NAG Library Function Document

nag_sum_cheby_series (c06dcc)

void	nag_sum_cheby_series (const double x[], Integer lx, double xmin, double xmax, const double c[], Integer n, Nag_Series s, double res[], NagError *fail)

3 Description

nag_sum_cheby_series (c06dcc) 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:

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

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_1d_cheb_interp (e01aec) (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_ps_lin_solve (d02uec).

4 References

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

5 Arguments

1: x[lx] – const doubleInput

On entry:

x \in X

, the set of arguments of the series.

Constraint:

xmin \leq x [i - 1] \leq xmax

, for

i = 1, 2, \dots, lx

2: lx – IntegerInput

On entry: the number of evaluation points in

X

Constraint:

lx \geq 1

3: xmin – doubleInput

4: xmax – doubleInput

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] – const doubleInput

On entry:

c [j - 1]

must contain the coefficient

c_{j}

of the Chebyshev series, for

j = 1, 2, \dots, n

6: n – IntegerInput

On entry:

n

, the number of terms in the series.

Constraint:

n \geq 1

7: s – Nag_SeriesInput

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

$s = Nag_SeriesGeneral$: The series is general.
$s = Nag_SeriesEven$: The series is even.
$s = Nag_SeriesOdd$: The series is odd.

Constraint:

s = Nag_SeriesGeneral

Nag_SeriesEven

Nag_SeriesOdd

8: res[lx] – doubleOutput

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

X

9: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $lx = ⟨value⟩$ .
Constraint: $lx \geq 1$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
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.
NE_REAL_2: On entry, $xmax = ⟨value⟩$ and $xmin = ⟨value⟩$ .
Constraint: $xmin < xmax$ .
NE_REAL_3: On entry, element $x [⟨value⟩] = ⟨value⟩$ , $xmin = ⟨value⟩$ and $xmax = ⟨value⟩$ .
Constraint: $xmin \leq x [i] \leq xmax$ , for all $i$ .

7 Accuracy

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

n

is not too large, nag_sum_cheby_series (c06dcc) yields results which differ little from the best attainable for the available machine precision.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken increases with

n

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]

NAG Library Function Documentnag_sum_cheby_series (c06dcc)

+− 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 Library Function Document

nag_sum_cheby_series (c06dcc)