nag_fit_1dcheb_integ (e02aj) determines the coefficients in the Chebyshev series representation of the indefinite integral of a polynomial given in Chebyshev series form.

Syntax

[aintc, ifail] = e02aj(n, xmin, xmax, a, ia1, qatm1, iaint1)

[aintc, ifail] = nag_fit_1dcheb_integ(n, xmin, xmax, a, ia1, qatm1, iaint1)

Description

nag_fit_1dcheb_integ (e02aj) forms the polynomial which is the indefinite integral of a given polynomial. Both the original polynomial and its integral are represented in Chebyshev series form. If supplied with 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

a_{i}^{'}

, for

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

, of the polynomial

q (x)

of degree

n + 1

, where

q (x) = \frac{1}{2} a_{0}^{'} + a_{1}^{'} T_{1} (\bar{x}) + \dots + a_{n + 1}^{'} T_{n + 1} (\bar{x}),

and

q (x) = \int p (x) d 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 integral to be with respect to the variable

x

. If the integral with respect to

\bar{x}

is required, set

x_{\max} = 1

and

x_{\min} = - 1

Values of the integral 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 for integrating with respect to

x

. Initially taking

a_{n + 1} = a_{n + 2} = 0

, the function forms successively

a_{i}^{'} = \frac{a_{i - 1} - a_{i + 1}}{2 i} \times \frac{x_{\max} - x_{\min}}{2}, i = n + 1, n, \dots, 1 .

The constant coefficient

a_{0}^{'}

is chosen so that

q (x)

is equal to a specified value, qatm1, at the lower end point of the interval on which it is defined, i.e.,

\bar{x} = - 1

, which corresponds to

x = x_{\min}

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 + 1

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 aintc, 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: $qatm1$ – double scalar

The value that the integrated polynomial is required to have at the lower end point of its interval of definition, i.e., at

\bar{x} = - 1

which corresponds to

x = x_{\min}

. Thus, qatm1 is a constant of integration and will normally be set to zero by you.

7: $iaint1$ – int64int32nag_int scalar

The index increment of aintc. Most frequently the Chebyshev coefficients are required in adjacent elements of aintc, and iaint1 must be set to

1

. However, if, for example, they are to be stored in

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

, then the value of iaint1 must be

3

. See also Further Comments.

Constraint:

iaint1 \geq 1

Optional Input Parameters

None.

Output Parameters

1: $aintc (laint)$ – double array: $laint = 1 + (np1) \times iaint1$ .
The Chebyshev coefficients of the integral $q (x)$ . (The integration is with respect to the variable $x$ , and the constant coefficient is chosen so that $q (x_{\min})$ equals qatm1). Specifically, element $i \times iaint1 + 1$ of aintc contains the coefficient $a_{i}^{'}$ , for $i = 0, 1, \dots, n + 1$ . A call of the function may have the array name aintc the same as a, provided that note is taken of the order in which elements are overwritten when choosing starting elements and increments ia1 and iaint1: i.e., the coefficients, $a_{0}, a_{1}, \dots, a_{i - 2}$ must be intact after coefficient $a_{i}^{'}$ is stored. In particular it is possible to overwrite the $a_{i}$ entirely by having $ia1 = iaint1$ , and the actual array for a and aintc identical.
2: $ifail$ – 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	$iaint1 < 1$ ,
or	$laint \leq np1 \times iaint1$ .

$ifail = - 99$: 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

In general there is a gain in precision in numerical integration, in this case associated with the division by

2 i

in the formula quoted in Description.

Further Comments

The time taken is approximately proportional to

n + 1

The increments ia1, iaint1 are included as arguments to give a degree of flexibility which, for example, allows a polynomial in two variables to be integrated 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 integral of the polynomial from

0.0

2.0

. (For the purpose of this example, xmin, xmax and the Chebyshev coefficients are simply supplied . Normally a program would read in or generate data and compute the fitted polynomial).

Open in the MATLAB editor: e02aj_example

function e02aj_example


fprintf('e02aj example results\n\n');

% Chebyshev coefficients for f(x) on [-0.5,2.5]
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);
qatm1 = 0;
iaint1 = int64(1);

% Get integral coefficients aint
[aint, ifail] = e02aj( ...
                       n, xmin, xmax, a, ia1, qatm1, iaint1);

% Evaluate integral from x=0 to x=2 by subtraction
xa = 0;
xb = 2;
[inta, ifail] = e02ak( ...
                       n+1, xmin, xmax, aint, ia1, xa);
[intb, ifail] = e02ak( ...
                       n+1, xmin, xmax, aint, ia1, xb);

res = intb - inta;

fprintf('Value of definite integral = %7.4f\n',res);

e02aj example results

Value of definite integral =  2.1515

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

Chapter Contents

Chapter Introduction

NAG Toolbox