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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_fit_1dcheb_deriv (e02ah)

## Purpose

nag_fit_1dcheb_deriv (e02ah) determines the coefficients in the Chebyshev series representation of the derivative of a polynomial given in Chebyshev series form.

## Description

nag_fit_1dcheb_deriv (e02ah) forms the polynomial which is the derivative of a given polynomial. Both the original polynomial and its derivative are represented in Chebyshev series form. Given the coefficients ${a}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n$, of a polynomial $p\left(x\right)$ of degree $n$, where
 $px=12a0+a1T1x-+⋯+anTnx-$
the function returns the coefficients ${\stackrel{-}{a}}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n-1$, of the polynomial $q\left(x\right)$ of degree $n-1$, where
 $qx=dpx dx =12a-0+a-1T1x-+⋯+a-n-1Tn-1x-.$
Here ${T}_{j}\left(\stackrel{-}{x}\right)$ denotes the Chebyshev polynomial of the first kind of degree $j$ with argument $\stackrel{-}{x}$. It is assumed that the normalized variable $\stackrel{-}{x}$ in the interval $\left[-1,+1\right]$ was obtained from your original variable $x$ in the interval $\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]$ by the linear transformation
 $x-=2x-xmax+xmin xmax-xmin$
and that you require the derivative to be with respect to the variable $x$. If the derivative with respect to $\stackrel{-}{x}$ is required, set ${x}_{\mathrm{max}}=1$ and ${x}_{\mathrm{min}}=-1$.
Values of the derivative can subsequently be computed, from the coefficients obtained, by using nag_fit_1dcheb_eval2 (e02ak).
The method employed is that of Chebyshev series (see Chapter 8 of Modern Computing Methods (1961)), modified to obtain the derivative with respect to $x$. Initially setting ${\stackrel{-}{a}}_{n+1}={\stackrel{-}{a}}_{n}=0$, the function forms successively
 $a-i-1=a-i+1+2xmax-xmin 2iai, i=n,n-1,…,1.$

## References

Modern Computing Methods (1961) Chebyshev-series NPL Notes on Applied Science 16 (2nd Edition) HMSO

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
$n$, the degree of the given polynomial $p\left(x\right)$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{xmin}$ – double scalar
3:     $\mathrm{xmax}$ – double scalar
The lower and upper end points respectively of the interval $\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]$. The Chebyshev series representation is in terms of the normalized variable $\stackrel{-}{x}$, where
 $x-=2x-xmax+xmin xmax-xmin .$
Constraint: ${\mathbf{xmax}}>{\mathbf{xmin}}$.
4:     $\mathrm{a}\left(\mathit{la}\right)$ – double array
la, the dimension of the array, must satisfy the constraint $\mathit{la}\ge 1+\left(\mathit{np1}-1\right)×{\mathbf{ia1}}$.
The Chebyshev coefficients of the polynomial $p\left(x\right)$. Specifically, element $\mathit{i}×{\mathbf{ia1}}$ of a must contain the coefficient ${a}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n$. Only these $n+1$ elements will be accessed.
Unchanged on exit, but see adif, below.
5:     $\mathrm{ia1}$int64int32nag_int scalar
The index increment of a. Most frequently the Chebyshev coefficients are stored in adjacent elements of a, and ia1 must be set to $1$. However, if for example, they are stored in ${\mathbf{a}}\left(1\right),{\mathbf{a}}\left(4\right),{\mathbf{a}}\left(7\right),\dots \text{}$, then the value of ia1 must be $3$. See also Further Comments.
Constraint: ${\mathbf{ia1}}\ge 1$.
6:     $\mathrm{iadif1}$int64int32nag_int scalar
The index increment of adif. Most frequently the Chebyshev coefficients are required in adjacent elements of adif, and iadif1 must be set to $1$. However, if, for example, they are to be stored in ${\mathbf{adif}}\left(1\right),{\mathbf{adif}}\left(4\right),{\mathbf{adif}}\left(7\right),\dots \text{}$, then the value of iadif1 must be $3$. See Further Comments.
Constraint: ${\mathbf{iadif1}}\ge 1$.

None.

### Output Parameters

1:     $\mathrm{patm1}$ – double scalar
The value of $p\left({x}_{\mathrm{min}}\right)$. If this value is passed to the integration function nag_fit_1dcheb_integ (e02aj) with the coefficients of $q\left(x\right)$, then the original polynomial $p\left(x\right)$ is recovered, including its constant coefficient.
2:     $\mathrm{adif}\left(\mathit{ladif}\right)$ – double array
$\mathit{ladif}=1+\left(\mathit{np1}-1\right)×{\mathbf{iadif1}}$.
The Chebyshev coefficients of the derived polynomial $q\left(x\right)$. (The differentiation is with respect to the variable $x$.) Specifically, element $\mathit{i}×{\mathbf{iadif1}}+1$ of adif contains the coefficient ${\stackrel{-}{a}}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n-1$. Additionally, element $n×{\mathbf{iadif1}}+1$ is set to zero. A call of the function may have the array name adif the same as a, provided that note is taken of the order in which elements are overwritten, when choosing the starting elements and increments ia1 and iadif1, i.e., the coefficients ${a}_{0},{a}_{1},\dots ,{a}_{i-1}$ must be intact after coefficient ${\stackrel{-}{a}}_{i}$ is stored. In particular, it is possible to overwrite the ${a}_{i}$ completely by having ${\mathbf{ia1}}={\mathbf{iadif1}}$, and the actual arrays for a and adif identical.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, $\mathit{np1}<1$, or ${\mathbf{xmax}}\le {\mathbf{xmin}}$, or ${\mathbf{ia1}}<1$, or $\mathit{la}\le \left(\mathit{np1}-1\right)×{\mathbf{ia1}}$, or ${\mathbf{iadif1}}<1$, or $\mathit{ladif}\le \left(\mathit{np1}-1\right)×{\mathbf{iadif1}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

There is always a loss of precision in numerical differentiation, in this case associated with the multiplication by $2i$ in the formula quoted in Description.

The time taken is approximately proportional to $n+1$.
The increments ia1, iadif1 are included as arguments to give a degree of flexibility which, for example, allows a polynomial in two variables to be differentiated with respect to either variable without rearranging the coefficients.

## Example

Suppose a polynomial has been computed in Chebyshev series form to fit data over the interval $\left[-0.5,2.5\right]$. The following program evaluates the first and second derivatives of this polynomial at $4$ equally spaced points over the interval. (For the purposes of this example, xmin, xmax and the Chebyshev coefficients are simply supplied . Normally a program would first read in or generate data and compute the fitted polynomial.)
```function e02ah_example

fprintf('e02ah example results\n\n');

xmin = -0.5; xmax = 2.5;
a = [2.53213  1.13032  0.2715  0.04434   0.00547   0.00054  4e-05];

n = int64(6);
ia1 = int64(1);

% Get first derivative
[patm1, adif, ifail] = e02ah( ...
n, xmin, xmax, a, ia1, iadif1);

% Get second derivative
[patm1, adif2, ifail] = e02ah( ...

% Evaluate polynomial and derivatives from coefficients adif and adif2 at x
m = 21;
dx =  (xmax-xmin)/(m-1);
x = [xmin:dx:xmax];

for i = 1:m
[fit0(i), ifail] = e02ak( ...
n, xmin, xmax, a, ia1, x(i));
[fit1(i), ifail] = e02ak( ...
[fit2(i), ifail] = e02ak( ...
end

sol = [x; fit0; fit1; fit2];
fprintf('     x           p(x)         p''(x)        p''''(x)\n');
fprintf('%9.4f    %9.4f    %9.4f    %9.4f\n',sol(1:4,1:5:21));

fig1 = figure;
plot(x,fit0,x,fit1,x,fit2);
title('Evaluation of Chebyshev Polynomial and its Derivatives');
xlabel('x');
ylabel('p(x),p''(x),p''''(x)');
legend('p(x)', 'p''(x)', 'p''''(x)','Location','NorthWest');

```
```e02ah example results

x           p(x)         p'(x)        p''(x)
-0.5000       0.3679       0.2453       0.1637
0.2500       0.6065       0.4043       0.2696
1.0000       1.0000       0.6667       0.4444
1.7500       1.6487       1.0992       0.7329
2.5000       2.7183       1.8119       1.2056
```