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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_fit_1dspline_deriv (e02bc)

▸▿ Contents

    1  Purpose
    2  Syntax
    3  Description
    4  References
▸▿ 5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
    9  Example

Purpose

nag_fit_1dspline_deriv (e02bc) evaluates a cubic spline and its first three derivatives from its B-spline representation.

Syntax

[s, ifail] = e02bc(lamda, c, x, left, 'ncap7', ncap7)
[s, ifail] = nag_fit_1dspline_deriv(lamda, c, x, left, 'ncap7', ncap7)

Description

nag_fit_1dspline_deriv (e02bc) evaluates the cubic spline sx and its first three derivatives at a prescribed argument x. It is assumed that sx is represented in terms of its B-spline coefficients ci, for i=1,2,…,n-+3 and (augmented) ordered knot set λi, for i=1,2,…,n-+7, (see nag_fit_1dspline_knots (e02ba)), i.e.,
sx = ∑ i=1q ci Nix .  
Here q=n-+3, n- is the number of intervals of the spline and Nix denotes the normalized B-spline of degree 3 (order 4) defined upon the knots λi,λi+1,…,λi+4. The prescribed argument x must satisfy
λ4 ≤ x ≤ λ n-+4 .  
At a simple knot λi (i.e., one satisfying λi-1<λi<λi+1), the third derivative of the spline is in general discontinuous. At a multiple knot (i.e., two or more knots with the same value), lower derivatives, and even the spline itself, may be discontinuous. Specifically, at a point x=u where (exactly) r knots coincide (such a point is termed a knot of multiplicity r), the values of the derivatives of order 4-j, for j=1,2,…,r, are in general discontinuous. (Here 1≤r≤4; r>4 is not meaningful.) You must specify whether the value at such a point is required to be the left- or right-hand derivative.
The method employed is based upon:
(i) carrying out a binary search for the knot interval containing the argument x (see Cox (1978)),
(ii) evaluating the nonzero B-splines of orders 1, 2, 3 and 4 by recurrence (see Cox (1972) and Cox (1978)),
(iii) computing all derivatives of the B-splines of order 4 by applying a second recurrence to these computed B-spline values (see de Boor (1972)),
(iv) multiplying the fourth-order B-spline values and their derivative by the appropriate B-spline coefficients, and summing, to yield the values of sx and its derivatives.
nag_fit_1dspline_deriv (e02bc) can be used to compute the values and derivatives of cubic spline fits and interpolants produced by nag_fit_1dspline_knots (e02ba).
If only values and not derivatives are required, nag_fit_1dspline_eval (e02bb) may be used instead of nag_fit_1dspline_deriv (e02bc), which takes about 50% longer than nag_fit_1dspline_eval (e02bb).

References

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
de Boor C (1972) On calculating with B-splines J. Approx. Theory 6 50–62

Parameters

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 and its derivatives are to be evaluated.
Constraint: lamda4≤x≤lamdancap7-3.
4:     left – int64int32nag_int scalar
Specifies whether left- or right-hand values of the spline and its derivatives are to be computed (see Description). Left- or right-hand values are formed according to whether left is equal or not equal to 1.
If x does not coincide with a knot, the value of left is immaterial.
If x=lamda4, right-hand values are computed.
If x=lamdancap7-3, left-hand values are formed, regardless of the value of left.

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 of the spline (which is 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: ncap7≥8.

Output Parameters

1:     s4 – double array
sj contains the value of the j-1th derivative of the spline at the argument x, for j=1,2,3,4. Note that s1 contains the value of the spline.
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
ncap7<8, i.e., the number of intervals is not positive.
   ifail=2
Either lamda4≥lamdancap7-3, i.e., the range over which sx is defined is null or negative in length, or x is an invalid argument, i.e., x<lamda4 or x>lamdancap7-3.
   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 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+3≤x≤λj+4. If cj,cj+1,cj+2 and cj+3 are all of the same sign, then the computed value of sx has relative error bounded by 20×machine precision. For full details see Cox (1978).
No complete error analysis is available for the computation of the derivatives of sx. However, for most practical purposes the absolute errors in the computed derivatives should be small.

Further Comments

The time taken is approximately linear in logn-+7.
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.

Example

Compute, at the 7 arguments x=0, 1, 2, 3, 4, 5, 6, the left- and right-hand values and first 3 derivatives of the cubic spline defined over the interval 0≤x≤6 having the 6 interior knots x=1, 3, 3, 3, 4, 4, the 8 additional knots 0, 0, 0, 0, 6, 6, 6, 6, and the 10 B-spline coefficients 10, 12, 13, 15, 22, 26, 24, 18, 14, 12.
The input data items (using the notation of Arguments) comprise the following values in the order indicated:
n- m
lamdaj, for j=1,2,…,ncap7
cj, for j=1,2,…,ncap7-4
xi, for i=1,2,…,m
This example program is written in a general form that will enable the values and derivatives of a cubic spline having an arbitrary number of knots to be evaluated at a set of arbitrary points. Any number of datasets may be supplied. The only changes required to the program relate to the dimensions of the arrays lamda and c.

Open in the MATLAB editor: e02bc_example

function e02bc_example


fprintf('e02bc example results\n\n');

knots = [ 1  3  3  3  4  4];
ncap = size(knots,2) + 1;
ncap7 = ncap + 7;

lamda = zeros(ncap7,1);
lamda(5:ncap+3) = knots;
lamda(ncap+4:ncap7) = 6;

% B-spline coefficients
c = zeros(ncap7,1);
c(1:ncap+3) = [10  12  13  15  22  26  24  18  14  12];

% Evaluate spline at values in lamda range
left  = int64(1);
right = int64(2);
k = 0;
for x = 0:0.2:6;
  k = k+1;
  [sl(:,k), ifail] = e02bc( ...
                            lamda, c, x, left);
  [sr(:,k), ifail] = e02bc( ...
                            lamda, c, x, right);
end
x = 0:0.2:6;
fprintf('Left hand values and derivatives\n');
fprintf('%5s%12s%12s%11s%11s\n', 'x', 'spline', '1st deriv', ...
        '2nd deriv', '3rd deriv');
sol = [ x; sl];
fprintf('%7.2f%11.4f%11.4f%11.4f%11.4f\n',sol(:,1:5:end));

fprintf('\nRight hand values and derivatives\n');
fprintf('%5s%12s%12s%11s%11s\n', 'x', 'spline', '1st deriv', ...
        '2nd deriv', '3rd deriv');
sol = [ x; sr];
fprintf('%7.2f%11.4f%11.4f%11.4f%11.4f\n', sol(:,1:5:end));

fig1 = figure;
plot(x,sl(1,:),x,sl(2,:),x,sl(3,:),x,sl(4,:));
xlabel('x');
title('Evaluation of Left-hand cubic spline and derivatives');
legend('cubic spline', '1st derivative', '2nd derivative', ...
       '3rd derivative', 'Location', 'NorthWest');
fig2 = figure;
plot(x,sr(1,:),x,sr(2,:),x,sr(3,:),x,sr(4,:));
xlabel('x');
title('Evaluation of Right-hand cubic spline and derivatives');
legend('cubic spline', '1st derivative', '2nd derivative', ...
       '3rd derivative', 'Location', 'NorthWest');


e02bc example results

Left hand values and derivatives
    x      spline   1st deriv  2nd deriv  3rd deriv
   0.00    10.0000     6.0000   -10.0000    10.6667
   1.00    12.7778     1.3333     0.6667    10.6667
   2.00    15.0972     3.9583     4.5833     3.9167
   3.00    22.0000    10.5000     8.5000     3.9167
   4.00    22.0000    -6.0000     0.0000    36.0000
   5.00    16.2500    -5.2500     1.5000     1.5000
   6.00    12.0000    -3.0000     3.0000     1.5000

Right hand values and derivatives
    x      spline   1st deriv  2nd deriv  3rd deriv
   0.00    10.0000     6.0000   -10.0000    10.6667
   1.00    12.7778     1.3333     0.6667     3.9167
   2.00    15.0972     3.9583     4.5833     3.9167
   3.00    22.0000    12.0000   -36.0000    36.0000
   4.00    22.0000    -6.0000     0.0000     1.5000
   5.00    16.2500    -5.2500     1.5000     1.5000
   6.00    12.0000    -3.0000     3.0000     1.5000
e02bc_fig1.png
e02bc_fig2.png
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