NAG Library Routine Document

e02bff  (dim1_spline_deriv_vector)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{start}$ – IntegerInput

On entry: indicates the completion state of the first phase of the algorithm.

$\mathbf{start}=0$

The enclosing interval numbers ${\mathit{ixloc}}_j$ for the abscissae $x_j$ contained in x have not been determined, and you wish to use the sorted mode of vectorization.

$\mathbf{start}=1$

The enclosing interval numbers ${\mathit{ixloc}}_j$ have been determined and are provided in ixloc, however the required permutation and interval related information has not been determined and you wish to use the sorted mode of vectorization.

$\mathbf{start}=2$

You wish to use the sorted mode of vectorization, and the entire first phase has been completed, with the enclosing interval numbers supplied in ixloc, and the required permutation and interval related information provided in iwrk (from a previous call to e02bff).

$\mathbf{start}=10$

The enclosing interval numbers ${\mathit{ixloc}}_j$ for the abscissae $x_j$ contained in x have not been determined, and you wish to use the unsorted mode of vectorization.

$\mathbf{start}=11$

The enclosing interval numbers ${\mathit{ixloc}}_j$ for the abscissae $x_j$ contained in x have been supplied in ixloc, and you wish to use the unsorted mode of vectorization.

Constraint: $\mathbf{start}=0$, $1$, $2$, $10$ or $11$.

Additional: $\mathbf{start}=0$ or $10$ should be used unless you are sure that the knot set is unchanged between calls.

2:     $\mathbf{ncap7}$ – 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 ${\lambda }_4$ to ${\lambda }_{\stackrel{-}{n}+4}$ over which the spline is defined). Note that if e02bef was used to generate the knots and spline coefficients then ncap7 should contain the same value as returned in n by e02bef.

Constraint: $\mathbf{ncap7}\ge 8$.

3:     $\mathbf{lamda}\left(\mathbf{ncap7}\right)$ – 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 $\mathbf{lamda}\left(\mathbf{ncap7}-3\right)>\mathbf{lamda}\left(4\right)$.

4:     $\mathbf{c}\left(\mathbf{ncap7}\right)$ – 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.

5:     $\mathbf{deriv}$ – IntegerInput

On entry: determines the maximum order of derivatives required, $D_{\mathrm{ord}}$.

If $\mathbf{deriv}<0$ left derivatives are calculated, otherwise right derivatives are calculated. For abscissae satisfying $x_j={\lambda }_4$ or $x_j={\lambda }_{\stackrel{-}{n}+4}$ only right-handed or left-handed computation will be used respectively. For abscissae which do not coincide exactly with a knot, the handedness of the computation is immaterial.

$\mathbf{deriv}=0$

No derivatives required.

$\mathbf{deriv}=\pm 1$

Only $s\left(x\right)$ and its first derivative are required.

$\mathbf{deriv}=\pm 2$

Only $s\left(x\right)$ and its first and second derivatives are required.

$\mathbf{deriv}=\pm 3$

$s\left(x\right)$ and its first, second and third derivatives are required.

Note: if $\left|\mathbf{deriv}\right|$ is greater than $3$ only the derivatives up to and including $3$ will be returned..

6:     $\mathbf{xord}$ – IntegerInput

On entry: indicates whether x is supplied in a sufficiently ordered manner. If x is sufficiently ordered e02bff will complete faster.

$\mathbf{xord}\ne 0$

The abscissae in x are ordered at least by ascending interval, in that any two abscissae contained in the same interval are only separated by abscissae in the same interval, and the intervals are arranged in ascending order. For example, $x_j<x_{\mathit{j}+1}$, for $\mathit{j}=1,2,\dots ,\mathbf{nx}-1$.

$\mathbf{xord}=0$

The abscissae in x are not sufficiently ordered.

7:     $\mathbf{x}\left(\mathbf{nx}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the abscissae $x_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n_x$. If $\mathbf{start}=0$ or $10$ then evaluations will only be performed for these $x_j$ satisfying ${\lambda }_4\le x_j\le {\lambda }_{\stackrel{-}{n}+4}$. Otherwise evaluation will be performed unless the corresponding element of ixloc contains an invalid interval number. Please note that if the $\mathbf{ixloc}\left(j\right)$ is a valid interval number then no check is made that $\mathbf{x}\left(j\right)$ actually lies in that interval.

Constraint: at least one abscissa must fall between $\mathbf{lamda}\left(4\right)$ and $\mathbf{lamda}\left(\mathbf{ncap7}-3\right)$.

8:     $\mathbf{ixloc}\left(\mathbf{nx}\right)$ – Integer arrayInput/Output

On entry: if $\mathbf{start}=1$, $2$ or $11$, if you wish $x_j$ to be evaluated, $\mathbf{ixloc}\left(j\right)$ must be the enclosing interval number ${\mathit{ixloc}}_j$ of the abscissae $x_j$ (see (1)). If you do not wish $x_j$ to be evaluated, you may set the interval number to be either less than $4$ or greater than $\stackrel{-}{n}+4$.

Otherwise, ixloc need not be set.

On exit: if $\mathbf{start}=1$, $2$ or $11$, ixloc is unchanged on exit.

Otherwise, $\mathbf{ixloc}\left(\mathit{j}\right)$, contains the enclosing interval number ${\mathit{ixloc}}_{\mathit{j}}$, for the abscissa supplied in $\mathbf{x}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,n_x$. Evaluations will only be performed for abscissae $x_j$ satisfying ${\lambda }_4\le x_j\le {\lambda }_{\stackrel{-}{n}+4}$. If evaluation is not performed $\mathbf{ixloc}\left(j\right)$ is set to $0$ if $x_j<{\lambda }_4$ or $\stackrel{-}{n}+7$ if $x_j>{\lambda }_{\stackrel{-}{n}+4}$.

Constraint: if $\mathbf{start}=1$, $2$ or $11$, at least one element of ixloc must be between $4$ and $\mathbf{ncap7}-3$.

9:     $\mathbf{nx}$ – IntegerInput

On entry: $n_x$, the total number of abscissae contained in x, including any that will not be evaluated.

Constraint: $\mathbf{nx}\ge 1$.

10:   $\mathbf{s}\left(\mathbf{lds},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array s must be at least $D_{\mathrm{ord}}+1$, see deriv for the definition of $D_{\mathrm{ord}}$.

On exit: if $x_j$ is valid, $\mathbf{s}\left(\mathit{j},\mathit{d}\right)$ will contain the ($\mathit{d}-1$)th derivative of $s\left(x\right)$, for $\mathit{d}=1,2,\dots ,D_{\mathrm{ord}}+1$ and $\mathit{j}=1,2,\dots ,n_x$. In particular, $\mathbf{s}\left(j,1\right)$ will contain the approximation of $s\left(x_j\right)$ for all legal values in x.

11:   $\mathbf{lds}$ – IntegerInput

On entry: the first dimension of the array s as declared in the (sub)program from which e02bff is called.

Constraint: $\mathbf{lds}\ge \mathbf{nx}$, regardless of the acceptability of the elements of x.

12:   $\mathbf{iwrk}\left(\mathbf{liwrk}\right)$ – Integer arrayInput/Output

On entry: if $\mathbf{start}=2$, iwrk must be unchanged from a previous call to e02bff with $\mathbf{start}=0$ or $1$.

Otherwise, iwrk need not be set.

On exit: if $\mathbf{start}=10$ or $11$, iwrk is unchanged on exit.

Otherwise, iwrk contains the required permutation of elements of x, if any, and information related to the division of the abscissae $x_j$ between the intervals derived from lamda.

13:   $\mathbf{liwrk}$ – IntegerInput

On entry: the dimension of the array iwrk as declared in the (sub)program from which e02bff is called.

Constraint: if $\mathbf{start}=0$, $1$ or $2$, $\mathbf{liwrk}\ge 3+3\times \mathbf{nx}$.

14:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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, because for this routine the values of the output arguments may be useful even if $\mathbf{ifail}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ 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

$\mathbf{ifail}=1$

On entry, at least one element of x has an enclosing interval number in ixloc outside the set allowed by the provided spline. The spline has been evaluated for all x with enclosing interval numbers inside the allowable set.
$〈\mathit{\text{value}}〉$ entries of x were indexed below the lower bound $〈\mathit{\text{value}}〉$.
$〈\mathit{\text{value}}〉$ entries of x were indexed above the upper bound $〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=2$

On entry, all elements of x had enclosing interval numbers in ixloc outside the domain allowed by the provided spline.
$〈\mathit{\text{value}}〉$ entries of x were indexed below the lower bound $〈\mathit{\text{value}}〉$.
$〈\mathit{\text{value}}〉$ entries of x were indexed above the upper bound $〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=11$

On entry, $\mathbf{start}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{start}=0$, $1$, $2$, $10$ or $11$.

$\mathbf{ifail}=12$

On entry, $\mathbf{start}=2$ and nx is not consistent with the previous call to e02bff.
On entry, $\mathbf{nx}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nx}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=21$

On entry, $\mathbf{ncap7}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ncap7}\ge 8$.

$\mathbf{ifail}=31$

On entry, $\mathbf{lamda}\left(4\right)=〈\mathit{\text{value}}〉$, $\mathbf{ncap7}=〈\mathit{\text{value}}〉$ and $\mathbf{lamda}\left(\mathbf{ncap7}-3\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lamda}\left(4\right)<\mathbf{lamda}\left(\mathbf{ncap7}-3\right)$.

$\mathbf{ifail}=91$

On entry, $\mathbf{nx}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nx}\ge 1$.

$\mathbf{ifail}=111$

On entry, $\mathbf{lds}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lds}\ge \mathbf{nx}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=131$

On entry, $\mathbf{liwrk}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{liwrk}\ge 3\times \mathbf{nx}+3=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (e02bffe.f90)

10.2

Program Data

Program Data (e02bffe.d)

10.3

Program Results

Program Results (e02bffe.r)