e02akf evaluates a polynomial from its Chebyshev series representation, allowing an arbitrary index increment for accessing the array of coefficients.

2 Specification

Fortran Interface

Subroutine e02akf (

np1, xmin, xmax, a, ia1, la, x, result, ifail)

Integer, Intent (In)	::	np1, ia1, la
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	xmin, xmax, a(la), x
Real (Kind=nag_wp), Intent (Out)	::	result

C Header Interface

#include <nag.h>

void	e02akf_ (const Integer np1, const double xmin, const double xmax, const double a[], const Integer ia1, const Integer la, const double x, double result, Integer ifail)

The routine may be called by the names e02akf or nagf_fit_dim1_cheb_eval2.

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}),

e02akf 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 routine 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: $np1$ – Integer Input

On entry:

n + 1

, where

n

is the degree of the given polynomial

p (\bar{x})

Constraint:

np1 \geq 1

2: $xmin$ – Real (Kind=nag_wp) Input

3: $xmax$ – Real (Kind=nag_wp) Input

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: $a (la)$ – Real (Kind=nag_wp) array Input

On entry: the Chebyshev coefficients of the polynomial

p (\bar{x})

. Specifically, element

i \times ia1 + 1

must contain the coefficient

a_{i}

, for

i = 0, 1, \dots, n

. Only these

n + 1

elements will be accessed.

5: $ia1$ – Integer Input

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 (1), a (4), a (7), \dots

, the value of ia1 must be

3

Constraint:

ia1 \geq 1

6: $la$ – Integer Input

On entry: the dimension of the array a as declared in the (sub)program from which e02akf is called.

Constraints:

$la \geq (np1 - 1) \times ia1 + 1$ .

7: $x$ – Real (Kind=nag_wp) Input

On entry: the argument

x

at which the polynomial is to be evaluated.

Constraint:

xmin \leq x \leq xmax

8: $result$ – Real (Kind=nag_wp) Output

On exit: the value of the polynomial

p (\bar{x})

9: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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:

$ifail = 1$: On entry, $ia1 = ⟨ value ⟩$ .
Constraint: $ia1 \geq 1$ .

On entry, $la = ⟨ value ⟩$ , $np1 = ⟨ value ⟩$ and $ia1 = ⟨ value ⟩$ .
Constraint: $la > (np1 - 1) \times ia1$ .

On entry, $np1 = ⟨ value ⟩$ .
Constraint: $np1 \geq 1$ .

On entry, $xmax = ⟨ value ⟩$ and $xmin = ⟨ value ⟩$ .
Constraint: $xmax > xmin$ .

$ifail = 2$: On entry, x does not lie in $[xmin, xmax]$ : $x = ⟨ value ⟩$ , $xmin = ⟨ value ⟩$ and $xmax = ⟨ value ⟩$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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

Background information to multithreading can be found in the Multithreading documentation.

e02akf is not threaded in any implementation.

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 in DATA statements. Normally a program would first read in or generate data and compute the fitted polynomial.)