NAG CL Interfacee02afc (dim1_​cheb_​glp)

Settings help

CL Name Style:

1Purpose

e02afc 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.

2Specification

 #include
 void e02afc (Integer nplus1, const double f[], double a[], NagError *fail)
The function may be called by the names: e02afc, nag_fit_dim1_cheb_glp or nag_1d_cheb_interp_fit.

3Description

e02afc computes the coefficients ${a}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n+1$, in the Chebyshev series
 $1 2 a 1 T 0 ( x ¯) + a 2 T 1 ( x ¯) + a 3 T 2 ( x ¯) + ⋯ + a n+1 T n ( x ¯) ,$
which interpolates the data ${f}_{r}$ at the points
 $x ¯ r = cos( (r-1) π /n) , r = 1 , 2 , … , n + 1 .$
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 you can choose the data abscissae, 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
 $x r = 1 2 ( x max - x min ) cos( (r-1) π /n) + 1 2 ( x max + x min ) , r = 1 , 2 , … , n + 1$
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
 $1 2 ε 1 2 + ε 2 2 + ε 3 2 + ⋯ + ε n 2 + 1 2 ε n+1 2 ,$
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 three-term 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), Cox and Hayes (1973).
Subsequent evaluation of the computed polynomial, perhaps truncated after an appropriate number of terms, should be carried out using e02aec.

4References

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 data-fitting package for the non-specialist 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 non-specialist 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

5Arguments

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: $\mathbf{f}\left[{\mathbf{nplus1}}\right]$const double Input
On entry: for $r=1,2,\dots ,n+1$, ${\mathbf{f}}\left[r-1\right]$ must contain ${f}_{r}$ the value of the dependent variable (ordinate) corresponding to the value
 $x ¯ r = cos( π (r-1) n )$
of the independent variable (abscissa) $\overline{x}$, or equivalently to the value
 $x r = 1 2 ( x max - x min ) cos( π (r-1) /n) + 1 2 ( x max + x min )$
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: $\mathbf{a}\left[{\mathbf{nplus1}}\right]$double Output
On exit: ${\mathbf{a}}\left[\mathit{j}-1\right]$ is the coefficient ${a}_{\mathit{j}}$ in the interpolating polynomial, for $\mathit{j}=1,2,\dots ,n+1$.
4: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_INT_ARG_LT
On entry, nplus1 must not be less than 2: ${\mathbf{nplus1}}=⟨\mathit{\text{value}}⟩$.

7Accuracy

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.

8Parallelism and Performance

e02afc is not threaded in any implementation.

The time taken by e02afc is approximately proportional to ${\left(n+1\right)}^{2}+30$.
For choice of degree when using the function for least squares approximation, see the E02 Chapter Introduction.

10Example

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 ${\overline{x}}_{r}=\mathrm{cos}\left(\left(r-1\right)\pi /10\right)$ 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 $\mathrm{cos}\left(\left(\mathit{r}-1\right)\pi /n\right)$, for $\mathit{r}=1,2,\dots ,n+1$ to be obtained for any $n$ ($\text{}={\mathbf{nplus1}}-1$). Note that e02aec is used to evaluate the interpolating polynomial. The program is self-starting in that any number of datasets can be supplied.

10.1Program Text

Program Text (e02afce.c)

10.2Program Data

Program Data (e02afce.d)

10.3Program Results

Program Results (e02afce.r)