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Chapter Introduction
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NAG Toolbox: nag_fit_1dspline_eval (e02bb)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_fit_1dspline_eval (e02bb) evaluates a cubic spline from its B-spline representation.


[s, ifail] = e02bb(lamda, c, x, 'ncap7', ncap7)
[s, ifail] = nag_fit_1dspline_eval(lamda, c, x, 'ncap7', ncap7)


nag_fit_1dspline_eval (e02bb) evaluates the cubic spline sx at a prescribed argument x from its augmented knot set λi, for i=1,2,,n+7, (see nag_fit_1dspline_knots (e02ba)) and from the coefficients ci, for i=1,2,,qin its B-spline representation
Here q=n-+3, where n- is the number of intervals of the spline, and Nix denotes the normalized B-spline of degree 3 defined upon the knots λi,λi+1,,λi+4. The prescribed argument x must satisfy λ4xλn-+4.
It is assumed that λjλj-1, for j=2,3,,n-+7, and λn-+4>λ4.
If x is a point at which 4 knots coincide, sx is discontinuous at x; in this case, s contains the value defined as x is approached from the right.
The method employed is that of evaluation by taking convex combinations due to de Boor (1972). For further details of the algorithm and its use see Cox (1972) and Cox and Hayes (1973).
It is expected that a common use of nag_fit_1dspline_eval (e02bb) will be the evaluation of the cubic spline approximations produced by nag_fit_1dspline_knots (e02ba). A generalization of nag_fit_1dspline_eval (e02bb) which also forms the derivative of sx is nag_fit_1dspline_deriv (e02bc). nag_fit_1dspline_deriv (e02bc) takes about 50% longer than nag_fit_1dspline_eval (e02bb).


Cox M G (1972) The numerical evaluation of B-splines J. Inst. Math. Appl. 10 134–149
Cox M G (1978) The numerical evaluation of a spline from its B-spline representation J. Inst. Math. Appl. 21 135–143
Cox M G and Hayes J G (1973) Curve fitting: a guide and suite of algorithms for the non-specialist user NPL Report NAC26 National Physical Laboratory
de Boor C (1972) On calculating with B-splines J. Approx. Theory 6 50–62


Compulsory Input Parameters

1:     lamdancap7 – double array
lamdaj must be set to the value of the jth member of the complete set of knots, λj, for j=1,2,,n-+7.
Constraint: the lamdaj must be in nondecreasing order with lamdancap7-3> lamda4.
2:     cncap7 – double array
The coefficient ci of the B-spline Nix, for i=1,2,,n-+3. The remaining elements of the array are not referenced.
3:     x – double scalar
The argument x at which the cubic spline is to be evaluated.
Constraint: lamda4xlamdancap7-3.

Optional Input Parameters

1:     ncap7 int64int32nag_int scalar
Default: the dimension of the arrays lamda, c. (An error is raised if these dimensions are not equal.)
n-+7, where n- is the number of intervals (one greater than the number of interior knots, i.e., the knots strictly within the range λ4 to λn-+4) over which the spline is defined.
Constraint: ncap78.

Output Parameters

1:     s – double scalar
The value of the spline, sx.
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:
The argument x does not satisfy lamda4xlamdancap7-3.
In this case the value of s is set arbitrarily to zero.
ncap7<8, i.e., the number of interior knots is negative.
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 computed value of sx has negligible error in most practical situations. Specifically, this value has an absolute error bounded in modulus by 18×cmax×machine precision, where cmax is the largest in modulus of cj,cj+1,cj+2 and cj+3, and j is an integer such that λj+3xλj+4. If cj,cj+1,cj+2 and cj+3 are all of the same sign, then the computed value of sx has a relative error not exceeding 20×machine precision in modulus. For further details see Cox (1978).

Further Comments

The time taken is approximately c×1+0.1×logn-+7 seconds, where c is a machine-dependent constant.
Note:  the function does not test all the conditions on the knots given in the description of lamda in Arguments, since to do this would result in a computation time approximately linear in n-+7 instead of logn-+7. All the conditions are tested in nag_fit_1dspline_knots (e02ba), however.


Evaluate at nine equally-spaced points in the interval 1.0x9.0 the cubic spline with (augmented) knots 1.0, 1.0, 1.0, 1.0, 3.0, 6.0, 8.0, 9.0, 9.0, 9.0, 9.0 and normalized cubic B-spline coefficients 1.0, 2.0, 4.0, 7.0, 6.0, 4.0, 3.0.
The example program is written in a general form that will enable a cubic spline with n- intervals, in its normalized cubic B-spline form, to be evaluated at m equally-spaced points in the interval lamda4xlamdan-+4. The program is self-starting in that any number of datasets may be supplied.
function e02bb_example

fprintf('e02bb example results\n\n');

knots = [3 6 8];
ncap = size(knots,2) + 1;
ncap7 = ncap + 7;

lamda = zeros(ncap7,1);
lamda(1:4) = 1;
lamda(5:7) = knots;
lamda(8:ncap7) = 9;

% B-spline coefficients
c = zeros(ncap7,1);
c(1:ncap+3) = [1  2  4   7   6    4    3];

% Evaluate spline at values in lamda range
a = lamda(4);
b = lamda(ncap+4);
for x = 1:9;
  [s(x), ifail] = e02bb( ...
                         lamda, c, x);
fprintf('    x      spline at x\n');
fprintf('%7.2f%14.4f\n',[ [1:9]; s]);

fig1 = figure;
hold on
title('Evaluation of cubic spline representation');
legend('Evaluation points','cubic spline','Location','NorthWest');
hold off;

e02bb example results

    x      spline at x
   1.00        1.0000
   2.00        2.3779
   3.00        3.6229
   4.00        4.8327
   5.00        5.8273
   6.00        6.3571
   7.00        6.1905
   8.00        5.1667
   9.00        3.0000

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