This function may be used to calculate values of a bicubic spline given in the form produced by e01dac, e02dcc and 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: $mx$ – Integer Input

2: $my$ – Integer Input

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 double Input

4: $y [my]$ – const double Input

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: $ff [mx \times my]$ – double Output

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 e02dfc follows a call to e01dac, e02dcc or e02ddc, in which case, members of the structure spline will have been set up correctly for input to e02dfc.

7: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

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

Background information to multithreading can be found in the Multithreading documentation.

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.

e02df: FL CL CPP AD PY MB

NAG CL Interfacee02dfc (dim2_​spline_​evalm)

▸▿ 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 CL Interface
e02dfc (dim2_spline_evalm)