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# NAG Toolbox: nag_fit_1dcheb_glp (e02af)

## Purpose

nag_fit_1dcheb_glp (e02af) 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.

## Syntax

[a, ifail] = e02af(f, 'nplus1', nplus1)
[a, ifail] = nag_fit_1dcheb_glp(f, 'nplus1', nplus1)

## Description

nag_fit_1dcheb_glp (e02af) computes the coefficients aj${a}_{\mathit{j}}$, for j = 1,2,,n + 1$\mathit{j}=1,2,\dots ,n+1$, in the Chebyshev series
 (1/2)a1T0(x) + a2T1(x) + a3T2(x) + ⋯ + an + 1Tn(x), $12a1T0(x-)+a2T1(x-)+a3T2(x-)+⋯+an+1Tn(x-),$
which interpolates the data fr${f}_{r}$ at the points
 xr = cos((r − 1)π / n) ,  r = 1,2, … ,n + 1. $x-r=cos((r-1)π/n) , r=1,2,…,n+1.$
Here Tj(x)${T}_{j}\left(\stackrel{-}{x}\right)$ denotes the Chebyshev polynomial of the first kind of degree j$j$ with argument x$\stackrel{-}{x}$. 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$x$ say, the values of x$x$ at which the data fr${f}_{r}$ are to be provided are
 xr = (1/2)(xmax − xmin)cos(π(r − 1) / n) + (1/2)(xmax + xmin),  r = 1,2, … ,n + 1 $xr=12(xmax-xmin)cos(π(r-1)/n)+12(xmax+xmin), r=1,2,…,n+1$
where xmax${x}_{\mathrm{max}}$ and xmin${x}_{\mathrm{min}}$ are respectively the upper and lower ends of the range of x$x$ over which you wish to interpolate.
Truncation of the resulting series after the term involving ai + 1${a}_{i+1}$, say, yields a least squares approximation to the data. This approximation, p(x)$p\left(\stackrel{-}{x}\right)$, say, is the polynomial of degree i$i$ which minimizes
 (1/2)ε12 + ε22 + ε32 + ⋯ + εn2 + (1/2)εn + 12, $12ε12+ε22+ε32+⋯+εn2+12εn+12,$
where the residual εr = p(xr)fr${\epsilon }_{\mathit{r}}=p\left({\stackrel{-}{x}}_{\mathit{r}}\right)-{f}_{\mathit{r}}$, for r = 1,2,,n + 1$\mathit{r}=1,2,\dots ,n+1$.
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) and Cox and Hayes (1973).
Subsequent evaluation of the computed polynomial, perhaps truncated after an appropriate number of terms, should be carried out using nag_fit_1dcheb_eval (e02ae).

## 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 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

## Parameters

### Compulsory Input Parameters

1:     f(nplus1) – double array
nplus1, the dimension of the array, must satisfy the constraint nplus12${\mathbf{nplus1}}\ge 2$.
For r = 1,2,,n + 1$r=1,2,\dots ,n+1$, f(r)${\mathbf{f}}\left(r\right)$ must contain fr${f}_{r}$ the value of the dependent variable (ordinate) corresponding to the value
 xr = cos(π(r − 1) / n) $x-r=cos(π(r-1)/n)$
of the independent variable (abscissa) x$\stackrel{-}{x}$, or equivalently to the value
 x(r) = (1/2)(xmax − xmin)cos(π(r − 1) / n) + (1/2)(xmax + xmin) $x(r)=12(xmax-xmin)cos(π(r-1)/n)+12(xmax+xmin)$
of your original variable x$x$. Here xmax${x}_{\mathrm{max}}$ and xmin${x}_{\mathrm{min}}$ are respectively the upper and lower ends of the range over which you wish to interpolate.

### Optional Input Parameters

1:     nplus1 – int64int32nag_int scalar
Default: The dimension of the array f.
The number n + 1$n+1$ of data points (one greater than the degree n$n$ of the interpolating polynomial).
Constraint: nplus12${\mathbf{nplus1}}\ge 2$.

None.

### Output Parameters

1:     a(nplus1) – double array
a(j)${\mathbf{a}}\left(\mathit{j}\right)$ is the coefficient aj${a}_{\mathit{j}}$ in the interpolating polynomial, for j = 1,2,,n + 1$\mathit{j}=1,2,\dots ,n+1$.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, nplus1 < 2${\mathbf{nplus1}}<2$.

## Accuracy

The rounding errors committed are such that the computed coefficients are exact for a slightly perturbed set of ordinates fr + δfr${f}_{r}+\delta {f}_{r}$. The ratio of the sum of the absolute values of the δfr$\delta {f}_{r}$ to the sum of the absolute values of the fr${f}_{r}$ is less than a small multiple of (n + 1)ε$\left(n+1\right)\epsilon$, where ε$\epsilon$ is the machine precision.

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

## Example

```function nag_fit_1dcheb_glp_example
f = [2.7182;
2.5884;
2.2456;
1.7999;
1.362;
1;
0.7341;
0.5555;
0.4452;
0.3863;
0.3678];
[a, ifail] = nag_fit_1dcheb_glp(f)
```
```

a =

2.5320
1.1303
0.2715
0.0443
0.0055
0.0005
0.0000
-0.0000
-0.0000
0.0000
-0.0000

ifail =

0

```
```function e02af_example
f = [2.7182;
2.5884;
2.2456;
1.7999;
1.362;
1;
0.7341;
0.5555;
0.4452;
0.3863;
0.3678];
[a, ifail] = e02af(f)
```
```

a =

2.5320
1.1303
0.2715
0.0443
0.0055
0.0005
0.0000
-0.0000
-0.0000
0.0000
-0.0000

ifail =

0

```

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