NAG Library Routine Document

e02ajf (dim1_cheb_integ)


    1  Purpose
    7  Accuracy


e02ajf determines the coefficients in the Chebyshev series representation of the indefinite integral of a polynomial given in Chebyshev series form.


Fortran Interface
Subroutine e02ajf ( np1, xmin, xmax, a, ia1, la, qatm1, aintc, iaint1, laint, ifail)
Integer, Intent (In):: np1, ia1, la, iaint1, laint
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: xmin, xmax, a(la), qatm1
Real (Kind=nag_wp), Intent (Out):: aintc(laint)
C Header Interface
#include nagmk26.h
void  e02ajf_ (const Integer *np1, const double *xmin, const double *xmax, const double a[], const Integer *ia1, const Integer *la, const double *qatm1, double aintc[], const Integer *iaint1, const Integer *laint, Integer *ifail)


e02ajf forms the polynomial which is the indefinite integral of a given polynomial. Both the original polynomial and its integral are represented in Chebyshev series form. If supplied with the coefficients ai, for i=0,1,,n, of a polynomial px of degree n, where
the routine returns the coefficients ai, for i=0,1,,n+1, of the polynomial qx of degree n+1, where
Here Tjx- denotes the Chebyshev polynomial of the first kind of degree j with argument x-. It is assumed that the normalized 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  
and that you require the integral to be with respect to the variable x. If the integral with respect to x- is required, set xmax=1 and xmin=-1.
Values of the integral can subsequently be computed, from the coefficients obtained, by using e02akf.
The method employed is that of Chebyshev series (see Chapter 8 of Modern Computing Methods (1961)), modified for integrating with respect to x. Initially taking an+1=an+2=0, the routine forms successively
ai=ai-1-ai+1 2i ×xmax-xmin2,  i=n+1,n,,1.  
The constant coefficient a0 is chosen so that qx is equal to a specified value, qatm1, at the lower end point of the interval on which it is defined, i.e., x-=-1, which corresponds to x=xmin.


Modern Computing Methods (1961) Chebyshev-series NPL Notes on Applied Science 16 (2nd Edition) HMSO


1:     np1 – IntegerInput
On entry: n+1, where n is the degree of the given polynomial px. Thus np1 is the number of coefficients in this polynomial.
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: xmax>xmin.
4:     ala – Real (Kind=nag_wp) arrayInput
On entry: the Chebyshev coefficients of the polynomial px. Specifically, element i×ia1+1 of a must contain the coefficient ai, for i=0,1,,n. Only these n+1 elements will be accessed.
Unchanged on exit, but see aintc, below.
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. See also Section 9.
Constraint: ia11.
6:     la – IntegerInput
On entry: the dimension of the array a as declared in the (sub)program from which e02ajf is called.
Constraint: la1+np1-1×ia1.
7:     qatm1 – Real (Kind=nag_wp)Input
On entry: the value that the integrated polynomial is required to have at the lower end point of its interval of definition, i.e., at x-=-1 which corresponds to x=xmin. Thus, qatm1 is a constant of integration and will normally be set to zero by you.
8:     aintclaint – Real (Kind=nag_wp) arrayOutput
On exit: the Chebyshev coefficients of the integral qx. (The integration is with respect to the variable x, and the constant coefficient is chosen so that qxmin equals qatm1). Specifically, element i×iaint1+1 of aintc contains the coefficient ai, for i=0,1,,n+1. A call of the routine may have the array name aintc the same as a, provided that note is taken of the order in which elements are overwritten when choosing starting elements and increments ia1 and iaint1: i.e., the coefficients, a0,a1,,ai-2 must be intact after coefficient ai is stored. In particular it is possible to overwrite the ai entirely by having ia1=iaint1, and the actual array for a and aintc identical.
9:     iaint1 – IntegerInput
On entry: the index increment of aintc. Most frequently the Chebyshev coefficients are required in adjacent elements of aintc, and iaint1 must be set to 1. However, if, for example, they are to be stored in aintc1,aintc4,aintc7,, the value of iaint1 must be 3. See also Section 9.
Constraint: iaint11.
10:   laint – IntegerInput
On entry: the dimension of the array aintc as declared in the (sub)program from which e02ajf is called.
Constraint: laint1+np1 ×iaint1.
11:   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,
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.


In general there is a gain in precision in numerical integration, in this case associated with the division by 2i in the formula quoted in Section 3.

Parallelism and Performance

e02ajf is not threaded in any implementation.

Further Comments

The time taken is approximately proportional to n+1.
The increments ia1, iaint1 are included as arguments to give a degree of flexibility which, for example, allows a polynomial in two variables to be integrated with respect to either variable without rearranging the coefficients.


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 integral of the polynomial from 0.0 to 2.0. (For the purpose of this example, xmin, xmax and the Chebyshev coefficients are simply supplied in DATA statements. Normally a program would read in or generate data and compute the fitted polynomial).

Program Text

Program Text (e02ajfe.f90)

Program Data


Program Results

Program Results (e02ajfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017