Integer, Intent (In)	::	np1, ia1, la, iadif1, ladif
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	xmin, xmax, a(la)
Real (Kind=nag_wp), Intent (Out)	::	patm1, adif(ladif)

C Header Interface

#include <nag.h>

void	e02ahf_ (const Integer np1, const double xmin, const double xmax, const double a[], const Integer ia1, const Integer la, double patm1, double adif[], const Integer iadif1, const Integer ladif, Integer *ifail)

The routine may be called by the names e02ahf or nagf_fit_dim1_cheb_deriv.

3 Description

e02ahf 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 routine 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 e02akf.

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 routine 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: $np1$ – Integer Input

On entry:

n + 1

, where

n

is the degree of the given polynomial

p (x)

. Thus np1 is the number of coefficients in this polynomial.

Constraint:

np1 \geq 1

2: $xmin$ – Real (Kind=nag_wp) Input

3: $xmax$ – Real (Kind=nag_wp) 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: $a (la)$ – Real (Kind=nag_wp) array Input

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.

Unchanged on exit, but see adif, below.

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

, the value of ia1 must be

3

. See also Section 9.

Constraint:

ia1 \geq 1

6: $la$ – Integer Input

On entry: the dimension of the array a as declared in the (sub)program from which e02ahf is called.

Constraints:

$la \geq 1 + (np1 - 1) \times ia1$ .

7: $patm1$ – Real (Kind=nag_wp) Output

On exit: the value of

p (x_{\min})

. If this value is passed to the integration routine e02ajf with the coefficients of

q (x)

, the original polynomial

p (x)

is recovered, including its constant coefficient.

8: $adif (ladif)$ – Real (Kind=nag_wp) array Output

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 + 1

of adif contains the coefficient

{\bar{a}}_{i}

, for

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

. Additionally, element

n \times iadif1 + 1

is set to zero. A call of the routine may have the array name adif the same as a, provided that note is taken of the order in which elements are overwritten, when choosing the starting elements and increments ia1 and iadif1, i.e., the coefficients

a_{0}, a_{1}, \dots, a_{i - 1}

must be intact after coefficient

{\bar{a}}_{i}

is stored. In particular, it is possible to overwrite the

a_{i}

completely by having

ia1 = iadif1

, and the actual arrays for a and adif identical.

9: $iadif1$ – 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 (1), adif (4), adif (7), \dots

, the value of iadif1 must be

3

. See Section 9.

Constraint:

iadif1 \geq 1

10: $ladif$ – Integer Input

On entry: the dimension of the array adif as declared in the (sub)program from which e02ahf is called.

Constraints:

$ladif \geq 1 + (np1 - 1) \times iadif1$ .

11: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $ia1 = ⟨ value ⟩$ .
Constraint: $ia1 \geq 1$ .

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

On entry, $la = ⟨ value ⟩$ , $np1 = ⟨ value ⟩$ and $ia1 = ⟨ value ⟩$ .
Constraint: $la > (np1 - 1) \times ia1$ .

On entry, $ladif = ⟨ value ⟩$ , $np1 = ⟨ value ⟩$ and $iadif1 = ⟨ value ⟩$ .
Constraint: $ladif > (np1 - 1) \times iadif1$ .

On entry, $np1 = ⟨ value ⟩$ .
Constraint: $np1 \geq 1$ .

On entry, $xmax = ⟨ value ⟩$ and $xmin = ⟨ value ⟩$ .
Constraint: $xmax > xmin$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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.

e02ahf 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 suppliedin DATA statements. Normally a program would first read in or generate data and compute the fitted polynomial.)