Integer, Intent (In)	::	m, px, py
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	iwrk(py-4)
Real (Kind=nag_wp), Intent (In)	::	x(m), y(m), lamda(px), mu(py), c((px-4)*(py-4))
Real (Kind=nag_wp), Intent (Out)	::	ff(m), wrk(py-4)

C Header Interface

#include <nag.h>

void	e02def_ (const Integer m, const Integer px, const Integer py, const double x[], const double y[], const double lamda[], const double mu[], const double c[], double ff[], double wrk[], Integer iwrk[], Integer ifail)

The routine may be called by the names e02def or nagf_fit_dim2_spline_evalv.

3 Description

e02def calculates values of the bicubic spline

s (x, y)

at prescribed points

(x_{r}, y_{r})

, for

r = 1, 2, \dots, m

, from its augmented knot sets

{λ}

and

{μ}

and from the coefficients

c_{i j}

, for

i = 1, 2, \dots, px - 4

and

j = 1, 2, \dots, py - 4

, in its B-spline representation

s (x, y) = \sum_{i j} c_{i j} M_{i} (x) N_{j} (y) .

Here

M_{i} (x)

and

N_{j} (y)

denote normalized cubic B-splines, the former defined on the knots

λ_{i}

λ_{i + 4}

and the latter on the knots

μ_{j}

μ_{j + 4}

This routine may be used to calculate values of a bicubic spline given in the form produced by e01daf, e02daf, e02dcf and e02ddf. It is derived from the routine B2VRE in Anthony et al. (1982).

4 References

Anthony G T, Cox M G and Hayes J G (1982) DASL – Data Approximation Subroutine Library National Physical Laboratory

Cox M G (1978) The numerical evaluation of a spline from its B-spline representation J. Inst. Math. Appl. 21 135–143

5 Arguments

1: $m$ – Integer Input

On entry:

m

, the number of points at which values of the spline are required.

Constraint:

m \geq 1

2: $px$ – Integer Input

3: $py$ – Integer Input

On entry: px and py must specify the total number of knots associated with the variables

x

and

y

respectively. They are such that

px - 8

and

py - 8

are the corresponding numbers of interior knots.

Constraint:

px \geq 8

and

py \geq 8

4: $x (m)$ – Real (Kind=nag_wp) array Input

5: $y (m)$ – Real (Kind=nag_wp) array Input

On entry: x and y must contain

x_{r}

and

y_{r}

, for

r = 1, 2, \dots, m

, respectively. These are the coordinates of the points at which values of the spline are required. The order of the points is immaterial.

Constraint:

x

and

y

must satisfy

lamda (4) \leq x (r) \leq lamda (px - 3)

and

mu (4) \leq y (r) \leq mu (py - 3), r = 1, 2, \dots, m .

The spline representation is not valid outside these intervals

6: $lamda (px)$ – Real (Kind=nag_wp) array Input

7: $mu (py)$ – Real (Kind=nag_wp) array Input

On entry: lamda and mu must contain the complete sets of knots

{λ}

and

{μ}

associated with the

x

and

y

variables respectively.

Constraint: the knots in each set must be in nondecreasing order, with

lamda (px - 3) > lamda (4)

and

mu (py - 3) > mu (4)

8: $c ((px - 4) \times (py - 4))$ – Real (Kind=nag_wp) array Input

On entry:

c ((py - 4) \times (i - 1) + j)

must contain the coefficient

c_{i j}

described in Section 3, for

i = 1, 2, \dots, px - 4

and

j = 1, 2, \dots, py - 4

9: $ff (m)$ – Real (Kind=nag_wp) array Output

On exit:

ff (r)

contains the value of the spline at the point

(x_{r}, y_{r})

, for

r = 1, 2, \dots, m

10: $wrk (py - 4)$ – Real (Kind=nag_wp) array Workspace

11: $iwrk (py - 4)$ – Integer array Workspace

12: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 1$ .

On entry, $px = ⟨ value ⟩$ .
Constraint: $px \geq 8$ .

On entry, $py = ⟨ value ⟩$ .
Constraint: $py \geq 8$ .

$ifail = 2$: On entry, the knots in lamda are not in nondecreasing order.

On entry, the knots in mu are not in nondecreasing order.

$ifail = 3$: On entry, point $(x (K), y (K))$ lies outside the rectangle bounded by $lamda (4)$ , $lamda (px - 3)$ , $mu (4)$ , $mu (py - 3)$ : $K = ⟨ value ⟩$ , $x (K) = ⟨ value ⟩$ and $y (K) = ⟨ value ⟩$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The method used to evaluate the B-splines is numerically stable, in the sense that each computed value of

s (x_{r}, y_{r})

can be regarded as the value that would have been obtained in exact arithmetic from slightly perturbed B-spline coefficients. See Cox (1978) for details.

8 Parallelism and Performance

e02def is not threaded in any implementation.

9 Further Comments

Computation time is approximately proportional to the number of points,

m

, at which the evaluation is required.

10 Example

This program reads in knot sets

lamda (1), \dots, lamda (px)

and

mu (1), \dots, mu (py)

, and a set of bicubic spline coefficients

c_{i j}

. Following these are a value for

m

and the coordinates

(x_{r}, y_{r})

, for

r = 1, 2, \dots, m

, at which the spline is to be evaluated.

e02de: FL CL CPP AD

NAG FL Interfacee02def (dim2_​spline_​evalv)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
e02def (dim2_spline_evalv)