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

Syntax

[patm1, adif, ifail] = e02ah(n, xmin, xmax, a, ia1, iadif1)

[patm1, adif, ifail] = nag_fit_1dcheb_deriv(n, xmin, xmax, a, ia1, iadif1)

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_{i}

, for

i = 0, 1, \dots, n

, of a polynomial

p (x)

of degree

n

, where

p (x) = \frac{1}{2} a_{0} + a_{1} T_{1} (\bar{x}) + \dots + a_{n} T_{n} (\bar{x})

the function returns the coefficients

{\bar{a}}_{i}

, for

i = 0, 1, \dots, n - 1

, of the polynomial

q (x)

of degree

n - 1

, where

q (x) = \frac{d p (x)}{d x} = \frac{1}{2} {\bar{a}}_{0} + {\bar{a}}_{1} T_{1} (\bar{x}) + \dots + {\bar{a}}_{n - 1} T_{n - 1} (\bar{x}) .

Here

T_{j} (\bar{x})

denotes the Chebyshev polynomial of the first kind of degree

j

with argument

\bar{x}

. It is assumed that the normalized variable

\bar{x}

in the interval

[- 1, + 1]

was obtained from your original variable

x

in the interval

[x_{\min}, x_{\max}]

by the linear transformation

\bar{x} = \frac{2 x - (x_{\max} + x_{\min})}{x_{\max} - x_{\min}}

and that you require the derivative to be with respect to the variable

x

. If the derivative with respect to

\bar{x}

is required, set

x_{\max} = 1

and

x_{\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

{\bar{a}}_{n + 1} = {\bar{a}}_{n} = 0

, the function forms successively

{\bar{a}}_{i - 1} = {\bar{a}}_{i + 1} + \frac{2}{x_{\max} - x_{\min}} 2 i a_{i}, i = n, n - 1, \dots, 1 .

References

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

Parameters

Compulsory Input Parameters

1: $n$ – int64int32nag_int scalar

n

, the degree of the given polynomial

p (x)

Constraint:

n \geq 0

2: $xmin$ – double scalar

3: $xmax$ – double scalar

The lower and upper end points respectively of the interval

[x_{\min}, x_{\max}]

. The Chebyshev series representation is in terms of the normalized variable

\bar{x}

, where

\bar{x} = \frac{2 x - (x_{\max} + x_{\min})}{x_{\max} - x_{\min}} .

Constraint:

xmax > xmin

4: $a (la)$ – double array

la, the dimension of the array, must satisfy the constraint

la \geq 1 + (np1 - 1) \times ia1

The Chebyshev coefficients of the polynomial

p (x)

. Specifically, element

i \times ia1

of a must contain the coefficient

a_{i}

, for

i = 0, 1, \dots, n

. Only these

n + 1

elements will be accessed.

Unchanged on exit, but see adif, below.

5: $ia1$ – 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

a (1), a (4), a (7), \dots

, then the value of ia1 must be

3

. See also Further Comments.

Constraint:

ia1 \geq 1

6: $iadif1$ – 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

adif (1), adif (4), adif (7), \dots

, then the value of iadif1 must be

3

. See Further Comments.

Constraint:

iadif1 \geq 1

Optional Input Parameters

None.

Output Parameters

1: $patm1$ – double scalar: The value of $p (x_{\min})$ . If this value is passed to the integration function nag_fit_1dcheb_integ (e02aj) with the coefficients of $q (x)$ , then the original polynomial $p (x)$ is recovered, including its constant coefficient.
2: $adif (ladif)$ – double array: $ladif = 1 + (np1 - 1) \times iadif1$ .
The Chebyshev coefficients of the derived polynomial $q (x)$ . (The differentiation is with respect to the variable $x$ .) Specifically, element $i \times iadif1 + 1$ of adif contains the coefficient ${\bar{a}}_{i}$ , for $i = 0, 1, \dots, n - 1$ . Additionally, element $n \times iadif1 + 1$ 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 ${\bar{a}}_{i}$ is stored. In particular, it is possible to overwrite the $a_{i}$ completely by having $ia1 = iadif1$ , and the actual arrays for a and adif identical.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$np1 < 1$ ,
or	$xmax \leq xmin$ ,
or	$ia1 < 1$ ,
or	$la \leq (np1 - 1) \times ia1$ ,
or	$iadif1 < 1$ ,
or	$ladif \leq (np1 - 1) \times iadif1$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

There is always a loss of precision in numerical differentiation, in this case associated with the multiplication by

2 i

in the formula quoted in Description.

Further Comments

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

[- 0.5, 2.5]

. 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.)

Open in the MATLAB editor: e02ah_example

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);
iadif1 = int64(1);

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

% Get second derivative
[patm1, adif2, ifail] = e02ah( ...
                               n, xmin, xmax, adif, ia1, iadif1);

% 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( ...
                             n-1, xmin, xmax, adif, iadif1, x(i));
  [fit2(i), ifail] = e02ak( ...
                             n-2, xmin, xmax, adif2, iadif1, x(i));
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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox