NAG Library Routine Document
e02akf
(dim1_cheb_eval2)
1
Purpose
e02akf evaluates a polynomial from its Chebyshev series representation, allowing an arbitrary index increment for accessing the array of coefficients.
2
Specification
Fortran Interface
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) 

3
Description
If supplied with the coefficients
${a}_{i}$, for
$\mathit{i}=0,1,\dots ,n$, of a polynomial
$p\left(\stackrel{}{x}\right)$ of degree
$n$, where
e02akf returns the value of
$p\left(\stackrel{}{x}\right)$ at a userspecified value of the variable
$x$. Here
${T}_{j}\left(\stackrel{}{x}\right)$ denotes the Chebyshev polynomial of the first kind of degree
$j$ with argument
$\stackrel{}{x}$. It is assumed that the independent variable
$\stackrel{}{x}$ in the interval
$\left[1,+1\right]$ was obtained from your original variable
$x$ in the interval
$\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]$ by the linear transformation
The coefficients
${a}_{i}$ 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 threeterm 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).
4
References
Clenshaw C W (1955) A note on the summation of Chebyshev series Math. Tables Aids Comput. 9 118–120
Cox M G (1973) A datafitting package for the nonspecialist 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 nonspecialist 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
5
Arguments
 1: $\mathbf{np1}$ – IntegerInput

On entry: $n+1$, where $n$ is the degree of the given polynomial $p\left(\stackrel{}{x}\right)$.
Constraint:
${\mathbf{np1}}\ge 1$.
 2: $\mathbf{xmin}$ – Real (Kind=nag_wp)Input
 3: $\mathbf{xmax}$ – Real (Kind=nag_wp)Input

On entry: the lower and upper end points respectively of the interval
$\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]$. The Chebyshev series representation is in terms of the normalized variable
$\stackrel{}{x}$, where
Constraint:
${\mathbf{xmin}}<{\mathbf{xmax}}$.
 4: $\mathbf{a}\left({\mathbf{la}}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the Chebyshev coefficients of the polynomial $p\left(\stackrel{}{x}\right)$. Specifically, element
$\mathit{i}\times {\mathbf{ia1}}+1$ must contain the coefficient ${a}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n$. Only these $n+1$ elements will be accessed.
 5: $\mathbf{ia1}$ – 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
${\mathbf{a}}\left(1\right),{\mathbf{a}}\left(4\right),{\mathbf{a}}\left(7\right),\dots \text{}$, the value of
ia1 must be
$3$.
Constraint:
${\mathbf{ia1}}\ge 1$.
 6: $\mathbf{la}$ – IntegerInput

On entry: the dimension of the array
a as declared in the (sub)program from which
e02akf is called.
Constraint:
${\mathbf{la}}\ge \left({\mathbf{np1}}1\right)\times {\mathbf{ia1}}+1$.
 7: $\mathbf{x}$ – Real (Kind=nag_wp)Input

On entry: the argument $x$ at which the polynomial is to be evaluated.
Constraint:
${\mathbf{xmin}}\le {\mathbf{x}}\le {\mathbf{xmax}}$.
 8: $\mathbf{result}$ – Real (Kind=nag_wp)Output

On exit: the value of the polynomial $p\left(\stackrel{}{x}\right)$.
 9: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ 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 $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry,  ${\mathbf{np1}}<1$, 
or  ${\mathbf{ia1}}<1$, 
or  ${\mathbf{la}}\le \left({\mathbf{np1}}1\right)\times {\mathbf{ia1}}$, 
or  ${\mathbf{xmin}}\ge {\mathbf{xmax}}$. 
 ${\mathbf{ifail}}=2$

x does not satisfy the restriction
${\mathbf{xmin}}\le {\mathbf{x}}\le {\mathbf{xmax}}$.
 ${\mathbf{ifail}}=99$
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.
 ${\mathbf{ifail}}=399$
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.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
The rounding errors are such that the computed value of the polynomial is exact for a slightly perturbed set of coefficients ${a}_{i}+\delta {a}_{i}$. The ratio of the sum of the absolute values of the $\delta {a}_{i}$ to the sum of the absolute values of the ${a}_{i}$ is less than a small multiple of $\left(n+1\right)\times \mathit{machineprecision}$.
8
Parallelism and Performance
e02akf is not threaded in any implementation.
The time taken is approximately proportional to $n+1$.
10
Example
Suppose a polynomial has been computed in Chebyshev series form to fit data over the interval
$\left[0.5,2.5\right]$. 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.)
10.1
Program Text
Program Text (e02akfe.f90)
10.2
Program Data
None.
10.3
Program Results
Program Results (e02akfe.r)