e02bb evaluates a cubic spline from its B-spline representation.

# Syntax

C#
public static void e02bb(
int ncap7,
double[] lamda,
double[] c,
double x,
out double s,
out int ifail
)
Visual Basic
Public Shared Sub e02bb ( _
ncap7 As Integer, _
lamda As Double(), _
c As Double(), _
x As Double, _
<OutAttribute> ByRef s As Double, _
<OutAttribute> ByRef ifail As Integer _
)
Visual C++
public:
static void e02bb(
int ncap7,
array<double>^ lamda,
array<double>^ c,
double x,
[OutAttribute] double% s,
[OutAttribute] int% ifail
)
F#
static member e02bb :
ncap7 : int *
lamda : float[] *
c : float[] *
x : float *
s : float byref *
ifail : int byref -> unit

#### Parameters

ncap7
Type: System..::..Int32
On entry: $\stackrel{-}{n}+7$, 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.
Constraint: ${\mathbf{ncap7}}\ge 8$.
lamda
Type: array<System..::..Double>[]()[][]
An array of size [ncap7]
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 $\mathit{j}=1,2,\dots ,\stackrel{-}{n}+7$.
Constraint: the ${\mathbf{lamda}}\left[j-1\right]$ must be in nondecreasing order with ${\mathbf{lamda}}\left[{\mathbf{ncap7}}-4\right]>\phantom{\rule{0ex}{0ex}}{\mathbf{lamda}}\left[3\right]$.
c
Type: array<System..::..Double>[]()[][]
An array of size [ncap7]
On entry: the coefficient ${c}_{\mathit{i}}$ of the B-spline ${N}_{\mathit{i}}\left(x\right)$, for $\mathit{i}=1,2,\dots ,\stackrel{-}{n}+3$. The remaining elements of the array are not referenced.
x
Type: System..::..Double
On entry: the argument $x$ at which the cubic spline is to be evaluated.
Constraint: ${\mathbf{lamda}}\left[3\right]\le {\mathbf{x}}\le {\mathbf{lamda}}\left[{\mathbf{ncap7}}-4\right]$.
s
Type: System..::..Double%
On exit: the value of the spline, $s\left(x\right)$.
ifail
Type: System..::..Int32%
On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

# Description

e02bb 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 not in this release)) and from the coefficients ${c}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$in its B-spline representation
 $sx=∑i=1qciNix.$
Here $q=\stackrel{-}{n}+3$, where $\stackrel{-}{n}$ 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 ${\lambda }_{4}\le x\le {\lambda }_{\stackrel{-}{n}+4}$.
It is assumed that ${\lambda }_{\mathit{j}}\ge {\lambda }_{\mathit{j}-1}$, for $\mathit{j}=2,3,\dots ,\stackrel{-}{n}+7$, and ${\lambda }_{\stackrel{-}{n}+4}>{\lambda }_{4}$.
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 e02bb will be the evaluation of the cubic spline approximations produced by (E02BAF not in this release). A generalization of e02bb which also forms the derivative of $s\left(x\right)$ is (E02BCF not in this release). (E02BCF not in this release) takes about $50%$ longer than 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

# Error Indicators and Warnings

Errors or warnings detected by the method:
${\mathbf{ifail}}=1$
The parameter x does not satisfy ${\mathbf{lamda}}\left[3\right]\le {\mathbf{x}}\le {\mathbf{lamda}}\left[{\mathbf{ncap7}}-4\right]$.
In this case the value of s is set arbitrarily to zero.
${\mathbf{ifail}}=2$
${\mathbf{ncap7}}<8$, i.e., the number of interior knots is negative.
${\mathbf{ifail}}=-9000$
An error occured, see message report.
${\mathbf{ifail}}=-8000$
Negative dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$

# Accuracy

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).

# Parallelism and Performance

None.

The time taken is approximately ${\mathbf{c}}×\left(1+0.1×\mathrm{log}\left(\stackrel{-}{n}+7\right)\right)$ seconds, where c is a machine-dependent constant.
Note:  the method does not test all the conditions on the knots given in the description of lamda in [Parameters], since to do this would result in a computation time approximately linear in $\stackrel{-}{n}+7$ instead of $\mathrm{log}\left(\stackrel{-}{n}+7\right)$. All the conditions are tested in (E02BAF not in this release), however.

# Example

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 $\stackrel{-}{n}$ intervals, in its normalized cubic B-spline form, to be evaluated at $m$ equally-spaced points in the interval ${\mathbf{lamda}}\left[3\right]\le x\le {\mathbf{lamda}}\left[\stackrel{-}{n}+3\right]$. The program is self-starting in that any number of datasets may be supplied.

Example program (C#): e02bbe.cs

Example program data: e02bbe.d

Example program results: e02bbe.r