NAG C Library Function Document

nag_1d_cheb_eval2 (e02akc)


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


#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)


If supplied with the coefficients ai, for i=0,1,,n, of a polynomial px- of degree n, where
nag_1d_cheb_eval2 (e02akc) returns the value of px- at a user-specified value of the variable x. Here Tjx- denotes the Chebyshev polynomial of the first kind of degree j with argument x-. It is assumed that the independent variable x- in the interval -1,+1 was obtained from your original variable x in the interval xmin,xmax by the linear transformation
x-=2x-xmax+xmin xmax-xmin .  
The coefficients ai 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).


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


1:     n IntegerInput
On entry: n, the degree of the given polynomial px-.
Constraint: n0.
2:     xmin doubleInput
3:     xmax doubleInput
On entry: the lower and upper end points respectively of the interval xmin,xmax. The Chebyshev series representation is in terms of the normalized variable x-, where
x-=2x-xmax+xmin xmax-xmin .  
Constraint: xmin<xmax.
4:     a[dim] const doubleInput
Note: the dimension, dim, of the array a must be at least n+1-1×ia1+1.
On entry: the Chebyshev coefficients of the polynomial px-. Specifically, element i×ia1 must contain the coefficient ai, for i=0,1,,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],, the value of ia1 must be 3.
Constraint: ia11.
6:     x doubleInput
On entry: the argument x at which the polynomial is to be evaluated.
Constraint: xminxxmax.
7:     result double *Output
On exit: the value of the polynomial px-.
8:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
On entry, ia1=value.
Constraint: ia11.
On entry, n+1=value.
Constraint: n+11.
On entry, n=value.
Constraint: n0.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, xmax=value and xmin=value.
Constraint: xmax>xmin.
On entry, x does not lie in xmin,xmax: x=value, xmin=value and xmax=value.


The rounding errors are such that the computed value of the polynomial is exact for a slightly perturbed set of coefficients ai+δai. The ratio of the sum of the absolute values of the δai to the sum of the absolute values of the ai is less than a small multiple of n+1×machine precision.

Parallelism and Performance

nag_1d_cheb_eval2 (e02akc) is not threaded in any implementation.

Further Comments

The time taken is approximately proportional to n+1.


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.)

Program Text

Program Text (e02akce.c)

Program Data


Program Results

Program Results (e02akce.r)

GnuplotProduced by GNUPLOT 4.6 patchlevel 3 0 0.5 1 1.5 2 2.5 3 −0.5 0 0.5 1 1.5 2 2.5 P(x) x Example Program Evaluation of Chebyshev Representation of Polynomial gnuplot_plot_1