NAG Library Routine Document

1Purpose

e02ahf determines the coefficients in the Chebyshev series representation of the derivative of a polynomial given in Chebyshev series form.

2Specification

Fortran Interface
 Subroutine e02ahf ( np1, xmin, xmax, a, ia1, la, adif,
 Integer, Intent (In) :: np1, ia1, la, iadif1, ladif Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: xmin, xmax, a(la) Real (Kind=nag_wp), Intent (Out) :: patm1, adif(ladif)
#include nagmk26.h
 void e02ahf_ (const Integer *np1, const double *xmin, const double *xmax, const double a[], const Integer *ia1, const Integer *la, double *patm1, double adif[], const Integer *iadif1, const Integer *ladif, Integer *ifail)

3Description

e02ahf forms the polynomial which is the derivative of a given polynomial. Both the original polynomial and its derivative are represented in Chebyshev series form. Given the coefficients ${a}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n$, of a polynomial $p\left(x\right)$ of degree $n$, where
 $px=12a0+a1T1x-+⋯+anTnx-$
the routine returns the coefficients ${\stackrel{-}{a}}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n-1$, of the polynomial $q\left(x\right)$ of degree $n-1$, where
 $qx=dpx dx =12a-0+a-1T1x-+⋯+a-n-1Tn-1x-.$
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 normalized 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
 $x-=2x-xmax+xmin xmax-xmin$
and that you require the derivative to be with respect to the variable $x$. If the derivative with respect to $\stackrel{-}{x}$ is required, set ${x}_{\mathrm{max}}=1$ and ${x}_{\mathrm{min}}=-1$.
Values of the derivative 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 to obtain the derivative with respect to $x$. Initially setting ${\stackrel{-}{a}}_{n+1}={\stackrel{-}{a}}_{n}=0$, the routine forms successively
 $a-i-1=a-i+1+2xmax-xmin 2iai, i=n,n-1,…,1.$

4References

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

5Arguments

1:     $\mathbf{np1}$ – IntegerInput
On entry: $n+1$, where $n$ is the degree of the given polynomial $p\left(x\right)$. Thus np1 is the number of coefficients in this polynomial.
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
 $x-=2x-xmax+xmin xmax-xmin .$
Constraint: ${\mathbf{xmax}}>{\mathbf{xmin}}$.
4:     $\mathbf{a}\left({\mathbf{la}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the Chebyshev coefficients of the polynomial $p\left(x\right)$. Specifically, element $\mathit{i}×{\mathbf{ia1}}$ of a must contain the coefficient ${a}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n$. Only these $n+1$ elements will be accessed.
Unchanged on exit, but see adif, below.
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 $3$. See also Section 9.
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 e02ahf is called.
Constraint: ${\mathbf{la}}\ge 1+\left({\mathbf{np1}}-1\right)×{\mathbf{ia1}}$.
7:     $\mathbf{patm1}$ – Real (Kind=nag_wp)Output
On exit: the value of $p\left({x}_{\mathrm{min}}\right)$. If this value is passed to the integration routine e02ajf with the coefficients of $q\left(x\right)$, the original polynomial $p\left(x\right)$ is recovered, including its constant coefficient.
8:     $\mathbf{adif}\left({\mathbf{ladif}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the Chebyshev coefficients of the derived polynomial $q\left(x\right)$. (The differentiation is with respect to the variable $x$.) Specifically, element $\mathit{i}×{\mathbf{iadif1}}+1$ of adif contains the coefficient ${\stackrel{-}{a}}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n-1$. Additionally, element $n×{\mathbf{iadif1}}+1$ is set to zero. A call of the routine may have the array name adif the same as a, provided that note is taken of the order in which elements are overwritten, when choosing the starting elements and increments ia1 and iadif1, i.e., the coefficients ${a}_{0},{a}_{1},\dots ,{a}_{i-1}$ must be intact after coefficient ${\stackrel{-}{a}}_{i}$ is stored. In particular, it is possible to overwrite the ${a}_{i}$ completely by having ${\mathbf{ia1}}={\mathbf{iadif1}}$, and the actual arrays for a and adif identical.
9:     $\mathbf{iadif1}$ – IntegerInput
On entry: the index increment of adif. Most frequently the Chebyshev coefficients are required in adjacent elements of adif, and iadif1 must be set to $1$. However, if, for example, they are to be stored in ${\mathbf{adif}}\left(1\right),{\mathbf{adif}}\left(4\right),{\mathbf{adif}}\left(7\right),\dots \text{}$, the value of iadif1 must be $3$. See Section 9.
Constraint: ${\mathbf{iadif1}}\ge 1$.
10:   $\mathbf{ladif}$ – IntegerInput
On entry: the dimension of the array adif as declared in the (sub)program from which e02ahf is called.
Constraint: ${\mathbf{ladif}}\ge 1+\left({\mathbf{np1}}-1\right)×{\mathbf{iadif1}}$.
11:   $\mathbf{ifail}$ – 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).

6Error 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 ${\mathbf{xmax}}\le {\mathbf{xmin}}$, or ${\mathbf{ia1}}<1$, or ${\mathbf{la}}\le \left({\mathbf{np1}}-1\right)×{\mathbf{ia1}}$, or ${\mathbf{iadif1}}<1$, or ${\mathbf{ladif}}\le \left({\mathbf{np1}}-1\right)×{\mathbf{iadif1}}$.
${\mathbf{ifail}}=-99$
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.

7Accuracy

There is always a loss of precision in numerical differentiation, in this case associated with the multiplication by $2i$ in the formula quoted in Section 3.

8Parallelism and Performance

e02ahf is not threaded in any implementation.

The time taken is approximately proportional to $n+1$.
The increments ia1, iadif1 are included as arguments to give a degree of flexibility which, for example, allows a polynomial in two variables to be differentiated with respect to either variable without rearranging the coefficients.

10Example

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 first and second derivatives of this polynomial at $4$ equally spaced points over the interval. (For the purposes of this example, xmin, xmax and the Chebyshev coefficients are simply supplied in DATA statements. Normally a program would first read in or generate data and compute the fitted polynomial.)

10.1Program Text

Program Text (e02ahfe.f90)

None.

10.3Program Results

Program Results (e02ahfe.r)

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