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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_interp_2d_spline_grid (e01da)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_interp_2d_spline_grid (e01da) computes a bicubic spline interpolating surface through a set of data values, given on a rectangular grid in the x-y plane.


[px, py, lamda, mu, c, ifail] = e01da(x, y, f, 'mx', mx, 'my', my)
[px, py, lamda, mu, c, ifail] = nag_interp_2d_spline_grid(x, y, f, 'mx', mx, 'my', my)


nag_interp_2d_spline_grid (e01da) determines a bicubic spline interpolant to the set of data points x q , y r , f q , r , for q=1,2,,mx and r=1,2,,my. The spline is given in the B-spline representation
such that
where 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, and the cij are the spline coefficients. These knots, as well as the coefficients, are determined by the function, which is derived from the function B2IRE in Anthony et al. (1982). The method used is described in Outline of Method Used.
For further information on splines, see Hayes and Halliday (1974) for bicubic splines and de Boor (1972) for normalized B-splines.
Values and derivatives of the computed spline can subsequently be computed by calling nag_fit_2dspline_evalv (e02de), nag_fit_2dspline_evalm (e02df) or nag_fit_2dspline_derivm (e02dh) as described in Evaluation of Computed Spline.


Anthony G T, Cox M G and Hayes J G (1982) DASL – Data Approximation Subroutine Library National Physical Laboratory
Cox M G (1975) An algorithm for spline interpolation J. Inst. Math. Appl. 15 95–108
de Boor C (1972) On calculating with B-splines J. Approx. Theory 6 50–62
Hayes J G and Halliday J (1974) The least squares fitting of cubic spline surfaces to general data sets J. Inst. Math. Appl. 14 89–103


Compulsory Input Parameters

1:     xmx – double array
2:     ymy – double array
xq and yr must contain xq, for q=1,2,,mx, and yr, for r=1,2,,my, respectively.
  • xq<xq+1, for q=1,2,,mx-1;
  • yr<yr+1, for r=1,2,,my-1.
3:     fmx×my – double array
fmy×q-1+r must contain fq,r, for q=1,2,,mx and r=1,2,,my.

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: mx4 and my4.

Output Parameters

1:     px int64int32nag_int scalar
2:     py int64int32nag_int scalar
px and py contain mx+4 and my+4, the total number of knots of the computed spline with respect to the x and y variables, respectively.
3:     lamdamx+4 – double array
4:     mumy+4 – double array
lamda contains the complete set of knots λi associated with the x variable, i.e., the interior knots lamda5,lamda6,,lamdapx-4, as well as the additional knots
needed for the B-spline representation.
5:     cmx×my – double array
The coefficients of the spline interpolant. cmy×i-1+j contains the coefficient cij described in Description.
6:     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<4,
On entry, either the values in the x array or the values in the y array are not in increasing order if not already there.
A system of linear equations defining the B-spline coefficients was singular; the problem is too ill-conditioned to permit solution.
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 main sources of rounding errors are in steps 2, 3, 6 and 7 of the algorithm described in Outline of Method Used. It can be shown (see Cox (1975)) that the matrix Ax formed in step 2 has elements differing relatively from their true values by at most a small multiple of 3ε, where ε is the machine precision. Ax is ‘totally positive’, and a linear system with such a coefficient matrix can be solved quite safely by elimination without pivoting. Similar comments apply to steps 6 and 7. Thus the complete process is numerically stable.

Further Comments


The time taken by nag_interp_2d_spline_grid (e01da) is approximately proportional to mxmy.

Outline of Method Used

The process of computing the spline consists of the following steps:
1. choice of the interior x-knots λ5, λ6,,λmx as λi=xi-2, for i=5,6,,mx,
2. formation of the system
where Ax is a band matrix of order mx and bandwidth 4, containing in its qth row the values at xq of the B-splines in x, f is the mx by my rectangular matrix of values fq,r, and E denotes an mx by my rectangular matrix of intermediate coefficients,
3. use of Gaussian elimination to reduce this system to band triangular form,
4. solution of this triangular system for E,
5. choice of the interior y knots μ5, μ6,,μmy as μi=yi-2, for i=5,6,,my,
6. formation of the system
where Ay is the counterpart of Ax for the y variable, and C denotes the mx by my rectangular matrix of values of cij,
7. use of Gaussian elimination to reduce this system to band triangular form,
8. solution of this triangular system for CT and hence C.
For computational convenience, steps 2 and 3, and likewise steps 6 and 7, are combined so that the formation of Ax and Ay and the reductions to triangular form are carried out one row at a time.

Evaluation of Computed Spline

The values of the computed spline at the points xk,yk , for k=1,2,,m, may be obtained in the double array ff (see nag_fit_2dspline_evalv (e02de)), of length at least m, by the following call:
[ff, ifail] = e02de(x, y, lamda, mu, c);
where M=m and the coordinates xk, yk are stored in Xk, Yk. LAMDA, MU and C have the same values as lamda, mu and c output from nag_interp_2d_spline_grid (e01da). (See nag_fit_2dspline_evalv (e02de).)
To evaluate the computed spline on an mx by my rectangular grid of points in the x-y plane, which is defined by the x coordinates stored in Xj, for j=1,2,,mx, and the y coordinates stored in Yk, for k=1,2,,my, returning the results in the double array ff (see nag_fit_2dspline_evalm (e02df)) which is of length at least mx×my, the following call may be used:
[fg, ifail] = e02df(x, y, lamda, mu, c);
where MX=mx, MY=my. LAMDA, MU and C have the same values as lamda, mu and c output from nag_interp_2d_spline_grid (e01da).
The result of the spline evaluated at grid point j,k  is returned in element ( MY×j-1+k ) of the array FG.


This example reads in values of mx, xq, for q=1,2,,mx, my and yr, for r=1,2,,my, followed by values of the ordinates fq,r defined at the grid points xq,yr.
It then calls nag_interp_2d_spline_grid (e01da) to compute a bicubic spline interpolant of the data values, and prints the values of the knots and B-spline coefficients. Finally it evaluates the spline at a small sample of points on a rectangular grid.
function e01da_example

fprintf('e01da example results\n\n');

x = [1.0   1.10    1.30    1.50    1.60    1.80    2.0];
f = [1.0   1.21    1.69    2.25    2.56    3.24    4.0;   
     1.1   1.31    1.79    2.35    2.66    3.34    4.1; 
     1.4   1.61    2.09    2.65    2.96    3.64    4.4; 
     1.7   1.91    2.39    2.95    3.26    3.94    4.7; 
     1.9   2.11    2.59    3.15    3.46    4.14    4.9;
     2.0   2.21    2.69    3.25    3.56    4.24    5.0];
y = [0.0;

[px, py, lamda, mu, c, ifail] = e01da( ...
                                       x, y, f);

% Display the knot sets, LAMDA and MU.

 fprintf('\n               i   knot lamda(i)      j     knot mu(j)\n');
j = 4:min(px,py)-3;
fprintf('%16d%12.4f%11d%12.4f\n',[j' lamda(j) j' mu(j)]');
if (px>py);
  j = py-2:px-3;
  fprintf('%16d%12.4f\n',[j' lamda(j)]');
elseif (px<py);
  j = px-2:py-3;
  fprintf('%16d%12.4f\n',[j' mu(j)]')

% Display the spline coefficients.
c = reshape(c, size(f')); 
disp('The B-Spline coefficients:');

% Evaluate spline on regular 6-by-6 mesh
dx = (x(end)-x(1))/5;
dy = (y(end)-y(1))/5;
tx = [x(1):dx:x(end)];
ty = [y(1):dy:y(end)]';

[ff, ifail] = e02df( ...
                     tx, ty, lamda, mu, c);

% Display evaluations as matrix
ff = reshape(ff,[6,6]);

matrix = 'General';
diag = 'Non-unit';
format = 'F8.3';
title  = 'Spline evaluated on a regular mesh (x across, y down):';
chlab  = 'Character';
rlabs  = cellstr(num2str(tx'));
clabs  = cellstr(num2str(ty));
ncols  = int64(80);
indent = int64(0);
[ifail] =  x04cb( ...
                  matrix, diag, ff, format, title, chlab, ...
                  rlabs, chlab, clabs, ncols, indent);

e01da example results

               i   knot lamda(i)      j     knot mu(j)
               4      1.0000          4      0.0000
               5      1.0000          5      0.0000
               6      2.0000          6      1.0000
               7      2.0000          7      1.0000
               8      2.0000

The B-Spline coefficients:
    1.0000    1.1333    1.3667    1.7000    1.9000    2.0000    1.2000
    1.3333    1.5667    1.9000    2.1000    2.2000    1.5833    1.7167
    1.9500    2.2833    2.4833    2.5833    2.1433    2.2767    2.5100
    2.8433    3.0433    3.1433    2.8667    3.0000    3.2333    3.5667
    3.7667    3.8667    3.4667    3.6000    3.8333    4.1667    4.3667
    4.4667    4.0000    4.1333    4.3667    4.7000    4.9000    5.0000

 Spline evaluated on a regular mesh (x across, y down):
            0     0.2     0.4     0.6     0.8       1
   1    1.000   1.440   1.960   2.560   3.240   4.000
 1.2    1.200   1.640   2.160   2.760   3.440   4.200
 1.4    1.400   1.840   2.360   2.960   3.640   4.400
 1.6    1.600   2.040   2.560   3.160   3.840   4.600
 1.8    1.800   2.240   2.760   3.360   4.040   4.800
   2    2.000   2.440   2.960   3.560   4.240   5.000

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