NAG Library Routine Document

E02BDF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     NCAP7 – 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: $\mathbf{NCAP7}\ge 8$.

2:     LAMDA(NCAP7) – REAL (KIND=nag_wp) arrayInput

On entry: $\mathbf{LAMDA}\left(\mathit{j}\right)$ 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 $\mathbf{LAMDA}\left(j\right)$ must be in nondecreasing order with $$\begin{gathered} \mathbf{LAMDA}\left(\mathbf{NCAP7}-3\right)> \\ \mathbf{LAMDA}\left(4\right) \end{gathered}$$ and satisfy $\mathbf{LAMDA}\left(1\right)=\mathbf{LAMDA}\left(2\right)=\mathbf{LAMDA}\left(3\right)=\mathbf{LAMDA}\left(4\right)$ and $$\begin{gathered} \mathbf{LAMDA}\left(\mathbf{NCAP7}-3\right)=\mathbf{LAMDA}\left(\mathbf{NCAP7}-2\right)= \\ \mathbf{LAMDA}\left(\mathbf{NCAP7}-1\right)=\mathbf{LAMDA}\left(\mathbf{NCAP7}\right) \end{gathered}$$.

3:     C(NCAP7) – REAL (KIND=nag_wp) arrayInput

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.

4:     DINT – REAL (KIND=nag_wp)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 $b={\lambda }_{\stackrel{-}{n}+4}$.

5:     IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

$\mathbf{NCAP7}<8$, i.e., the number of intervals is not positive.

$\mathbf{IFAIL}=2$

At least one of the following restrictions on the knots is violated:

• $\mathbf{LAMDA}\left(\mathbf{NCAP7}-3\right)>\mathbf{LAMDA}\left(4\right)$,
• $\mathbf{LAMDA}\left(j\right)\ge \mathbf{LAMDA}\left(j-1\right)$,

for $j=2,3,\dots ,\mathbf{NCAP7}$, with equality in the cases $j=2,3,4,\mathbf{NCAP7}-2,\mathbf{NCAP7}-1$, and NCAP7.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (e02bdfe.f90)

9.2  Program Data

Program Data (e02bdfe.d)

9.3  Program Results

Program Results (e02bdfe.r)