Evaluates the coefficients of a Chebyshev series polynomial

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      - e02 Introduction
      - e02adc (nag_1d_cheb_fit)
      - e02aec (nag_1d_cheb_eval)
      - e02afc (nag_1d_cheb_interp_fit)
      - e02agc (nag_1d_cheb_fit_constr)
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NAG[e02aec] NAG[nag_1d_cheb_eval] - Evaluates the coefficients of a Chebyshev series polynomial

Calling Sequence

e02aec(a, xcap, p, 'nplus1'=nplus1, 'fail'=fail)

nag_1d_cheb_eval(. . .)

Parameters

a - Vector(1..nplus1, datatype=float[8]);

On entry: must be set to the value of the th coefficient in the series, for .

xcap - float;

On entry: , the argument at which the polynomial is to be evaluated. It should lie in the range to , but a value just outside this range is permitted (see Section [Further Comments]) to allow for possible rounding errors committed in the transformation from to discussed in Section [Description]. Provided the recommended form of the transformation is used, a successful exit is thus assured whenever the value of lies in the range to .

p - assignable;

Note: On exit the variable p will have a value of type float.

On exit: the value of the polynomial.

'nplus1'=nplus1 - integer; (optional)

Default value: the first dimension of the array a.

On entry: the number of terms in the series (i.e., one greater than the degree of the polynomial).

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_1d_cheb_eval (e02aec) evaluates a polynomial from its Chebyshev series representation.

Description

nag_1d_cheb_eval (e02aec) evaluates the polynomial

for any value of satisfying . Here denotes the Chebyshev polynomial of the first kind of degree with argument . The value of is prescribed by the user.

In practice, the variable will usually have been obtained from an original variable , where and

x=(((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

x=(2,x,-,x[min],-,x[max])/(x[max],-,x[min])

since the former guarantees that the computed value of differs from its true value by at most , where is the machine precision, whereas the latter has no such guarantee.

The method employed is based upon 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), Cox and Hayes (1973).

	Error Indicators and Warnings
	"NE_INT_ARG_LT" On entry, nplus1 must not be less than 1: . "NE_INVALID_XCAP" On entry, , where is the machine precision. In this case the value of p is set arbitrarily to zero.

	Accuracy
	The rounding errors committed are such that the computed value of the polynomial is exact for a slightly perturbed set of coefficients . The ratio of the sum of the absolute values of the to the sum of the absolute values of the is less than a small multiple of machine precision.

	Further Comments
	The time taken by nag_1d_cheb_eval (e02aec) is approximately proportional to . It is expected that a common use of nag_1d_cheb_eval (e02aec) will be the evaluation of the polynomial approximations produced by e02adc (nag_1d_cheb_fit) and e02afc (nag_1d_cheb_interp_fit).