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

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 functions, for example they may be those returned by the interpolation function nag_interp_1d_cheb (e01ae) (in vector a), by a least squares fitting function in Chapter E02, or as the solution of a boundary value problem by nag_ode_bvp_coll_nth (d02ja), nag_ode_bvp_coll_sys (d02jb) or nag_ode_bvp_ps_lin_solve (d02ue).

References

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

Parameters

Compulsory Input Parameters

1: $x (lx)$ – double array

x \in X

, the set of arguments of the series.

Constraint:

xmin \leq x (i) \leq xmax

, for

i = 1, 2, \dots, lx

2: $xmin$ – double scalar

3: $xmax$ – double scalar

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

4: $c (n)$ – double array

c (j)

must contain the coefficient

c_{j}

of the Chebyshev series, for

j = 1, 2, \dots, n

5: $s$ – int64int32nag_int scalar

Determines the series (see Description).

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

Constraint:

s = 1

2

3

Optional Input Parameters

1: $lx$ – int64int32nag_int scalar: Default: the dimension of the array x.
The number of evaluation points in $X$ .

Constraint: $lx \geq 1$ .
2: $n$ – int64int32nag_int scalar: Default: the dimension of the array c.
$n$ , the number of terms in the series.

Constraint: $n \geq 1$ .

Output Parameters

1: $res (lx)$ – double array: The Chebyshev series evaluated at the set of points $X$ .
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $lx \geq 1$ .

$ifail = 2$: Constraint: $n \geq 1$ .

$ifail = 3$: Constraint: $s = 1$ , $2$ or $3$ .

$ifail = 4$: Constraint: $xmin < xmax$ .

$ifail = 5$: Constraint: $xmin \leq x (i) \leq xmax$ , for all $i$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

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

n

is not too large, nag_sum_chebyshev (c06dc) yields results which differ little from the best attainable for the available machine precision.

Further Comments

The time taken increases with

n

nag_sum_chebyshev (c06dc) has been prepared in the present form to complement a number of integral equation solving functions which use Chebyshev series methods, e.g., nag_inteq_fredholm2_split (d05aa) and nag_inteq_fredholm2_smooth (d05ab).

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]

Open in the MATLAB editor: c06dc_example

function c06dc_example


fprintf('c06dc example results\n\n');

x = [0.5, 1.0, -0.2];
xmin = -1;
xmax =  1;
s = int64(1);
c = [1.0, 1.0, 0.5, 0.25];

[res, ifail] = c06dc(x, xmin, xmax, c, s);
fprintf('\nSums: \n');
disp(res);

c06dc example results


Sums: 
    0.5000
    2.2500
   -0.0180

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox