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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_fit_1dspline_eval (e02bb)

## Purpose

nag_fit_1dspline_eval (e02bb) evaluates a cubic spline from its B-spline representation.

## Syntax

[s, ifail] = e02bb(lamda, c, x, 'ncap7', ncap7)
[s, ifail] = nag_fit_1dspline_eval(lamda, c, x, 'ncap7', ncap7)

## Description

nag_fit_1dspline_eval (e02bb) evaluates the cubic spline s(x)$s\left(x\right)$ at a prescribed argument x$x$ from its augmented knot set λi${\lambda }_{\mathit{i}}$, for i = 1,2,,n + 7$\mathit{i}=1,2,\dots ,n+7$, (see nag_fit_1dspline_knots (e02ba)) and from the coefficients ci${c}_{\mathit{i}}$, for i = 1,2,,q$\mathit{i}=1,2,\dots ,q$in its B-spline representation
 q s(x) = ∑ ciNi(x). i = 1
$s(x)=∑i=1qciNi(x).$
Here q = n + 3$q=\stackrel{-}{n}+3$, where n$\stackrel{-}{n}$ is the number of intervals of the spline, and Ni(x)${N}_{i}\left(x\right)$ denotes the normalized B-spline of degree 3$3$ defined upon the knots λi,λi + 1,,λi + 4${\lambda }_{i},{\lambda }_{i+1},\dots ,{\lambda }_{i+4}$. The prescribed argument x$x$ must satisfy λ4xλn + 4${\lambda }_{4}\le x\le {\lambda }_{\stackrel{-}{n}+4}$.
It is assumed that λjλj1${\lambda }_{\mathit{j}}\ge {\lambda }_{\mathit{j}-1}$, for j = 2,3,,n + 7$\mathit{j}=2,3,\dots ,\stackrel{-}{n}+7$, and λn + 4 > λ4${\lambda }_{\stackrel{-}{n}+4}>{\lambda }_{4}$.
If x$x$ is a point at which 4$4$ knots coincide, s(x)$s\left(x\right)$ is discontinuous at x$x$; in this case, s contains the value defined as x$x$ 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 nag_fit_1dspline_eval (e02bb) will be the evaluation of the cubic spline approximations produced by nag_fit_1dspline_knots (e02ba). A generalization of nag_fit_1dspline_eval (e02bb) which also forms the derivative of s(x)$s\left(x\right)$ is nag_fit_1dspline_deriv (e02bc). nag_fit_1dspline_deriv (e02bc) takes about 50%$50%$ longer than nag_fit_1dspline_eval (e02bb).

## 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

## Parameters

### Compulsory Input Parameters

1:     lamda(ncap7) – double array
ncap7, the dimension of the array, must satisfy the constraint ncap78${\mathbf{ncap7}}\ge 8$.
lamda(j)${\mathbf{lamda}}\left(\mathit{j}\right)$ must be set to the value of the j$\mathit{j}$th member of the complete set of knots, λj${\lambda }_{\mathit{j}}$, for j = 1,2,,n + 7$\mathit{j}=1,2,\dots ,\stackrel{-}{n}+7$.
Constraint: the lamda(j)${\mathbf{lamda}}\left(j\right)$ must be in nondecreasing order with lamda(ncap73) > lamda(4)${\mathbf{lamda}}\left({\mathbf{ncap7}}-3\right)>{\mathbf{lamda}}\left(4\right)$.
2:     c(ncap7) – double array
ncap7, the dimension of the array, must satisfy the constraint ncap78${\mathbf{ncap7}}\ge 8$.
The coefficient ci${c}_{\mathit{i}}$ of the B-spline Ni(x)${N}_{\mathit{i}}\left(x\right)$, for i = 1,2,,n + 3$\mathit{i}=1,2,\dots ,\stackrel{-}{n}+3$. The remaining elements of the array are not referenced.
3:     x – double scalar
The argument x$x$ at which the cubic spline is to be evaluated.
Constraint: lamda(4)xlamda(ncap73)${\mathbf{lamda}}\left(4\right)\le {\mathbf{x}}\le {\mathbf{lamda}}\left({\mathbf{ncap7}}-3\right)$.

### Optional Input Parameters

1:     ncap7 – int64int32nag_int scalar
Default: The dimension of the arrays lamda, c. (An error is raised if these dimensions are not equal.)
n + 7$\stackrel{-}{n}+7$, where n$\stackrel{-}{n}$ is the number of intervals (one greater than the number of interior knots, i.e., the knots strictly within the range λ4${\lambda }_{4}$ to λn + 4${\lambda }_{\stackrel{-}{n}+4}$) over which the spline is defined.
Constraint: ncap78${\mathbf{ncap7}}\ge 8$.

None.

### Output Parameters

1:     s – double scalar
The value of the spline, s(x)$s\left(x\right)$.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
The parameter x does not satisfy lamda(4)xlamda(ncap73)${\mathbf{lamda}}\left(4\right)\le {\mathbf{x}}\le {\mathbf{lamda}}\left({\mathbf{ncap7}}-3\right)$.
In this case the value of s is set arbitrarily to zero.
ifail = 2${\mathbf{ifail}}=2$
ncap7 < 8${\mathbf{ncap7}}<8$, i.e., the number of interior knots is negative.

## Accuracy

The computed value of s(x)$s\left(x\right)$ has negligible error in most practical situations. Specifically, this value has an absolute error bounded in modulus by 18 × cmax × machine precision, where cmax${c}_{\mathrm{max}}$ is the largest in modulus of cj,cj + 1,cj + 2${c}_{j},{c}_{j+1},{c}_{j+2}$ and cj + 3${c}_{j+3}$, and j$j$ is an integer such that λj + 3xλj + 4${\lambda }_{j+3}\le x\le {\lambda }_{j+4}$. If cj,cj + 1,cj + 2${c}_{j},{c}_{j+1},{c}_{j+2}$ and cj + 3${c}_{j+3}$ are all of the same sign, then the computed value of s(x)$s\left(x\right)$ has a relative error not exceeding 20 × machine precision in modulus. For further details see Cox (1978).

The time taken is approximately c × (1 + 0.1 × log(n + 7))${\mathbf{c}}×\left(1+0.1×\mathrm{log}\left(\stackrel{-}{n}+7\right)\right)$ 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 [Parameters], since to do this would result in a computation time approximately linear in n + 7$\stackrel{-}{n}+7$ instead of log(n + 7)$\mathrm{log}\left(\stackrel{-}{n}+7\right)$. All the conditions are tested in nag_fit_1dspline_knots (e02ba), however.

## Example

```function nag_fit_1dspline_eval_example
lamda = [1;
1;
1;
1;
3;
6;
8;
9;
9;
9;
9];
c = [1;
2;
4;
7;
6;
4;
3;
0;
0;
0;
0];
x = 1;
[s, ifail] = nag_fit_1dspline_eval(lamda, c, x)
```
```

s =

1

ifail =

0

```
```function e02bb_example
lamda = [1;
1;
1;
1;
3;
6;
8;
9;
9;
9;
9];
c = [1;
2;
4;
7;
6;
4;
3;
0;
0;
0;
0];
x = 1;
[s, ifail] = e02bb(lamda, c, x)
```
```

s =

1

ifail =

0

```