=lamda[\text{ncap7}-1]$.

(errno $2$)

On entry, $\text{ncap7}=⟨value⟩$, $lamda[\text{ncap7}-3]=⟨value⟩$ and $lamda[\text{ncap7}-2]=⟨value⟩$.

Constraint: $lamda[\text{ncap7}-3]=lamda[\text{ncap7}-2]$.

(errno $2$)

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

Constraint: $lamda[\text{ncap7}-4]=lamda[\text{ncap7}-3]$.

(errno $2$)

On entry, $lamda\left[2\right]=⟨value⟩$ and $lamda\left[3\right]=⟨value⟩$.

Constraint: $lamda\left[2\right]=lamda\left[3\right]$.

(errno $2$)

On entry, $lamda\left[1\right]=⟨value⟩$ and $lamda\left[2\right]=⟨value⟩$.

Constraint: $lamda\left[1\right]=lamda\left[2\right]$.

(errno $2$)

On entry, $lamda\left[0\right]=⟨value⟩$ and $lamda\left[1\right]=⟨value⟩$.

Constraint: $lamda\left[0\right]=lamda\left[1\right]$.

(errno $2$)

On entry, $lamda\left[3\right]=⟨value⟩$, $\text{NCAP4}=⟨value⟩$ and $lamda[\text{NCAP4}-1]=⟨value⟩$.

Constraint: $lamda\left[3\right]<lamda[\text{NCAP4}-1]$.

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_integ computes the definite integral of the cubic spline $s\left(x\right)$ between the limits $x=a$ and $x=b$, where $a$ and $b$ are respectively the lower and upper limits of the range over which $s\left(x\right)$ is defined. It is assumed that $s\left(x\right)$ is represented in terms of its B-spline coefficients $c_i$, for $\text{i}=1,2,\dots ,\overline{n}+3$ and (augmented) ordered knot set ${\lambda }_i$, for $\text{i}=1,2,\dots ,\overline{n}+7$, with ${\lambda }_i=a$, for $\text{i}=1,2,\dots ,4$ and ${\lambda }_i=b$, for $\text{i}=\overline{n}+4,\dots ,\overline{n}+7$, (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 method employed uses the formula given in Section 3 of Cox (1975).

dim1_spline_integ can be used to determine the definite integrals of cubic spline fits and interpolants produced by dim1_spline_knots().

References

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