NAG Library Function Document

nag_1d_cheb_deriv (e02ahc)

+− 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_deriv (e02ahc) determines the coefficients in the Chebyshev series representation of the derivative of a polynomial given in Chebyshev series form.

2 Specification

#include <nag.h>

#include <nage02.h>

void	nag_1d_cheb_deriv (Integer n, double xmin, double xmax, const double a[], Integer ia1, double patm1, double adif[], Integer iadif1, NagError fail)

3 Description

nag_1d_cheb_deriv (e02ahc) 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_1d_cheb_eval2 (e02akc).

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 .

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: patm1 – double *Output

On exit: the value of

p (x_{\min})

. If this value is passed to the integration function nag_1d_cheb_intg (e02ajc) with the coefficients of

q (x)

, then the original polynomial

p (x)

is recovered, including its constant coefficient.

7: adif[ $\dim$ ] – doubleOutput

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

(1 + (n + 1 - 1) \times iadif1)

On exit: the Chebyshev coefficients of the derived polynomial

q (x)

. (The differentiation is with respect to the variable

x

.) Specifically, element

i \times iadif1

of adif contains the coefficient

{\bar{a}}_{i}

, for

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

. Additionally, element

n \times iadif1

is set to zero.

8: iadif1 – IntegerInput

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

, then the value of iadif1 must be

3

. See Section 9.

Constraint:

iadif1 \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, $iadif1 = ⟨value⟩$ .
Constraint: $iadif1 \geq 1$ .
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

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 Section 3.

8 Parallelism and Performance

Not applicable.

9 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.

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 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.)