NAG C Library Function Document

nag_1d_cheb_intg (e02ajc) determines the coefficients in the Chebyshev series representation of the indefinite integral of a polynomial given in Chebyshev series form.

2

Specification

#include <nag.h>

#include <nage02.h>

void	nag_1d_cheb_intg (Integer n, double xmin, double xmax, const double a[], Integer ia1, double qatm1, double aintc[], Integer iaint1, NagError *fail)

3

Description

nag_1d_cheb_intg (e02ajc) 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_1d_cheb_eval2 (e02akc).

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}

4

References

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

5

Arguments

1: $n$ – IntegerInput

On entry:

n

, the degree of the given polynomial

p (x)

Constraint:

n \geq 0

2: $xmin$ – doubleInput

3: $xmax$ – doubleInput

On entry: 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: $a [\dim]$ – const doubleInput

Note: the dimension, dim, of the array a must be at least

(1 + (n + 1 - 1) \times ia1)

On entry: 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.

5: $ia1$ – IntegerInput

On entry: 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 [0], a [3], a [6], \dots

, the value of ia1 must be

3

. See also Section 9.

Constraint:

ia1 \geq 1

6: $qatm1$ – doubleInput

On entry: 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: $aintc [\dim]$ – doubleOutput

Note: the dimension, dim, of the array aintc must be at least

(1 + (n + 1) \times iaint1)

On exit: 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

of aintc contains the coefficient

a_{i}^{'}

, for

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

8: $iaint1$ – IntegerInput

On entry: 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 [0], aintc [3], aintc [6], \dots

, the value of iaint1 must be

3

. See also Section 9.

Constraint:

iaint1 \geq 1

9: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $ia1 = 〈value〉$ .
Constraint: $ia1 \geq 1$ .

On entry, $iaint1 = 〈value〉$ .
Constraint: $iaint1 \geq 1$ .

On entry, $n + 1 = 〈value〉$ .
Constraint: $n + 1 \geq 1$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_2: On entry, $xmax = 〈value〉$ and $xmin = 〈value〉$ .
Constraint: $xmax > xmin$ .

7

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 Section 3.

8

Parallelism and Performance

nag_1d_cheb_intg (e02ajc) is not threaded in any implementation.

9

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.

10

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).