# NAG CPP Interfacenagcpp::fit::dim1_spline_eval (e02bb)

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## 1Purpose

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

## 2Specification

```#include "e02/nagcpp_e02bb.hpp"
```
```template <typename LAMDA, typename C>

void function dim1_spline_eval(const LAMDA &lamda, const C &c, const double x, double &s, OptionalE02BB opt)```
```template <typename LAMDA, typename C>

void function dim1_spline_eval(const LAMDA &lamda, const C &c, const double x, double &s)```

## 3Description

dim1_spline_eval evaluates the cubic spline $s\left(x\right)$ at a prescribed argument $x$ from its augmented knot set ${\lambda }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n+7$, (see e02baf (no CPP interface)) and from the coefficients ${c}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$ in its B-spline representation
 $s(x)=∑i=1qciNi(x).$
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 ${\lambda }_{\mathit{j}}\ge {\lambda }_{\mathit{j}-1}$, for $\mathit{j}=2,3,\dots ,\overline{n}+7$, 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 e02baf (no CPP interface). A generalization of dim1_spline_eval which also forms the derivative of $s\left(x\right)$ is e02bcf (no CPP interface). e02bcf (no CPP interface) takes about $50%$ longer than dim1_spline_eval.

## 4References

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

## 5Arguments

1: $\mathbf{lamda}\left({\mathbf{ncap7}}\right)$double array Input
On entry: ${\mathbf{lamda}}\left(\mathit{j}-1\right)$ must be set to the value of the $\mathit{j}$th member of the complete set of knots, ${\lambda }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\overline{n}+7$.
Constraint: the ${\mathbf{lamda}}\left(j-1\right)$ must be in nondecreasing order with ${\mathbf{lamda}}\left({\mathbf{ncap7}}-4\right)>{\mathbf{lamda}}\left(3\right)$.
2: $\mathbf{c}\left({\mathbf{ncap7}}\right)$double array Input
On entry: the coefficient ${c}_{\mathit{i}}$ of the B-spline ${N}_{\mathit{i}}\left(x\right)$, for $\mathit{i}=1,2,\dots ,\overline{n}+3$. The remaining elements of the array are not referenced.
3: $\mathbf{x}$double Input
On entry: the argument $x$ at which the cubic spline is to be evaluated.
Constraint: ${\mathbf{lamda}}\left(3\right)\le {\mathbf{x}}\le {\mathbf{lamda}}\left({\mathbf{ncap7}}-4\right)$.
4: $\mathbf{s}$double Output
On exit: the value of the spline, $s\left(x\right)$.
5: $\mathbf{opt}$OptionalE02BB Input/Output
Optional parameter container, derived from Optional.

1: $\mathbf{ncap7}$
$\stackrel{-}{n}+7$, where $\stackrel{-}{n}$ is the number of intervals (one greater than the number of interior knots, i.e., the knots strictly within the range ${\lambda }_{4}$ to ${\lambda }_{\stackrel{-}{n}+4}$) over which the spline is defined.

## 6Exceptions and Warnings

Errors or warnings detected by the function:
All errors and warnings have an associated numeric error code field, errorid, stored either as a member of the thrown exception object (see errorid), or as a member of opt.ifail, depending on how errors and warnings are being handled (see Error Handling for more details).
Raises: ErrorException
$\mathbf{errorid}=1$
On entry, ${\mathbf{x}}=⟨\mathit{value}⟩$, ${\mathbf{ncap7}}=⟨\mathit{value}⟩$
and ${\mathbf{lamda}}\left[{\mathbf{ncap7}}-4\right]=⟨\mathit{value}⟩$.
Constraint: ${\mathbf{x}}\le {\mathbf{lamda}}\left[{\mathbf{ncap7}}-4\right]$.
$\mathbf{errorid}=1$
On entry, ${\mathbf{x}}=⟨\mathit{value}⟩$ and
${\mathbf{lamda}}\left[3\right]=⟨\mathit{value}⟩$.
Constraint: ${\mathbf{x}}\ge {\mathbf{lamda}}\left[3\right]$.
$\mathbf{errorid}=2$
On entry, ${\mathbf{ncap7}}=⟨\mathit{value}⟩$.
Constraint: ${\mathbf{ncap7}}\ge 8$.
$\mathbf{errorid}=10601$
On entry, argument $⟨\mathit{\text{value}}⟩$ must be a vector of size $⟨\mathit{\text{value}}⟩$ array.
Supplied argument has $⟨\mathit{\text{value}}⟩$ dimensions.
$\mathbf{errorid}=10601$
On entry, argument $⟨\mathit{\text{value}}⟩$ must be a vector of size $⟨\mathit{\text{value}}⟩$ array.
Supplied argument was a vector of size $⟨\mathit{\text{value}}⟩$.
$\mathbf{errorid}=10601$
On entry, argument $⟨\mathit{\text{value}}⟩$ must be a vector of size $⟨\mathit{\text{value}}⟩$ array.
The size for the supplied array could not be ascertained.
$\mathbf{errorid}=10602$
On entry, the raw data component of $⟨\mathit{\text{value}}⟩$ is null.
$\mathbf{errorid}=10603$
On entry, unable to ascertain a value for $⟨\mathit{\text{value}}⟩$.
$\mathbf{errorid}=-99$
An unexpected error has been triggered by this routine.
$\mathbf{errorid}=-399$
Your licence key may have expired or may not have been installed correctly.
$\mathbf{errorid}=-999$
Dynamic memory allocation failed.

## 7Accuracy

The computed value of $s\left(x\right)$ has negligible error in most practical situations. Specifically, this value has an absolute error bounded in modulus by , where ${c}_{\mathrm{max}}$ is the largest in modulus of ${c}_{j},{c}_{j+1},{c}_{j+2}$ and ${c}_{j+3}$, and $j$ is an integer such that ${\lambda }_{j+3}\le x\le {\lambda }_{j+4}$. If ${c}_{j},{c}_{j+1},{c}_{j+2}$ and ${c}_{j+3}$ are all of the same sign, then the computed value of $s\left(x\right)$ has a relative error not exceeding in modulus. For further details see Cox (1978).

## 8Parallelism and Performance

Please see the description for the underlying computational routine in this section of the FL Interface documentation.

The time taken is approximately ${\mathbf{c}}×\left(1+0.1×\mathrm{log}\left(\overline{n}+7\right)\right)$ seconds, where c is a machine-dependent constant.
Note:  the function does not test all the conditions on the knots given in the description of lamda in Section 5, since to do this would result in a computation time approximately linear in $\overline{n}+7$ instead of $\mathrm{log}\left(\overline{n}+7\right)$. All the conditions are tested in e02baf (no CPP interface), however.

## 10Example

Evaluate at nine equally-spaced points in the interval $1.0\le x\le 9.0$ the cubic spline with (augmented) knots $1.0$, $1.0$, $1.0$, $1.0$, $3.0$, $6.0$, $8.0$, $9.0$, $9.0$, $9.0$, $9.0$ and normalized cubic B-spline coefficients $1.0$, $2.0$, $4.0$, $7.0$, $6.0$, $4.0$, $3.0$.
The example program is written in a general form that will enable a cubic spline with $\overline{n}$ intervals, in its normalized cubic B-spline form, to be evaluated at $m$ equally-spaced points in the interval ${\mathbf{lamda}}\left(3\right)\le x\le {\mathbf{lamda}}\left(\overline{n}+3\right)$. The program is self-starting in that any number of datasets may be supplied.

### 10.1Example Program

Source File Data Results
ex_e02bb.cpp None ex_e02bb.r