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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_fit_2dspline_evalm (e02df)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


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.


[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)


nag_fit_2dspline_evalm (e02df) calculates values of the bicubic spline sx,y on a rectangular grid of points in the x-y plane, from its augmented knot sets λ and μ and from the coefficients cij, for i=1,2,,px-4 and j=1,2,,py-4, in its B-spline representation
sx,y = ij cij Mix Njy .  
Here Mix and Njy denote normalized cubic B-splines, the former defined on the knots λi to λi+4 and the latter on the knots μj to μj+4.
The points in the grid are defined by coordinates xq, for q=1,2,,mx, along the x axis, and coordinates yr, for r=1,2,,my, 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).


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


Compulsory Input Parameters

1:     xmx – double array
2:     ymy – double array
x and y must contain xq, for q=1,2,,mx, and yr, for r=1,2,,my, 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
lamda4 xq < xq+1 lamdapx-3 ,   q=1,2,,mx-1  
mu4 yr < yr+1 mupy-3 ,   r= 1,2,,my- 1 .  
The spline representation is not valid outside these intervals.
3:     lamdapx – double array
4:     mupy – 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 lamdapx-3>lamda4 and mupy-3>mu4.
5:     cpx-4×py-4 – double array
c py-4 × i-1 +j  must contain the coefficient cij described in Description, for i=1,2,,px-4 and j=1,2,,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 mx and my respectively, the number of points along the x and y axis that define the rectangular grid.
Constraint: mx1 and my1.
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: px8 and py8.

Output Parameters

1:     ffmx×my – double array
ffmy×q-1+r contains the value of the spline at the point xq,yr, for q=1,2,,mx and r=1,2,,my.
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:
On entry,mx<1,
On entry,lwrk is too small,
orliwrk is too small.
On entry, the knots in array lamda, or those in array mu, are not in nondecreasing order, or lamdapx-3lamda4, or mupy-3mu4.
On entry, the restriction lamda4x1<<xmxlamdapx-3, or the restriction mu4y1<<ymymupy-3, is violated.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


The method used to evaluate the B-splines is numerically stable, in the sense that each computed value of sxr,yr 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 mxmy+4mx+my.


This example reads in knot sets lamda1,,lamdapx and mu1,,mupy, and a set of bicubic spline coefficients cij. Following these are values for mx and the x coordinates xq, for q=1,2,,mx, and values for my and the y coordinates yr, for r=1,2,,my, defining the grid of points on which the spline is to be evaluated.
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

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

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