Evaluation of fitted polynomial in one variable from Chebyshev series form

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NAG[e02akc] NAG[nag_1d_cheb_eval2] - Evaluation of fitted polynomial in one variable from Chebyshev series form

Calling Sequence

e02akc(n, xmin, xmax, a, ia1, x, result, 'fail'=fail)

nag_1d_cheb_eval2(. . .)

Parameters

n - integer;

On entry: , the degree of the given polynomial .

Constraint: . .

xmin - float;
xmax - float;

On entry: the lower and upper end points respectively of the interval . The Chebyshev-series representation is in terms of the normalized variable , where

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

Constraint: . .

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

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

On entry: the Chebyshev coefficients of the polynomial . Specifically, element must contain the coefficient , for . Only these elements will be accessed.

ia1 - integer;

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 , then the value of ia1 must be 3.

Constraint: . .

x - float;

On entry: the argument at which the polynomial is to be evaluated.

Constraint: . .

result - assignable;

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

On exit: the value of the polynomial .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	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.

Description

If supplied with the coefficients , for , of a polynomial of degree , where

nag_1d_cheb_eval2 (e02akc) returns the value of at a user-specified value of the variable . Here denotes the Chebyshev polynomial of the first kind of degree with argument . It is assumed that the independent variable in the interval was obtained from your original variable in the interval by the linear transformation

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

The coefficients 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 upon 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).

	Error Indicators and Warnings
	"NE_BAD_PARAM" On entry, argument had an illegal value. "NE_INT" On entry, . Constraint: . On entry, . Constraint: . "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 consult NAG for assistance. "NE_REAL_2" On entry, : , . "NE_REAL_3" On entry, x does not lie in : , , .

	Accuracy
	The rounding errors 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 .

	Further Comments
	The time taken is approximately proportional to .

Examples

>	n := 6: xmin := -0.5: xmax := 2.5: ia1 := 1: x := -0.5: a := Vector([2.53213, 1.13032, 0.2715, 0.04434, 0.00547, 0.00054, 4e-05], datatype=float[8]): NAG:-e02akc(n, xmin, xmax, a, ia1, x, result):