NAG C Library Function Document

nag_2d_spline_eval_rect (e02dfc) calculates values of a bicubic spline from its B-spline representation. The spline is evaluated at all points on a rectangular grid.

2

Specification

#include <nag.h>

#include <nage02.h>

void	nag_2d_spline_eval_rect (Integer mx, Integer my, const double x[], const double y[], double ff[], Nag_2dSpline spline, NagError fail)

3

Description

nag_2d_spline_eval_rect (e02dfc) calculates values of the bicubic spline

s (x, y)

on a rectangular grid of points in the

x

y

plane, from its augmented knot sets

\{λ\}

and

\{μ\}

and from the coefficients

c_{i j}

, for

i = 1, 2, \dots, spline \to nx - 4

and

j = 1, 2, \dots, spline \to ny - 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}

The points in the grid are defined by coordinates

x_{q}

, for

q = 1, 2, \dots, m_{x}

, along the

x

axis, and coordinates

y_{r}

, for

r = 1, 2, \dots, m_{y}

along the

y

axis.

This function may be used to calculate values of a bicubic spline given in the form produced by nag_2d_spline_interpolant (e01dac), nag_2d_spline_fit_grid (e02dcc) and nag_2d_spline_fit_scat (e02ddc). 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: $mx$ – IntegerInput

2: $my$ – IntegerInput

On entry: mx and my must specify

m_{x}

and

m_{y}

respectively, the number of points along the

x

and

y

axes that define the rectangular grid.

Constraint:

mx \geq 1

and

my \geq 1

3: $x [mx]$ – const doubleInput

4: $y [my]$ – const doubleInput

On entry: x and y must contain

x_{q}

, for

q = 1, 2, \dots, m_{x}

, and

y_{r}

, for

r = 1, 2, \dots, m_{y}

, respectively. These are the

x

and

y

coordinates that define the rectangular grid of points at which values of the spline are required.

Constraint: x and y must satisfy

spline \to lamda [3] \leq x [q - 1] < x [q] \leq spline \to lamda [spline \to nx - 4]

, for

q = 1, 2, \dots, m_{x} - 1

, and

spline \to mu [3] \leq y [r - 1] < y [r] \leq spline \to mu [spline \to ny - 4]

, for

r = 1, 2, \dots, m_{y} - 1

.
The spline representation is not valid outside these intervals.

5: $ff [mx \times my]$ – doubleOutput

On exit:

ff [my \times (q - 1) + r - 1]

contains the value of the spline at the point

(x_{q}, y_{r})

, for

q = 1, 2, \dots, m_{x}

and

r = 1, 2, \dots, m_{y}

6: $spline$ – Nag_2dSpline *

Pointer to structure of type Nag_2dSpline with the following members:

nx – IntegerInput: On entry: $nx$ must specify the total number of knots associated with the variable $x$ . It is such that $nx - 8$ is the number of interior knots.

Constraint: $nx \geq 8$ .

lamda – double *Input: On entry: a pointer to which memory of size $nx$ must be allocated. $lamda$ must contain the complete sets of knots $\{λ\}$ associated with the $x$ variable.

Constraint: the knots must be in nondecreasing order, with $lamda [nx - 4] > lamda [3]$ .

ny – IntegerInput: On entry: $ny$ must specify the total number of knots associated with the variable $y$ .
It is such that $ny - 8$ is the number of interior knots.

Constraint: $ny \geq 8$ .

mu – double *Input: On entry: a pointer to which memory of size $ny$ must be allocated. $mu$ must contain the complete sets of knots $\{μ\}$ associated with the $y$ variable.

Constraint: the knots must be in nondecreasing order, with $mu [ny - 4] > mu [3]$ .

c – double *Input: On entry: a pointer to which memory of size $(nx - 4) \times (ny - 4)$ must be allocated. $c [(ny - 4) \times (i - 1) + j - 1]$ must contain the coefficient $c_{i j}$ described in Section 3, for $i = 1, 2, \dots, nx - 4$ and $j = 1, 2, \dots, ny - 4$ .

In normal usage, the call to nag_2d_spline_eval_rect (e02dfc) follows a call to nag_2d_spline_interpolant (e01dac), nag_2d_spline_fit_grid (e02dcc) or nag_2d_spline_fit_scat (e02ddc), in which case, members of the structure spline will have been set up correctly for input to nag_2d_spline_eval_rect (e02dfc).

7: $fail$ – 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_ALLOC_FAIL: Dynamic memory allocation failed.
NE_END_KNOTS_CONS: On entry, the end knots must satisfy $〈value〉$ : $〈value〉 = 〈value〉$ , $〈value〉 = 〈value〉$ .
NE_INT_ARG_LT: On entry, $mx = 〈value〉$ .
Constraint: $mx \geq 1$ .

On entry, $my = 〈value〉$ .
Constraint: $my \geq 1$ .

On entry, $spline \to nx$ must not be less than 8: $spline \to nx = 〈value〉$ .

On entry, $spline \to ny$ must not be less than 8: $spline \to ny = 〈value〉$ .
NE_KNOTS_COORD_CONS: On entry, the end knots and coordinates must satisfy $spline \to lamda [3] \leq x [0]$ and $x [mx - 1] \leq spline \to lamda [spline \to nx - 4]$ . $spline \to lamda [3] = 〈value〉$ , $x [0] = 〈value〉$ , $x [〈value〉] = 〈value〉$ , $spline \to lamda [〈value〉] = 〈value〉$ .

On entry, the end knots and coordinates must satisfy $spline \to mu [3] \leq y [0]$ and $y [my - 1] \leq spline \to mu [spline \to ny - 4]$ . $spline \to mu [3] = 〈value〉$ , $y [0] = 〈value〉$ , $y [〈value〉] = 〈value〉$ , $spline \to mu [〈value〉] = 〈value〉$ .
NE_NOT_INCREASING: The sequence $spline \to lamda$ is not increasing: $spline \to lamda [〈value〉] = 〈value〉$ , $spline \to lamda [〈value〉] = 〈value〉$ .
The sequence $spline \to mu$ is not increasing: $spline \to mu [〈value〉] = 〈value〉$ , $spline \to mu [〈value〉] = 〈value〉$ .
NE_NOT_STRICTLY_INCREASING: The sequence x is not strictly increasing: $x [〈value〉] = 〈value〉$ , $x [〈value〉] = 〈value〉$ .
The sequence y is not strictly increasing: $y [〈value〉] = 〈value〉$ , $y [〈value〉] = 〈value〉$ .

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

nag_2d_spline_eval_rect (e02dfc) is not threaded in any implementation.

9

Further Comments

Computation time is approximately proportional to

m_{x} m_{y} + 4 (m_{x} + m_{y})

10

Example

This program reads in knot sets

spline \to lamda [0], \dots, spline \to lamda [spline \to nx - 1] ​ and ​ spline \to mu [0]

\dots

spline \to mu [spline \to ny - 1]

, and a set of bicubic spline coefficients

c_{i j}

. Following these are values for

m_{x}

and the

x

coordinates

x_{q}

, for

q = 1, 2, \dots, m_{x}

, and values for

m_{y}

and the

y

coordinates

y_{r}

, for

r = 1, 2, \dots, m_{y}

, defining the grid of points on which the spline is to be evaluated.

NAG C Library Function Document

nag_2d_spline_eval_rect (e02dfc)

▸▿ 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

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