NAG Library Routine Document

C06DBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

C06DBF returns the value of the sum of a Chebyshev series through the routine name.

2 Specification

FUNCTION C06DBF (

X, C, N, S)

REAL (KIND=nag_wp) C06DBF

INTEGER	N, S
REAL (KIND=nag_wp)	X, C(N)

3 Description

C06DBF evaluates the sum of a Chebyshev series of one of three forms according to the value of the parameter S:

\begin{array}{l} S = 1 : & 0.5 c_{1} + \sum_{j = 2}^{n} c_{j} T_{j - 1} (x), \\ S = 2 : & 0.5 c_{1} + \sum_{j = 2}^{n} c_{j} T_{2 j - 2} (x), \\ S = 3 : & \sum_{j = 1}^{n} c_{j} T_{2 j - 1} (x) \end{array}

where

x

lies in the range

- 1.0 \leq x \leq 1.0

. Here

T_{r} (x)

is the Chebyshev polynomial of order

r

x

, defined by

\cos (r y)

where

\cos y = x

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

4 References

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

5 Parameters

1: X – REAL (KIND=nag_wp)Input: On entry: the argument $x$ of the series.
Constraint: $- 1.0 \leq X \leq 1.0$ .
2: C(N) – REAL (KIND=nag_wp) arrayInput: On entry: $C (j)$ must contain the coefficient $c_{j}$ of the Chebyshev series, for $j = 1, 2, \dots, n$ .
3: N – INTEGERInput: On entry: $n$ , the number of terms in the series.
4: S – INTEGERInput: On entry: must have the value $1$ , $2$ or $3$ according to whether the series is general, even or odd respectively (see Section 3). For all other values of S, the routine behaves as though $S = 2$ .

6 Error Indicators and Warnings

If an error is detected in an input parameter C06DBF will act as if a soft noisy exit has been requested (see Section 3.3.4 in the Essential Introduction).

7 Accuracy

There may be a loss of significant figures due to cancellation between terms. However, provided that

n

is not too large, C06DBF yields results which differ little from the best attainable for the available machine precision.

8 Further Comments

The time taken increases with

n

C06DBF 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.