NAG Library Routine Document
E02BAF
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
SUBROUTINE E02BAF ( 
M, NCAP7, X, Y, W, LAMDA, WORK1, WORK2, C, SS, IFAIL) 
INTEGER 
M, NCAP7, IFAIL 
REAL (KIND=nag_wp) 
X(M), Y(M), W(M), LAMDA(NCAP7), WORK1(M), WORK2(4*NCAP7), C(NCAP7), SS 

3 Description
E02BAF determines a least squares cubic spline approximation $s\left(x\right)$ to the set of data points $\left({x}_{\mathit{r}},{y}_{\mathit{r}}\right)$ with weights ${w}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$. The value of ${\mathbf{NCAP7}}=\stackrel{}{n}+7$, where $\stackrel{}{n}$ is the number of intervals of the spline (one greater than the number of interior knots), and the values of the knots ${\lambda}_{5},{\lambda}_{6},\dots ,{\lambda}_{\stackrel{}{n}+3}$, interior to the data interval, are prescribed by you.
$s\left(x\right)$ has the property that it minimizes
$\theta $, the sum of squares of the weighted residuals
${\epsilon}_{\mathit{r}}$, for
$\mathit{r}=1,2,\dots ,m$, where
The routine produces this minimizing value of
$\theta $ and the coefficients
${c}_{1},{c}_{2},\dots ,{c}_{q}$, where
$q=\stackrel{}{n}+3$, in the Bspline representation
Here
${N}_{i}\left(x\right)$ denotes the normalized Bspline of degree
$3$ defined upon the knots
${\lambda}_{i},{\lambda}_{i+1},\dots ,{\lambda}_{i+4}$.
In order to define the full set of Bsplines required, eight additional knots ${\lambda}_{1},{\lambda}_{2},{\lambda}_{3},{\lambda}_{4}$ and ${\lambda}_{\stackrel{}{n}+4},{\lambda}_{\stackrel{}{n}+5},{\lambda}_{\stackrel{}{n}+6},{\lambda}_{\stackrel{}{n}+7}$ are inserted automatically by the routine. The first four of these are set equal to the smallest ${x}_{r}$ and the last four to the largest ${x}_{r}$.
The representation of $s\left(x\right)$ in terms of Bsplines is the most compact form possible in that only $\stackrel{}{n}+3$ coefficients, in addition to the $\stackrel{}{n}+7$ knots, fully define $s\left(x\right)$.
The method employed involves forming and then computing the least squares solution of a set of
$m$ linear equations in the coefficients
${c}_{\mathit{i}}$, for
$\mathit{i}=1,2,\dots ,\stackrel{}{n}+3$. The equations are formed using a recurrence relation for Bsplines 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
${N}_{i}\left(x\right)$ 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
$s\left(x\right)$ from its Bspline representation may be carried out using
E02BBF. If derivatives of
$s\left(x\right)$ are also required,
E02BCF may be used.
E02BDF can be used to compute the definite integral of
$s\left(x\right)$.
4 References
Cox M G (1972) The numerical evaluation of Bsplines J. Inst. Math. Appl. 10 134–149
Cox M G (1974) A datafitting package for the nonspecialist 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 nonspecialist user NPL Report NAC26 National Physical Laboratory
de Boor C (1972) On calculating with Bsplines J. Approx. Theory 6 50–62
Gentleman W M (1973) Leastsquares 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 leastsquares 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 Parameters
 1: M – INTEGERInput
On entry: the number $m$ of data points.
Constraint:
${\mathbf{M}}\ge \mathit{mdist}\ge 4$, where $\mathit{mdist}$ is the number of distinct $x$ values in the data.
 2: NCAP7 – INTEGERInput
On entry: $\stackrel{}{n}+7$, where $\stackrel{}{n}$ 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 ${x}_{1}$ to ${x}_{m}$) over which the spline is defined.
Constraint:
$8\le {\mathbf{NCAP7}}\le \mathit{mdist}+4$, where $\mathit{mdist}$ is the number of distinct $x$ values in the data.
 3: X(M) – REAL (KIND=nag_wp) arrayInput
On entry: the values
${x}_{\mathit{r}}$ of the independent variable (abscissa), for $\mathit{r}=1,2,\dots ,m$.
Constraint:
${x}_{1}\le {x}_{2}\le \cdots \le {x}_{m}$.
 4: Y(M) – REAL (KIND=nag_wp) arrayInput
On entry: the values
${y}_{\mathit{r}}$ of the dependent variable (ordinate), for $\mathit{r}=1,2,\dots ,m$.
 5: W(M) – REAL (KIND=nag_wp) arrayInput
On entry: the values
${w}_{\mathit{r}}$ of the weights, for
$\mathit{r}=1,2,\dots ,m$. For advice on the choice of weights, see the
E02 Chapter Introduction.
Constraint:
${\mathbf{W}}\left(\mathit{r}\right)>0.0$, for $\mathit{r}=1,2,\dots ,m$.
 6: LAMDA(NCAP7) – REAL (KIND=nag_wp) arrayInput/Output
On entry: ${\mathbf{LAMDA}}\left(\mathit{i}\right)$ must be set to the $\left(\mathit{i}4\right)$th (interior) knot, ${\lambda}_{\mathit{i}}$, for $\mathit{i}=5,6,\dots ,\stackrel{}{n}+3$.
Constraint:
${\mathbf{X}}\left(1\right)<{\mathbf{LAMDA}}\left(5\right)\le {\mathbf{LAMDA}}\left(6\right)\le \cdots \le {\mathbf{LAMDA}}\left({\mathbf{NCAP7}}4\right)<{\mathbf{X}}\left({\mathbf{M}}\right)$.
On exit: the input values are unchanged, and
${\mathbf{LAMDA}}\left(\mathit{i}\right)$, for
$\mathit{i}=1,2,3,4$,
${\mathbf{NCAP7}}3$,
${\mathbf{NCAP7}}2$,
${\mathbf{NCAP7}}1$,
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) arrayWorkspace
 8: WORK2($4\times {\mathbf{NCAP7}}$) – REAL (KIND=nag_wp) arrayWorkspace
 9: C(NCAP7) – REAL (KIND=nag_wp) arrayOutput
On exit: the coefficient
${c}_{\mathit{i}}$ of the Bspline ${N}_{\mathit{i}}\left(x\right)$, for $\mathit{i}=1,2,\dots ,\stackrel{}{n}+3$. The remaining elements of the array are not used.
 10: SS – REAL (KIND=nag_wp)Output
On exit: the residual sum of squares, $\theta $.
 11: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$
The knots fail to satisfy the condition
${\mathbf{X}}\left(1\right)<{\mathbf{LAMDA}}\left(5\right)\le {\mathbf{LAMDA}}\left(6\right)\le \cdots \le {\mathbf{LAMDA}}\left({\mathbf{NCAP7}}4\right)<{\mathbf{X}}\left({\mathbf{M}}\right)$.
Thus the knots are not in correct order or are not interior to the data interval.
 ${\mathbf{IFAIL}}=2$

The weights are not all strictly positive.
 ${\mathbf{IFAIL}}=3$
The values of ${\mathbf{X}}\left(\mathit{r}\right)$, for $\mathit{r}=1,2,\dots ,{\mathbf{M}}$, are not in nondecreasing order.
 ${\mathbf{IFAIL}}=4$
${\mathbf{NCAP7}}<8$ (so the number of interior knots is negative) or ${\mathbf{NCAP7}}>\mathit{mdist}+4$, where $\mathit{mdist}$ is the number of distinct $x$ values in the data (so there cannot be a unique solution).
 ${\mathbf{IFAIL}}=5$
The conditions specified by
Schoenberg and Whitney (1953) fail to hold for at least one subset of the distinct data abscissae. That is, there is no subset of
${\mathbf{NCAP7}}4$ strictly increasing values,
${\mathbf{X}}\left(R\left(1\right)\right),{\mathbf{X}}\left(R\left(2\right)\right),\dots ,{\mathbf{X}}\left(R\left({\mathbf{NCAP7}}4\right)\right)$, among the abscissae such that
 ${\mathbf{X}}\left(R\left(1\right)\right)<{\mathbf{LAMDA}}\left(1\right)<{\mathbf{X}}\left(R\left(5\right)\right)$,
 ${\mathbf{X}}\left(R\left(2\right)\right)<{\mathbf{LAMDA}}\left(2\right)<{\mathbf{X}}\left(R\left(6\right)\right)$,
 $\vdots $
 ${\mathbf{X}}\left(R\left({\mathbf{NCAP7}}8\right)\right)<{\mathbf{LAMDA}}\left({\mathbf{NCAP7}}8\right)<{\mathbf{X}}\left(R\left({\mathbf{NCAP7}}4\right)\right)$.
This means that there is no unique solution: there are regions containing too many knots compared with the number of data points.
7 Accuracy
The rounding errors committed are such that the computed coefficients are exact for a slightly perturbed set of ordinates
${y}_{r}+\delta {y}_{r}$. The ratio of the rootmeansquare value for the
$\delta {y}_{r}$ to the rootmeansquare value of the
${y}_{r}$ can be expected to be less than a small multiple of
$\kappa \times m\times \mathit{machineprecision}$, where
$\kappa $ is a condition number for the problem. Values of
$\kappa $ for
$20$–
$30$ 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
${x}_{1}$ and
${x}_{m}$ used in the routine by various choices of noncoincident exterior knots gave values of
$\kappa $ between
$16$ and
$180$. Again see
Cox (1975) for further details.) In general we would not expect
$\kappa $ to be large unless the choice of knots results in nearviolation 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.
The time taken is approximately $C\times \left(2m+\stackrel{}{n}+7\right)$ seconds, where $C$ is a machinedependent constant.
Multiple knots are permitted as long as their multiplicity does not exceed
$4$, i.e., the complete set of knots must satisfy
${\lambda}_{\mathit{i}}<{\lambda}_{\mathit{i}+4}$, for
$\mathit{i}=1,2,\dots ,\stackrel{}{n}+3$, (see
Section 6). At a knot of multiplicity one (the usual case),
$s\left(x\right)$ and its first two derivatives are continuous. At a knot of multiplicity two,
$s\left(x\right)$ and its first derivative are continuous. At a knot of multiplicity three,
$s\left(x\right)$ is continuous, and at a knot of multiplicity four,
$s\left(x\right)$ is generally discontinuous.
The routine can be used efficiently for cubic spline interpolation, i.e., if $m=\stackrel{}{n}+3$. The abscissae must then of course satisfy ${x}_{1}<{x}_{2}<\cdots <{x}_{m}$. Recommended values for the knots in this case are ${\lambda}_{\mathit{i}}={x}_{\mathit{i}2}$, for $\mathit{i}=5,6,\dots ,\stackrel{}{n}+3$.
9 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 halfway between each pair of adjacent data points.
The example program is written in a general form that will enable a cubic spline approximation with
$\stackrel{}{n}$ intervals (
$\stackrel{}{n}1$ 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 selfstarting in that any number of datasets can be supplied.
9.1 Program Text
Program Text (e02bafe.f90)
9.2 Program Data
Program Data (e02bafe.d)
9.3 Program Results
Program Results (e02bafe.r)