NAG Library Routine Document

E02AEF

+− 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

E02AEF evaluates a polynomial from its Chebyshev series representation.

2 Specification

SUBROUTINE E02AEF (

NPLUS1, A, XCAP, P, IFAIL)

INTEGER	NPLUS1, IFAIL
REAL (KIND=nag_wp)	A(NPLUS1), XCAP, P

3 Description

E02AEF evaluates the polynomial

\frac{1}{2} a_{1} T_{0} (\bar{x}) + a_{2} T_{1} (\bar{x}) + a_{3} T_{2} (\bar{x}) + \dots + a_{n + 1} T_{n} (\bar{x})

for any value of

\bar{x}

satisfying

- 1 \leq \bar{x} \leq 1

. Here

T_{j} (\bar{x})

denotes the Chebyshev polynomial of the first kind of degree

j

with argument

\bar{x}

. The value of

n

is prescribed by you.

In practice, the variable

\bar{x}

will usually have been obtained from an original variable

x

, where

x_{\min} \leq x \leq x_{\max}

and

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

Note that this form of the transformation should be used computationally rather than the mathematical equivalent

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

since the former guarantees that the computed value of

\bar{x}

differs from its true value by at most

4 ε

, where

ε

is the machine precision, whereas the latter has no such guarantee.

The method employed is based on the three-term recurrence relation due to Clenshaw (1955), with modifications to give greater numerical stability due to Reinsch and Gentleman (see Gentleman (1969)).

For further details of the algorithm and its use see Cox (1974) 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 (1974) A data-fitting package for the non-specialist user Software for Numerical Mathematics (ed D J Evans) Academic Press

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 Parameters

1: NPLUS1 – INTEGERInput: On entry: the number $n + 1$ of terms in the series (i.e., one greater than the degree of the polynomial).
Constraint: $NPLUS1 \geq 1$ .
2: A(NPLUS1) – REAL (KIND=nag_wp) arrayInput: On entry: $A (i)$ must be set to the value of the $i$ th coefficient in the series, for $i = 1, 2, \dots, n + 1$ .
3: XCAP – REAL (KIND=nag_wp)Input: On entry: $\bar{x}$ , the argument at which the polynomial is to be evaluated. It should lie in the range $- 1$ to $+ 1$ , but a value just outside this range is permitted (see Section 6) to allow for possible rounding errors committed in the transformation from $x$ to $\bar{x}$ discussed in Section 3. Provided the recommended form of the transformation is used, a successful exit is thus assured whenever the value of $x$ lies in the range $x_{\min}$ to $x_{\max}$ .
4: P – REAL (KIND=nag_wp)Output: On exit: the value of the polynomial.
5: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . 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:

$IFAIL = 1$: $ABS (XCAP) > 1.0 + 4 ε$ , where $ε$ is the machine precision. In this case the value of P is set arbitrarily to zero.

$IFAIL = 2$

On entry,

NPLUS1 < 1

7 Accuracy

The rounding errors committed 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 Further Comments

The time taken is approximately proportional to

n + 1

It is expected that a common use of E02AEF will be the evaluation of the polynomial approximations produced by E02ADF and E02AFF.

9 Example

Evaluate at

11

equally-spaced points in the interval

- 1 \leq \bar{x} \leq 1

the polynomial of degree

4

with Chebyshev coefficients,

2.0

0.5

0.25

0.125

0.0625

The example program is written in a general form that will enable a polynomial of degree

n

in its Chebyshev series form to be evaluated at

m