NAG FL Interface
e02aff (dim1_cheb_glp)
1
Purpose
e02aff computes the coefficients of a polynomial, in its Chebyshev series form, which interpolates (passes exactly through) data at a special set of points. Least squares polynomial approximations can also be obtained.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
nplus1 
Integer, Intent (Inout) 
:: 
ifail 
Real (Kind=nag_wp), Intent (In) 
:: 
f(nplus1) 
Real (Kind=nag_wp), Intent (Out) 
:: 
a(nplus1) 

C Header Interface
#include <nag.h>
void 
e02aff_ (const Integer *nplus1, const double f[], double a[], Integer *ifail) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
e02aff_ (const Integer &nplus1, const double f[], double a[], Integer &ifail) 
}

The routine may be called by the names e02aff or nagf_fit_dim1_cheb_glp.
3
Description
e02aff computes the coefficients
${a}_{\mathit{j}}$, for
$\mathit{j}=1,2,\dots ,n+1$, in the Chebyshev series
which interpolates the data
${f}_{r}$ at the points
Here
${T}_{j}\left(\overline{x}\right)$ denotes the Chebyshev polynomial of the first kind of degree
$j$ with argument
$\overline{x}$. The use of these points minimizes the risk of unwanted fluctuations in the polynomial and is recommended when the data abscissae can be chosen by you, e.g., when the data is given as a graph. For further advantages of this choice of points, see
Clenshaw (1962).
In terms of your original variables,
$x$ say, the values of
$x$ at which the data
${f}_{r}$ are to be provided are
where
${x}_{\mathrm{max}}$ and
${x}_{\mathrm{min}}$ are respectively the upper and lower ends of the range of
$x$ over which you wish to interpolate.
Truncation of the resulting series after the term involving
${a}_{i+1}$, say, yields a least squares approximation to the data. This approximation,
$p\left(\overline{x}\right)$, say, is the polynomial of degree
$i$ which minimizes
where the residual
${\epsilon}_{\mathit{r}}=p\left({\overline{x}}_{\mathit{r}}\right){f}_{\mathit{r}}$, for
$\mathit{r}=1,2,\dots ,n+1$.
The method employed is based on the application of the threeterm recurrence relation due to
Clenshaw (1955) for the evaluation of the defining expression for the Chebyshev coefficients (see, for example,
Clenshaw (1962)). The modifications to this recurrence relation suggested by Reinsch and Gentleman (see
Gentleman (1969)) are used to give greater numerical stability.
For further details of the algorithm and its use see
Cox (1974) and
Cox and Hayes (1973).
Subsequent evaluation of the computed polynomial, perhaps truncated after an appropriate number of terms, should be carried out using
e02aef.
4
References
Clenshaw C W (1955) A note on the summation of Chebyshev series Math. Tables Aids Comput. 9 118–120
Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO
Cox M G (1974) A datafitting package for the nonspecialist user Software for Numerical Mathematics (ed D J Evans) Academic Press
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
Gentleman W M (1969) An error analysis of Goertzel's (Watt's) method for computing Fourier coefficients Comput. J. 12 160–165
5
Arguments

1:
$\mathbf{nplus1}$ – Integer
Input

On entry: the number $n+1$ of data points (one greater than the degree $n$ of the interpolating polynomial).
Constraint:
${\mathbf{nplus1}}\ge 2$.

2:
$\mathbf{f}\left({\mathbf{nplus1}}\right)$ – Real (Kind=nag_wp) array
Input

On entry: for
$r=1,2,\dots ,n+1$,
${\mathbf{f}}\left(r\right)$ must contain
${f}_{r}$ the value of the dependent variable (ordinate) corresponding to the value
of the independent variable (abscissa)
$\overline{x}$, or equivalently to the value
of your original variable
$x$. Here
${x}_{\mathrm{max}}$ and
${x}_{\mathrm{min}}$ are respectively the upper and lower ends of the range over which you wish to interpolate.

3:
$\mathbf{a}\left({\mathbf{nplus1}}\right)$ – Real (Kind=nag_wp) array
Output

On exit: ${\mathbf{a}}\left(\mathit{j}\right)$ is the coefficient ${a}_{\mathit{j}}$ in the interpolating polynomial, for $\mathit{j}=1,2,\dots ,n+1$.

4:
$\mathbf{ifail}$ – Integer
Input/Output

On entry:
ifail must be set to
$0$,
$1\text{or}1$. If you are unfamiliar with this argument you should refer to
Section 4 in the Introduction to the NAG Library FL Interface 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 argument, 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
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry, ${\mathbf{nplus1}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nplus1}}\ge 2$.
 ${\mathbf{ifail}}=99$
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.
 ${\mathbf{ifail}}=399$
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.
 ${\mathbf{ifail}}=999$
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 ${f}_{r}+\delta {f}_{r}$. The ratio of the sum of the absolute values of the $\delta {f}_{r}$ to the sum of the absolute values of the ${f}_{r}$ is less than a small multiple of $\left(n+1\right)\epsilon $, where $\epsilon $ is the machine precision.
8
Parallelism and Performance
e02aff is not threaded in any implementation.
The time taken is approximately proportional to ${\left(n+1\right)}^{2}+30$.
For choice of degree when using the routine for least squares approximation, see
Section 3.2 in the
E02 Chapter Introduction.
10
Example
Determine the Chebyshev coefficients of the polynomial which interpolates the data ${\overline{x}}_{\mathit{r}},{f}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,11$, where ${\overline{x}}_{r}=\mathrm{cos}\left(\pi \times \left(r1\right)/10\right)$ and ${f}_{r}={e}^{{\overline{x}}_{r}}$. Evaluate, for comparison with the values of ${f}_{\mathit{r}}$, the resulting Chebyshev series at ${\overline{x}}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,11$.
The example program supplied is written in a general form that will enable polynomial interpolations of arbitrary data at the cosine points
$\mathrm{cos}\left(\pi \times \left(\mathit{r}1\right)/n\right)$, for
$\mathit{r}=1,2,\dots ,n+1$, to be obtained for any
$n$ (
$\text{}={\mathbf{nplus1}}1$). Note that
e02aef is used to evaluate the interpolating polynomial. The program is selfstarting in that any number of datasets can be supplied.
10.1
Program Text
10.2
Program Data
10.3
Program Results