NAG Library Function Document

nag_1d_cheb_eval2 (e02akc)

+− 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

1 Purpose

nag_1d_cheb_eval2 (e02akc) evaluates a polynomial from its Chebyshev series representation, allowing an arbitrary index increment for accessing the array of coefficients.

2 Specification

#include <nag.h>

#include <nage02.h>

void	nag_1d_cheb_eval2 (Integer n, double xmin, double xmax, const double a[], Integer ia1, double x, double result, NagError fail)

3 Description

If supplied with the coefficients

a_{i}

, for

i = 0, 1, \dots, n

, of a polynomial

p (\bar{x})

of degree

n

, where

p (\bar{x}) = \frac{1}{2} a_{0} + a_{1} T_{1} (\bar{x}) + \dots + a_{n} T_{n} (\bar{x}),

nag_1d_cheb_eval2 (e02akc) returns the value of

p (\bar{x})

at a user-specified value of the variable

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 independent 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}} .

The coefficients

a_{i}

may be supplied in the array a, with any increment between the indices of array elements which contain successive coefficients. This enables the function to be used in surface fitting and other applications, in which the array might have two or more dimensions.

The method employed is based on the three-term recurrence relation due to Clenshaw (see Clenshaw (1955)), with modifications due to Reinsch and Gentleman (see Gentleman (1969)). For further details of the algorithm and its use see Cox (1973) and Cox and Hayes (1973).

4 References

Clenshaw C W (1955) A note on the summation of Chebyshev series Math. Tables Aids Comput. 9 118–120

Cox M G (1973) A data-fitting package for the non-specialist user NPL Report NAC 40 National Physical Laboratory

Cox M G and Hayes J G (1973) Curve fitting: a guide and suite of algorithms for the non-specialist user NPL Report NAC26 National Physical Laboratory

Gentleman W M (1969) An error analysis of Goertzel's (Watt's) method for computing Fourier coefficients Comput. J. 12 160–165

5 Arguments

1: n – IntegerInput

On entry:

n

, the degree of the given polynomial

p (\bar{x})

Constraint:

n \geq 0

2: xmin – doubleInput

3: 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

4: a[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array a must be at least

((n + 1 - 1) \times ia1 + 1)

On entry: the Chebyshev coefficients of the polynomial

p (\bar{x})

. Specifically, element

i \times ia1

must contain the coefficient

a_{i}

, for

i = 0, 1, \dots, n

. Only these

n + 1

elements will be accessed.

5: ia1 – IntegerInput

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 [0], a [3], a [6], \dots

, then the value of ia1 must be

3

Constraint:

ia1 \geq 1

6: x – doubleInput

On entry: the argument

x

at which the polynomial is to be evaluated.

Constraint:

xmin \leq x \leq xmax

7: result – double *Output

On exit: the value of the polynomial

p (\bar{x})

8: 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, $ia1 = ⟨value⟩$ .
Constraint: $ia1 \geq 1$ .
On entry, $n + 1 = ⟨value⟩$ .
Constraint: $n + 1 \geq 1$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
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: $xmax > xmin$ .
NE_REAL_3: On entry, x does not lie in $[xmin, xmax]$ : $x = ⟨value⟩$ , $xmin = ⟨value⟩$ and $xmax = ⟨value⟩$ .

7 Accuracy

The rounding errors are such that the computed value of the polynomial is exact for a slightly perturbed set of coefficients

a_{i} + δ a_{i}

. The ratio of the sum of the absolute values of the

δ a_{i}

to the sum of the absolute values of the

a_{i}

is less than a small multiple of

(n + 1) \times machine precision

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken is approximately proportional to

n + 1

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 polynomial at

4

equally spaced points over the interval. (For the purposes of this example, xmin, xmax and the Chebyshev coefficients are supplied . Normally a program would first read in or generate data and compute the fitted polynomial.)