NAG Library Routine Document

e02akf (dim1_cheb_eval2)


    1  Purpose
    7  Accuracy


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


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 nagmk26.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)


If supplied with the coefficients ai, for i=0,1,,n, of a polynomial px- of degree n, where
e02akf 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 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).


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:     np1 – IntegerInput
On entry: n+1, where n is the degree of the given polynomial px-.
Constraint: np11.
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 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:     ala – Real (Kind=nag_wp) arrayInput
On entry: the Chebyshev coefficients of the polynomial px-. Specifically, element i×ia1+1 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 a1,a4,a7,, the value of ia1 must be 3.
Constraint: ia11.
6:     la – IntegerInput
On entry: the dimension of the array a as declared in the (sub)program from which e02akf is called.
Constraint: lanp1-1 ×ia1+1.
7:     x – Real (Kind=nag_wp)Input
On entry: the argument x at which the polynomial is to be evaluated.
Constraint: xminxxmax.
8:     result – Real (Kind=nag_wp)Output
On exit: the value of the polynomial px-.
9:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry,np1<1,
x does not satisfy the restriction xminxxmax.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 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 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


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

e02akf 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 in DATA statements. Normally a program would first read in or generate data and compute the fitted polynomial.)

Program Text

Program Text (e02akfe.f90)

Program Data


Program Results

Program Results (e02akfe.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
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017