Integer, Intent (In)	::	lx, n, s
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(lx), xmin, xmax, c(n)
Real (Kind=nag_wp), Intent (Out)	::	res(lx)

C Header Interface

#include <nag.h>

void	c06dcf_ (const double x[], const Integer lx, const double xmin, const double xmax, const double c[], const Integer n, const Integer s, double res[], Integer ifail)

The routine may be called by the names c06dcf or nagf_sum_chebyshev.

3 Description

c06dcf evaluates, at each point in a given set

X

, the sum of a Chebyshev series of one of three forms according to the value of the parameter s:

$s = 1$ :	$0.5 c_{1} + \sum_{j = 2}^{n} c_{j} T_{j - 1} (\bar{x})$
$s = 2$ :	$0.5 c_{1} + \sum_{j = 2}^{n} c_{j} T_{2 j - 2} (\bar{x})$
$s = 3$ :	$\sum_{j = 1}^{n} c_{j} T_{2 j - 1} (\bar{x})$

where

\bar{x}

lies in the range

- 1.0 \leq \bar{x} \leq 1.0

. Here

T_{r} (x)

is the Chebyshev polynomial of order

r

\bar{x}

, defined by

\cos (r y)

where

\cos y = \bar{x}

It is assumed that the independent variable

\bar{x}

in the interval

[- 1.0, + 1.0]

was obtained from your original variable

x \in X

, a set of real numbers in the interval

[x_{\min}, x_{\max}]

, by the linear transformation

\bar{x} = \frac{2 x - (x_{\max} + x_{\min})}{x_{\max} - x_{\min}} .

The method used is based upon a three-term recurrence relation; for details see Clenshaw (1962).

The coefficients

c_{j}

are normally generated by other routines, for example they may be those returned by the interpolation routine e01aef (in vector a), by a least squares fitting routine in Chapter E02, or as the solution of a boundary value problem by d02jaf, d02jbf or d02uef.

4 References

Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO

5 Arguments

1: $x (lx)$ – Real (Kind=nag_wp) array Input

On entry:

x \in X

, the set of arguments of the series.

Constraint:

xmin \leq x (i) \leq xmax

, for

i = 1, 2, \dots, lx

2: $lx$ – Integer Input

On entry: the number of evaluation points in

X

Constraint:

lx \geq 1

3: $xmin$ – Real (Kind=nag_wp) Input

4: $xmax$ – 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:

xmin < xmax

5: $c (n)$ – Real (Kind=nag_wp) array Input

On entry:

c (j)

must contain the coefficient

c_{j}

of the Chebyshev series, for

j = 1, 2, \dots, n

6: $n$ – Integer Input

On entry:

n

, the number of terms in the series.

Constraint:

n \geq 1

7: $s$ – Integer Input

On entry: determines the series (see Section 3).

$s = 1$: The series is general.
$s = 2$: The series is even.
$s = 3$: The series is odd.

Constraint:

s = 1

2

3

8: $res (lx)$ – Real (Kind=nag_wp) array Output

On exit: the Chebyshev series evaluated at the set of points

X

9: $ifail$ – 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, $lx = ⟨ value ⟩$ .
Constraint: $lx \geq 1$ .

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

$ifail = 3$: On entry, $s = ⟨ value ⟩$ .
Constraint: $s = 1$ , $2$ or $3$ .

$ifail = 4$: On entry, $xmax = ⟨ value ⟩$ and $xmin = ⟨ value ⟩$ .
Constraint: $xmin < xmax$ .

$ifail = 5$: On entry, element $x (⟨ value ⟩) = ⟨ value ⟩$ , $xmin = ⟨ value ⟩$ and $xmax = ⟨ value ⟩$ .
Constraint: $xmin \leq x (i) \leq xmax$ , for all $i$ .

$ifail = - 99$: 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 may be a loss of significant figures due to cancellation between terms. However, provided that

n

is not too large, c06dcf yields results which differ little from the best attainable for the available machine precision.

8 Parallelism and Performance

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

c06dcf is not threaded in any implementation.

9 Further Comments

The time taken increases with

n

c06dcf has been prepared in the present form to complement a number of integral equation solving routines which use Chebyshev series methods, e.g., d05aaf and d05abf.

10 Example

This example evaluates

0.5 + T_{1} (x) + 0.5 T_{2} (x) + 0.25 T_{3} (x)

at the points

X = [0.5, 1.0, - 0.2]

c06dc: FL CL CPP AD PY MB

NAG FL Interfacec06dcf (chebyshev)

▸▿ 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

NAG FL Interface
c06dcf (chebyshev)