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 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: $n$ – Integer Input

On entry:

n

, the degree of the given polynomial

p (x)

Constraint:

n \geq 0

2: $xmin$ – double Input

3: $xmax$ – double Input

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 double Input

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

(1 + (1 - 1) \times ia1 1 + n \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: $ia1$ – Integer Input

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: $patm1$ – double * Output

On exit: the value of

p (x_{\min})

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

q (x)

, the original polynomial

p (x)

is recovered, including its constant coefficient.

7: $adif [\dim]$ – double Output

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

(1 + (1 - 1) \times iadif1 1 + n \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: $iadif1$ – Integer Input

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

, 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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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, $iadif1 = ⟨ value ⟩$ .
Constraint: $iadif1 \geq 1$ .

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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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

Background information to multithreading can be found in the Multithreading documentation.

e02ahc is not threaded in any implementation.

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.)