NAG Library Function Document

nag_1d_spline_intg (e02bdc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{spline}$ – Nag_Spline *

Pointer to structure of type Nag_Spline with the following members:

n – IntegerInput

On entry: $\stackrel{-}{n}+7$, where $\stackrel{-}{n}$ is the number of intervals of the spline (which is one greater than the number of interior knots, i.e., the knots strictly within the range $a$ to $b$) over which the spline is defined.

Constraint: $\mathbf{spline}\mathbf{\to }\mathbf{n}\ge 8$.

lamda – double *Input

On entry: a pointer to which memory of size $\mathbf{spline}\mathbf{\to }\mathbf{n}$ must be allocated. $\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[j-1\right]$ must be set to the value of the $j$th member of the complete set of knots, ${\lambda }_j$ for $j=1,2,\dots ,\stackrel{-}{n}+7$.

Constraint: the ${\lambda }_j$ must be in nondecreasing order with $\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[\mathbf{spline}\mathbf{\to }\mathbf{n}-4\right]>\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[3\right]$ and satisfy

$\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[0\right]=\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[1\right]=\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[2\right]=\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[3\right]$ 

and

$\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[\mathbf{spline}\mathbf{\to }\mathbf{n}-4\right]=\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[\mathbf{spline}\mathbf{\to }\mathbf{n}-3\right]=\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[\mathbf{spline}\mathbf{\to }\mathbf{n}-2\right]=\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[\mathbf{spline}\mathbf{\to }\mathbf{n}-1\right]$ 

c – double *Input

On entry: a pointer to which memory of size $\mathbf{spline}\mathbf{\to }\mathbf{n}-4$ must be allocated. $\mathbf{spline}\mathbf{\to }\mathbf{c}$ holds the coefficient $c_i$ of the B-spline $N_i\left(x\right)$, for $i=1,2,\dots ,\stackrel{-}{n}+3$.

2:    $\mathbf{integral}$ – double *Output

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

3:    $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_INT_ARG_LT

On entry, $\mathbf{spline}\mathbf{\to }\mathbf{n}$ must not be less than 8: $\mathbf{spline}\mathbf{\to }\mathbf{n}=〈\mathit{\text{value}}〉$.

NE_KNOTS_CONS

On entry, the knots must satisfy the following constraints:
$\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[\mathbf{spline}\mathbf{\to }\mathbf{n}-4\right]>\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[3\right]$, $\mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[\mathit{j}\right]\ge \mathbf{spline}\mathbf{\to }\mathbf{lamda}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,\mathbf{spline}\mathbf{\to }\mathbf{n}-1$, with equality in the cases $j=1,2,3$, $\mathbf{spline}\mathbf{\to }\mathbf{n}-3$, $\mathbf{spline}\mathbf{\to }\mathbf{n}-2$ and $\mathbf{spline}\mathbf{\to }\mathbf{n}-1$.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (e02bdce.c)

10.2

Program Data

Program Data (e02bdce.d)

10.3

Program Results

Program Results (e02bdce.r)