NAG Library Function Document

nag_1d_cheb_intg (e02ajc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

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

, then 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

, then 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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.
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

Not applicable.

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).