nag_fit_2dspline_evalm (e02df) calculates values of a bicubic spline from its B-spline representation. The spline is evaluated at all points on a rectangular grid.

Syntax

[ff, ifail] = e02df(x, y, lamda, mu, c, 'mx', mx, 'my', my, 'px', px, 'py', py)

[ff, ifail] = nag_fit_2dspline_evalm(x, y, lamda, mu, c, 'mx', mx, 'my', my, 'px', px, 'py', py)

Description

nag_fit_2dspline_evalm (e02df) 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, 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}

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_interp_2d_spline_grid (e01da), nag_fit_2dspline_panel (e02da), nag_fit_2dspline_grid (e02dc) and nag_fit_2dspline_sctr (e02dd). It is derived from the function B2VRE in Anthony et al. (1982).

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

Parameters

Compulsory Input Parameters

1: $x (mx)$ – double array

2: $y (my)$ – double array

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

lamda (4) \leq x (q) < x (q + 1) \leq lamda (px - 3), q = 1, 2, \dots, m_{x} - 1

and

mu (4) \leq y (r) < y (r + 1) \leq mu (py - 3), r = 1, 2, \dots, m_{y} - 1 .

The spline representation is not valid outside these intervals.

3: $lamda (px)$ – double array

4: $mu (py)$ – double array

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)

5: $c ((px - 4) \times (py - 4))$ – double array

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

must contain the coefficient

c_{i j}

described in Description, for

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

and

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

Optional Input Parameters

1: $mx$ – int64int32nag_int scalar
2: $my$ – int64int32nag_int scalar: Default: the dimension of the arrays x, y. (An error is raised if these dimensions are not equal.)
mx and my must specify $m_{x}$ and $m_{y}$ respectively, the number of points along the $x$ and $y$ axis that define the rectangular grid.

Constraint: $mx \geq 1$ and $my \geq 1$ .
3: $px$ – int64int32nag_int scalar
4: $py$ – int64int32nag_int scalar: Default: For px, the dimension of the array lamda. For py, the dimension of the array mu.
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 $py \geq 8$ .

Output Parameters

1: $ff (mx \times my)$ – double array: $ff (my \times (q - 1) + r)$ contains the value of the spline at the point $(x_{q}, y_{r})$ , for $q = 1, 2, \dots, m_{x}$ and $r = 1, 2, \dots, m_{y}$ .
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$mx < 1$ ,
or	$my < 1$ ,
or	$py < 8$ ,
or	$px < 8$ .

$ifail = 2$

On entry,	lwrk is too small,
or	liwrk is too small.

$ifail = 3$: On entry, the knots in array lamda, or those in array mu, are not in nondecreasing order, or $lamda (px - 3) \leq lamda (4)$ , or $mu (py - 3) \leq mu (4)$ .

$ifail = 4$: On entry, the restriction $lamda (4) \leq x (1) < \dots < x (mx) \leq lamda (px - 3)$ , or the restriction $mu (4) \leq y (1) < \dots < y (my) \leq mu (py - 3)$ , is violated.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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.

Further Comments

Computation time is approximately proportional to

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

Example

This example 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 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.

Open in the MATLAB editor: e02df_example

function e02df_example


fprintf('e02df example results\n\n');

% knots
lamda = [1   1   1   1   1.3  1.5  1.6  2   2   2   2];
mu    = [0   0   0   0   0.4  0.7  1    1   1   1];
% Coefficients
c     = [1       1.1333  1.3667  1.7   1.9     2  ...
         1.2     1.3333  1.5667  1.9     2.1   2.2 ... 
         1.5833  1.7167  1.95    2.2833  2.4833  2.5833 ...
         2.1433  2.2767  2.51    2.8433  3.0433  3.1433 ...
         2.8667  3       3.2333  3.5667  3.7667  3.8667 ...
         3.4667  3.6     3.8333  4.1667  4.3667  4.4667 ...
         4       4.1333  4.3667  4.7     4.9     5];

%Evaluation points 
x = [1.0  1.1  1.3  1.4  1.5  1.7  2.0];
y = [0 : 0.2 : 1];

% Spline evaluation
[ff, ifail] = e02df( ...
                     x, y, lamda, mu, c);

fprintf(' Spline evaluated on x-y grid (x across, y down):\n');
ff = reshape(ff,[6,7]);
fprintf('%5s%9.1f%9.1f%9.1f%9.1f%9.1f%9.1f%9.1f\n',' ',x);
for i = 1:6
  fprintf('%5.1f%9.3f%9.3f%9.3f%9.3f%9.3f%9.3f%9.3f\n',y(i),ff(i,:));
end

e02df example results

 Spline evaluated on x-y grid (x across, y down):
           1.0      1.1      1.3      1.4      1.5      1.7      2.0
  0.0    1.000    1.210    1.690    1.960    2.250    2.890    4.000
  0.2    1.200    1.410    1.890    2.160    2.450    3.090    4.200
  0.4    1.400    1.610    2.090    2.360    2.650    3.290    4.400
  0.6    1.600    1.810    2.290    2.560    2.850    3.490    4.600
  0.8    1.800    2.010    2.490    2.760    3.050    3.690    4.800
  1.0    2.000    2.210    2.690    2.960    3.250    3.890    5.000

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox