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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_fit_1dspline_deriv_vector (e02bf)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_fit_1dspline_deriv_vector (e02bf) evaluates a cubic spline and up to its first three derivatives from its B-spline representation at a vector of points. nag_fit_1dspline_deriv_vector (e02bf) can be used to compute the values and derivatives of cubic spline fits and interpolants produced by reference to nag_interp_1d_spline (e01ba), nag_fit_1dspline_knots (e02ba) and nag_fit_1dspline_auto (e02be).


[s, ixloc, iwrk, ifail] = e02bf(start, lamda, c, deriv, xord, x, ixloc, iwrk, 'ncap7', ncap7, 'nx', nx, 'liwrk', liwrk)
[s, ixloc, iwrk, ifail] = nag_fit_1dspline_deriv_vector(start, lamda, c, deriv, xord, x, ixloc, iwrk, 'ncap7', ncap7, 'nx', nx, 'liwrk', liwrk)


nag_fit_1dspline_deriv_vector (e02bf) evaluates the cubic spline sx and optionally derivatives up to order 3 for a vector of points xj, for j=1,2,,nx. 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) and nag_fit_1dspline_auto (e02be)), 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 knots λ5,λ6,,λn-+3 are the interior knots. The remaining knots, λ1, λ2, λ3, λ4 and λn-+4, λn-+5, λn-+6, λn+7- are the exterior knots. The knots λ4 and λn-+4 are the boundaries of the spline.
Only abscissae satisfying,
λ4 xj λn-+4 ,  
will be evaluated. 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 1r4; r>4 is not meaningful.) The maximum order of the derivatives to be evaluated Dord, and the left- or right-handedness of the computation when an abscissa corresponds exactly to an interior knot, are determined by the value of deriv.
Each abscissa (point at which the spline is to be evaluated) xj contained in x has an associated enclosing interval number, ixlocj either supplied or returned in ixloc (see argument start). A simple call to nag_fit_1dspline_deriv_vector (e02bf) would set start=0 and the contents of ixloc need never be set nor referenced, and the following description on modes of operation can be ignored. However, where efficiency is an important consideration, the following description will help to choose the appropriate mode of operation.
The interval numbers are used to determine which B-splines must be evaluated for a given abscissa, and are defined as
ixlocj = 0 xj < λ1 4 λ4 = xj k λk < xj < λk+1 k λ4 < λk = xj left derivatives k xj = λk+1 < λ n-+4 right derivatives or no derivatives n-+4 λn-+4 = xj >n-+7 xj > λn-+7 (1)
The algorithm has two modes of vectorization, termed here sorted and unsorted, which are selectable by the argument start.
Furthermore, if the supplied abscissae are sufficiently ordered, as indicated by the argument xord, the algorithm will take advantage of significantly faster methods for the determination of both the interval numbers and the subsequent spline evaluations.
The sorted mode has two phases, a sorting phase and an evaluation phase. This mode is recommended if there are many abscissae to evaluate relative to the number of intervals of the spline, or the abscissae are distributed relatively densely over a subsection of the spline. In the first phase, ixlocj is determined for each xj and a permutation is calculated to sort the xj by interval number. The first phase may be either partially or completely by-passed using the argument start if the enclosing segments and/or the subsequent ordering are already known a priori, for example if multiple spline coefficients c are to be evaluated over the same set of knots lamda.
In the second phase of the sorted mode, spline approximations are evaluated by segment, so that non-abscissa dependent calculations over a segment may be reused in the evaluation for all abscissae belonging to a specific segment. For example, all third derivatives of all abscissae in the same segment will be identical.
In the unsorted mode of vectorization, no a priori segment sorting is performed, and if the abscissae are not sufficiently ordered, the evaluation at an abscissa will be independent of evaluations at other abscissae; also non-abscissa dependent calculations over a segment will be repeated for each abscissa in a segment. This may be quicker if the number of abscissa is small in comparison to the number of knots in the spline, and they are distributed sparsely throughout the domain of the spline. This is effectively a direct vectorization of nag_fit_1dspline_eval (e02bb) and nag_fit_1dspline_deriv (e02bc), although if the enclosing interval numbers ixlocj are known, these may again be provided.
If the abscissae are sufficiently ordered, then once the first abscissa in a segment is known, an efficient algorithm will be used to determine the location of the final abscissa in this segment. The spline will subsequently be evaluated in a vectorized manner for all the abscissae indexed between the first and last of the current segment.
If no derivatives are required, the spline evaluation is calculated by taking convex combinations due to de Boor (1972). Otherwise, the calculation of sx and its derivatives is based upon,
(i) evaluating the nonzero B-splines of orders 1, 2, 3 and 4 by recurrence (see Cox (1972) and Cox (1978)),
(ii) 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)),
(iii) 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.
The method of convex combinations is significantly faster than the recurrence based method. If higher derivatives of order 2 or 3 are not required, as much computation as possible is avoided.


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


Compulsory Input Parameters

1:     start int64int32nag_int scalar
Indicates the completion state of the first phase of the algorithm.
The enclosing interval numbers ixlocj for the abscissae xj contained in x have not been determined, and you wish to use the sorted mode of vectorization.
The enclosing interval numbers ixlocj have been determined and are provided in ixloc, however the required permutation and interval related information has not been determined and you wish to use the sorted mode of vectorization.
You wish to use the sorted mode of vectorization, and the entire first phase has been completed, with the enclosing interval numbers supplied in ixloc, and the required permutation and interval related information provided in iwrk (from a previous call to nag_fit_1dspline_deriv_vector (e02bf)).
The enclosing interval numbers ixlocj for the abscissae xj contained in x have not been determined, and you wish to use the unsorted mode of vectorization.
The enclosing interval numbers ixlocj for the abscissae xj contained in x have been supplied in ixloc, and you wish to use the unsorted mode of vectorization.
Constraint: start=0, 1, 2, 10 or 11.
Additional: start=0 or 10 should be used unless you are sure that the knot set is unchanged between calls.
2:     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
3:     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.
4:     deriv int64int32nag_int scalar
The order of derivatives required.
If deriv<0 left derivatives are calculated, otherwise right derivatives are calculated. For abscissae satisfying xj=λ4 or xj=λn-+4 only right-handed or left-handed computation will be used respectively. For abscissae which do not coincide exactly with a knot, the handedness of the computation is immaterial.
No derivatives required.
Only sx and its first derivative are required.
Only sx and its first and second derivatives are required.
sx and its first, second and third derivatives are required.
Note: if deriv is greater than 3 only the derivatives up to and including 3 will be returned.
5:     xord int64int32nag_int scalar
Indicates whether x is supplied in a sufficiently ordered manner. If x is sufficiently ordered nag_fit_1dspline_deriv_vector (e02bf) will complete faster.
The abscissae in x are ordered at least by ascending interval, in that any two abscissae contained in the same interval are only separated by abscissae in the same interval, and the intervals are arranged in ascending order. For example, xj<xj+1, for j=1,2,,nx-1.
The abscissae in x are not sufficiently ordered.
6:     xnx – double array
The abscissae xj, for j=1,2,,nx. If start=0 or 10 then evaluations will only be performed for these xj satisfying λ4xjλn-+4. Otherwise evaluation will be performed unless the corresponding element of ixloc contains an invalid interval number. Please note that if the ixlocj is a valid interval number then no check is made that xj actually lies in that interval.
Constraint: at least one abscissa must fall between lamda4 and lamdancap7-3.
7:     ixlocnx int64int32nag_int array
If start=1, 2 or 11, if you wish xj to be evaluated, ixlocj must be the enclosing interval number ixlocj of the abscissae xj (see (1)). If you do not wish xj to be evaluated, you may set the interval number to be either less than 4 or greater than n-+4.
Otherwise, ixloc need not be set.
Constraint: if start=1, 2 or 11, at least one element of ixloc must be between 4 and ncap7-3.
8:     iwrkliwrk int64int32nag_int array
If start=2, iwrk must be unchanged from a previous call to nag_fit_1dspline_deriv_vector (e02bf) with start=0 or 1.
Otherwise, iwrk need not be set.

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). Note that if nag_fit_1dspline_auto (e02be) was used to generate the knots and spline coefficients then ncap7 should contain the same value as returned in n by nag_fit_1dspline_auto (e02be).
Constraint: ncap78.
2:     nx int64int32nag_int scalar
Default: the dimension of the array x and the dimension of the array ixloc. (An error is raised if these dimensions are not equal.)
nx, the total number of abscissae contained in x, including any that will not be evaluated.
Constraint: nx1.
3:     liwrk int64int32nag_int scalar
Default: the dimension of the array iwrk.
The dimension of the array iwrk.
Constraint: if start=0, 1 or 2, liwrk3+3×nx.

Output Parameters

1:     slds: – double array
The first dimension of the array s will be nx, regardless of the acceptability of the elements of x.
The second dimension of the array s will be minderiv,3 +1 .
If xj is valid, sjd will contain the (d-1)th derivative of sx, for d=1,2,,Dord+1 and j=1,2,,nx. In particular, sj1 will contain the approximation of sxj for all legal values in x.
2:     ixlocnx int64int32nag_int array
If start=1, 2 or 11, ixloc is unchanged on exit.
Otherwise, ixlocj, contains the enclosing interval number ixlocj, for the abscissa supplied in xj, for j=1,2,,nx. Evaluations will only be performed for abscissae xj satisfying λ4xjλn-+4. If evaluation is not performed ixlocj is set to 0 if xj<λ4 or n-+7 if xj>λn-+4.
3:     iwrkliwrk int64int32nag_int array
If start=10 or 11, iwrk is unchanged on exit.
Otherwise, iwrk contains the required permutation of elements of x, if any, and information related to the division of the abscissae xj between the intervals derived from lamda.
4:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_fit_1dspline_deriv_vector (e02bf) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:
On entry, at least one element of x has an enclosing interval number in ixloc outside the set allowed by the provided spline.
On entry, all elements of x had enclosing interval numbers in ixloc outside the domain allowed by the provided spline.
Constraint: start=0, 1, 2, 10 or 11.
Constraint: ncap78.
Constraint: lamda4< lamdancap7-3.
Constraint: nx1.
ldsnx is too small.
liwrknx is too small.
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+3<xλj+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. Note that this is in comparison to the derivatives of the spline, which may or may not be comparable to the derivatives of the function that has been approximated by the spline.

Further Comments

If using the sorted mode of vectorization, the time required for the first phase to determine the enclosing intervals is approximately proportional to Onx logn-. The time required to then generate the required permutations and interval information is Onx if x is ordered sufficiently, or at worst O nx minnx,n- log minnx,n-  if x is not ordered. The time required by the second phase is then proportional to Onx.
If using the unsorted mode of vectorization, the time required is proportional to O nx logn-  if the enclosing interval numbers are not provided, or Onx  if they are provided. However, the repeated calculation of various quantities will typically make this slower than the sorted mode when the ratio of abscissae to knots is high, or the abscissae are densely distributed over a relatively small subset of the intervals of the spline.
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 with a linear dependency upon n- instead of logn-. All the conditions are tested in nag_fit_1dspline_knots (e02ba) and nag_fit_1dspline_auto (e02be), however.


This example fits a spline through a set of data points using nag_fit_1dspline_auto (e02be) and then evaluates the spline at a set of supplied abscissae.
function e02bf_example

fprintf('e02bf example results\n\n');

% Data to fit
npts = 15;
x(1:8)    = [ 0;    0.5;   1;     1.5;  2;    2.5;   3;    4];
y(1:8)    = [-1.1; -0.372; 0.431; 1.69; 2.11; 3.1;   4.23; 4.35];
x(9:npts) = [4.5;   5;     5.5;   6;    7;    7.5;   8];
y(9:npts) = [4.81;  4.61;  4.79;  5.23; 6.35; 7.19;  7.97];

% Input parameters for fit
w(1:npts) = 1;
w(3)      = 1.5;
cstart    = 'c';
sfac      = 0.001;
nest      = int64(npts + 4);
lamda     = zeros(nest, 1);
wrk       = zeros(4*npts + 16*nest + 41, 1);
iwrk1     = zeros(nest, 1, 'int64');

% Determine the spline approximation
[n, lamda, c, fp, wrk, iwrk1, ifail] = ...
          cstart, x, y, w, sfac, nest, lamda, wrk, iwrk1);

% Interpolation points
nip   = 20;
xe    = [6.5178; 7.2463; 1.0159; 7.3070; 5.0589; 0.7803; 2.2280; 4.3751; ...
         7.6601; 7.7191; 1.2609; 7.7647; 7.6573; 3.8830; 6.4022; 1.1351; ...
         3.3741; 7.3259; 6.3377; 7.6759];
xe = sort(xe);
% Input parameters for interpolation
ixloc = zeros(nip, 1, 'int64');
iwrk2 = zeros(3+3*nip, 1, 'int64');
xord  = int64(0);
start = int64(0);
deriv = int64(3);

% Evaluate the spline and derivatives
[s, ixloc, iwrk2, ifail] = e02bf(...
        start, lamda, c, deriv, xord, xe, ixloc, iwrk2);

% Output the results
sd2 = min(abs(deriv),3) + 1;
for r = 1:nip
  if ixloc(r) >= 4 && ixloc(r) <= n
    fprintf('%8.4f%7d%10.4f%10.4f%10.4f%10.4f\n', ...
            xe(r), ixloc(r), s(r, 1:sd2));
    fprintf('%8.4f%7d\n', x(r), ixloc(r));

e02bf example results

       x  ixloc      s(x)     ds/dx   d2s/dx2   d3s/dx3
  0.7803      4    0.0067    1.6216    2.5007    7.5980
  1.0159      5    0.4747    2.4179    3.8175  -22.1715
  1.1351      5    0.7838    2.7154    1.1746  -22.1715
  1.2609      5    1.1273    2.6878   -1.6146  -22.1715
  2.2280      7    2.4751    1.9559    3.0615   -6.6690
  3.3741      9    4.4165   -0.1181   -2.0644   10.2964
  3.8830      9    4.3152    0.1646    3.1754   10.2964
  4.3751     10    4.7199    0.8519   -3.0718  -19.8662
  5.0589     12    4.6105   -0.1036    2.9075   -4.4467
  6.3377     14    5.5563    0.9931    0.3321    1.3065
  6.4022     14    5.6211    1.0172    0.4163    1.3065
  6.5178     14    5.7418    1.0741    0.5674    1.3065
  7.2463     15    6.7486    1.7074    0.4905   -2.8697
  7.3070     15    6.8531    1.7319    0.3163   -2.8697
  7.3259     15    6.8859    1.7374    0.2621   -2.8697
  7.6573     15    7.4586    1.6667   -0.6889   -2.8697
  7.6601     15    7.4633    1.6647   -0.6970   -2.8697
  7.6759     15    7.4895    1.6534   -0.7423   -2.8697
  7.7191     15    7.5602    1.6186   -0.8663   -2.8697
  7.7647     15    7.6330    1.5761   -0.9971   -2.8697

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