NAG FL Interface
e02baf (dim1_​spline_​knots)

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1 Purpose

e02baf computes a weighted least squares approximation to an arbitrary set of data points by a cubic spline with knots prescribed by you. Cubic spline interpolation can also be carried out.

2 Specification

Fortran Interface
Subroutine e02baf ( m, ncap7, x, y, w, lamda, work1, work2, c, ss, ifail)
Integer, Intent (In) :: m, ncap7
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x(m), y(m), w(m)
Real (Kind=nag_wp), Intent (Inout) :: lamda(ncap7)
Real (Kind=nag_wp), Intent (Out) :: work1(m), work2(4*ncap7), c(ncap7), ss
C Header Interface
#include <nag.h>
void  e02baf_ (const Integer *m, const Integer *ncap7, const double x[], const double y[], const double w[], double lamda[], double work1[], double work2[], double c[], double *ss, Integer *ifail)
The routine may be called by the names e02baf or nagf_fit_dim1_spline_knots.

3 Description

e02baf determines a least squares cubic spline approximation s(x) to the set of data points (xr,yr) with weights wr, for r=1,2,,m. The value of ncap7=n¯+7, where n¯ is the number of intervals of the spline (one greater than the number of interior knots), and the values of the knots λ5,λ6,,λn¯+3, interior to the data interval, are prescribed by you.
s(x) has the property that it minimizes θ, the sum of squares of the weighted residuals εr, for r=1,2,,m, where
The routine produces this minimizing value of θ and the coefficients c1,c2,,cq, where q=n¯+3, in the B-spline representation
Here Ni(x) denotes the normalized B-spline of degree 3 defined upon the knots λi,λi+1,,λi+4.
In order to define the full set of B-splines required, eight additional knots λ1,λ2,λ3,λ4 and λn¯+4,λn¯+5,λn¯+6,λn¯+7 are inserted automatically by the routine. The first four of these are set equal to the smallest xr and the last four to the largest xr.
The representation of s(x) in terms of B-splines is the most compact form possible in that only n¯+3 coefficients, in addition to the n¯+7 knots, fully define s(x).
The method employed involves forming and then computing the least squares solution of a set of m linear equations in the coefficients ci, for i=1,2,,n¯+3. The equations are formed using a recurrence relation for B-splines that is unconditionally stable (see Cox (1972) and de Boor (1972)), even for multiple (coincident) knots. The least squares solution is also obtained in a stable manner by using orthogonal transformations, viz. a variant of Givens rotations (see Gentleman (1974) and Gentleman (1973)). This requires only one equation to be stored at a time. Full advantage is taken of the structure of the equations, there being at most four nonzero values of Ni(x) for any value of x and hence at most four coefficients in each equation.
For further details of the algorithm and its use see Cox (1974), Cox (1975) and Cox and Hayes (1973).
Subsequent evaluation of s(x) from its B-spline representation may be carried out using e02bbf. If derivatives of s(x) are also required, e02bcf may be used. e02bdf can be used to compute the definite integral of s(x).

4 References

Cox M G (1972) The numerical evaluation of B-splines J. Inst. Math. Appl. 10 134–149
Cox M G (1974) A data-fitting package for the non-specialist user Software for Numerical Mathematics (ed D J Evans) Academic Press
Cox M G (1975) Numerical methods for the interpolation and approximation of data by spline functions PhD Thesis City University, London
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
Gentleman W M (1973) Least squares computations by Givens transformations without square roots J. Inst. Math. Applic. 12 329–336
Gentleman W M (1974) Algorithm AS 75. Basic procedures for large sparse or weighted linear least squares problems Appl. Statist. 23 448–454
Schoenberg I J and Whitney A (1953) On Polya frequency functions III Trans. Amer. Math. Soc. 74 246–259

5 Arguments

1: m Integer Input
On entry: the number m of data points.
Constraint: mmdist4, where mdist is the number of distinct x values in the data.
2: ncap7 Integer Input
On entry: 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 x1 to xm) over which the spline is defined.
Constraint: 8ncap7mdist+4, where mdist is the number of distinct x values in the data.
3: x(m) Real (Kind=nag_wp) array Input
On entry: the values xr of the independent variable (abscissa), for r=1,2,,m. The values must satisfy the Schoenberg–Whitney conditions (see Section 9).
Constraint: x1x2xm.
4: y(m) Real (Kind=nag_wp) array Input
On entry: the values yr of the dependent variable (ordinate), for r=1,2,,m.
5: w(m) Real (Kind=nag_wp) array Input
On entry: the values wr of the weights, for r=1,2,,m. For advice on the choice of weights, see the E02 Chapter Introduction.
Constraint: w(r)>0.0, for r=1,2,,m.
6: lamda(ncap7) Real (Kind=nag_wp) array Input/Output
On entry: lamda(i) must be set to the (i-4)th (interior) knot, λi, for i=5,6,,n¯+3.
Constraint: x(1)<lamda(5)lamda(6)lamda(ncap7-4)<x(m).
On exit: the input values are unchanged, and lamda(i), for i=1,2,3,4, ncap7-3, ncap7-2, ncap7-1, ncap7 contains the additional (exterior) knots introduced by the routine. For advice on the choice of knots, see Section 3.3 in the E02 Chapter Introduction.
7: work1(m) Real (Kind=nag_wp) array Workspace
8: work2(4×ncap7) Real (Kind=nag_wp) array Workspace
9: c(ncap7) Real (Kind=nag_wp) array Output
On exit: the coefficient ci of the B-spline Ni(x), for i=1,2,,n¯+3. The remaining elements of the array are not used.
10: ss Real (Kind=nag_wp) Output
On exit: the residual sum of squares, θ.
11: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, J=value, lamda(J)=value and lamda(J+1)=value.
Constraint: lamda(J)lamda(J+1).
On entry, lamda(5)=value and x(1)=value.
Constraint: lamda(5)>x(1).
On entry, ncap7=value, lamda(ncap7-4)=value, m=value and x(m)=value.
Constraint: lamda(ncap7-4)<x(m).
On entry, i=value and w(i)=value.
Constraint: w(i)>0.0.
On entry, the x values are not in nondecreasing order. I=value, x(I)=value, J=value and xdist(J)=value.
Constraint: x(I)xdist(J), where xdist is the set of distinct x-values.
On entry, ncap7=value.
Constraint: ncap78.
On entry, ncap7=value and m=value.
Constraint: ncap7m+4.
On entry, ncap7=value and mdist=value.
Constraint: ncap7mdist+4, where mdist is the number of distinct x-values.
On entry, the Schoenberg–Whitney conditions fail to hold for at least one subset of the distinct data abscissae. I=value, xdist(I)=value, J=value and lamda(J)=value.
Constraint: xdist(I)<lamda(J), where xdist is the set of distinct x-values.
On entry, the Schoenberg–Whitney conditions fail to hold for at least one subset of the distinct data abscissae. J=value, xdist(J)=value, J+4=value and lamda(J+4)=value.
Constraint: xdist(J)<lamda(J+4), where xdist is the set of distinct x-values.
On entry, the Schoenberg–Whitney conditions fail to hold for at least one subset of the distinct data abscissae. L=value, xdist(L)=value, I=value and lamda(I)=value.
Constraint: xdist(L)>lamda(I), where xdist is the set of distinct x-values.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The rounding errors committed are such that the computed coefficients are exact for a slightly perturbed set of ordinates yr+δyr. The ratio of the root-mean-square value for the δyr to the root-mean-square value of the yr can be expected to be less than a small multiple of κ×m×machine precision, where κ is a condition number for the problem. Values of κ for 2030 practical datasets all proved to lie between 4.5 and 7.8 (see Cox (1975)). (Note that for these datasets, replacing the coincident end knots at the end points x1 and xm used in the routine by various choices of non-coincident exterior knots gave values of κ between 16 and 180. Again see Cox (1975) for further details.) In general we would not expect κ to be large unless the choice of knots results in near-violation of the Schoenberg–Whitney conditions.
A cubic spline which adequately fits the data and is free from spurious oscillations is more likely to be obtained if the knots are chosen to be grouped more closely in regions where the function (underlying the data) or its derivatives change more rapidly than elsewhere.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
e02baf is not threaded in any implementation.

9 Further Comments

The time taken is approximately C×(2m+n¯+7) seconds, where C is a machine-dependent constant.
Multiple knots are permitted as long as their multiplicity does not exceed 4, i.e., the complete set of knots must satisfy λi<λi+4, for i=1,2,,n¯+3, (see Section 6). At a knot of multiplicity one (the usual case), s(x) and its first two derivatives are continuous. At a knot of multiplicity two, s(x) and its first derivative are continuous. At a knot of multiplicity three, s(x) is continuous, and at a knot of multiplicity four, s(x) is generally discontinuous.
The routine can be used efficiently for cubic spline interpolation, i.e., if m=n¯+3. The abscissae must then of course satisfy x1<x2<<xm. Recommended values for the knots in this case are λi=xi-2, for i=5,6,,n¯+3.
The Schoenberg–Whitney conditions (see Schoenberg and Whitney (1953)) state that there must be a subset of ncap7-4 strictly increasing values, x(R(1)), x(R(2)), , x(R(ncap7-4)), among the abscissae such that
If this condition is not satisfied, then there is no unique solution: there are regions containing too many knots compared with the number of data points.

10 Example

Determine a weighted least squares cubic spline approximation with five intervals (four interior knots) to a set of 14 given data points. Tabulate the data and the corresponding values of the approximating spline, together with the residual errors, and also the values of the approximating spline at points half-way between each pair of adjacent data points.
The example program is written in a general form that will enable a cubic spline approximation with n¯ intervals (n¯-1 interior knots) to be obtained to m data points, with arbitrary positive weights, and the approximation to be tabulated. Note that e02bbf is used to evaluate the approximating spline. The program is self-starting in that any number of datasets can be supplied.

10.1 Program Text

Program Text (e02bafe.f90)

10.2 Program Data

Program Data (e02bafe.d)

10.3 Program Results

Program Results (e02bafe.r)
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 −4 −2 0 2 4 6 8 10 0 2 4 6 8 10 12 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 cubic spline fit residual Cubic Spline Approximation Residual at Data Points x data points data points gnuplot_plot_2 gnuplot_plot_3 Example Program Weighted Least-squares Cubic Spline Approximation to a Set of 14 Data Points