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NAG Toolbox: nag_fit_1dcheb_deriv (e02ah)

Purpose

nag_fit_1dcheb_deriv (e02ah) determines the coefficients in the Chebyshev series representation of the derivative of a polynomial given in Chebyshev series form.

Syntax

[patm1, adif, ifail] = e02ah(n, xmin, xmax, a, ia1, iadif1)
[patm1, adif, ifail] = nag_fit_1dcheb_deriv(n, xmin, xmax, a, ia1, iadif1)

Description

nag_fit_1dcheb_deriv (e02ah) 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 aiai, for i = 0,1,,ni=0,1,,n, of a polynomial p(x)p(x) of degree nn, where
p(x) = (1/2)a0 + a1T1(x) + + anTn(x)
p(x)=12a0+a1T1(x-)++anTn(x-)
the function returns the coefficients aia-i, for i = 0,1,,n1i=0,1,,n-1, of the polynomial q(x)q(x) of degree n1n-1, where
q(x) = (dp(x))/(dx) = (1/2)a0 + a1T1(x) + + an1Tn1(x).
q(x)=dp(x) dx =12a-0+a-1T1(x-)++a-n-1Tn-1(x-).
Here Tj(x)Tj(x-) denotes the Chebyshev polynomial of the first kind of degree jj with argument xx-. It is assumed that the normalized variable xx- in the interval [1, + 1][-1,+1] was obtained from your original variable xx in the interval [xmin,xmax][xmin,xmax] by the linear transformation
x = (2x(xmax + xmin))/(xmaxxmin)
x-=2x-(xmax+xmin) xmax-xmin
and that you require the derivative to be with respect to the variable xx. If the derivative with respect to xx- is required, set xmax = 1xmax=1 and xmin = 1xmin=-1.
Values of the derivative can subsequently be computed, from the coefficients obtained, by using nag_fit_1dcheb_eval2 (e02ak).
The method employed is that of Chebyshev series (see Chapter 8 of Modern Computing Methods (1961)), modified to obtain the derivative with respect to xx. Initially setting an + 1 = an = 0a-n+1=a-n=0, the function forms successively
ai1 = ai + 1 + 2/(xmaxxmin)2iai,  i = n,n1,,1.
a-i-1=a-i+1+2xmax-xmin 2iai,  i=n,n-1,,1.

References

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

Parameters

Compulsory Input Parameters

1:     n – int64int32nag_int scalar
nn, the degree of the given polynomial p(x)p(x).
Constraint: n0n0.
2:     xmin – double scalar
3:     xmax – double scalar
The lower and upper end points respectively of the interval [xmin,xmax][xmin,xmax]. The Chebyshev series representation is in terms of the normalized variable xx-, where
x = (2x(xmax + xmin))/(xmaxxmin).
x-=2x-(xmax+xmin) xmax-xmin .
Constraint: xmax > xminxmax>xmin.
4:     a(la) – double array
la, the dimension of the array, must satisfy the constraint la1 + (np11) × ia1la1+(np1-1)×ia1.
The Chebyshev coefficients of the polynomial p(x)p(x). Specifically, element i × ia1i×ia1 of a must contain the coefficient aiai, for i = 0,1,,ni=0,1,,n. Only these n + 1n+1 elements will be accessed.
Unchanged on exit, but see adif, below.
5:     ia1 – int64int32nag_int scalar
The index increment of a. Most frequently the Chebyshev coefficients are stored in adjacent elements of a, and ia1 must be set to 11. However, if for example, they are stored in a(1),a(4),a(7),a1,a4,a7,, then the value of ia1 must be 33. See also Section [Further Comments].
Constraint: ia11ia11.
6:     iadif1 – int64int32nag_int scalar
The index increment of adif. Most frequently the Chebyshev coefficients are required in adjacent elements of adif, and iadif1 must be set to 11. However, if, for example, they are to be stored in adif(1),adif(4),adif(7),adif1,adif4,adif7,, then the value of iadif1 must be 33. See Section [Further Comments].
Constraint: iadif11iadif11.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

np1 la ladif

Output Parameters

1:     patm1 – double scalar
The value of p(xmin)p(xmin). If this value is passed to the integration function nag_fit_1dcheb_integ (e02aj) with the coefficients of q(x)q(x), then the original polynomial p(x)p(x) is recovered, including its constant coefficient.
2:     adif(ladif) – double array
ladif1 + (np11) × iadif1ladif1+(np1-1) ×iadif1.
The Chebyshev coefficients of the derived polynomial q(x)q(x). (The differentiation is with respect to the variable xx.) Specifically, element i × iadif1 + 1i×iadif1+1 of adif contains the coefficient aia-i, for i = 0,1,,n1i=0,1,,n-1. Additionally, element n × iadif1 + 1n×iadif1+1 is set to zero. A call of the function 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 a0,a1,,ai1a0,a1,,ai-1 must be intact after coefficient aia-i is stored. In particular, it is possible to overwrite the aiai completely by having ia1 = iadif1ia1=iadif1, and the actual arrays for a and adif identical.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry,np1 < 1np1<1,
orxmaxxminxmaxxmin,
oria1 < 1ia1<1,
orla(np11) × ia1la(np1-1)×ia1,
oriadif1 < 1iadif1<1,
orladif(np11) × iadif1ladif(np1-1)×iadif1.

Accuracy

There is always a loss of precision in numerical differentiation, in this case associated with the multiplication by 2i2i in the formula quoted in Section [Description].

Further Comments

The time taken is approximately proportional to n + 1n+1.
The increments ia1, iadif1 are included as parameters 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.

Example

function nag_fit_1dcheb_deriv_example
n = int64(6);
xmin = -0.5;
xmax = 2.5;
a = [2.53213;
     1.13032;
     0.2715;
     0.04434;
     0.00547;
     0.00054;
     4e-05];
ia1 = int64(1);
iadif1 = int64(1);
[patm1, adif, ifail] = nag_fit_1dcheb_deriv(n, xmin, xmax, a, ia1, iadif1)
 

patm1 =

    0.3679


adif =

    1.6881
    0.7535
    0.1810
    0.0295
    0.0036
    0.0003
         0


ifail =

                    0


function e02ah_example
n = int64(6);
xmin = -0.5;
xmax = 2.5;
a = [2.53213;
     1.13032;
     0.2715;
     0.04434;
     0.00547;
     0.00054;
     4e-05];
ia1 = int64(1);
iadif1 = int64(1);
[patm1, adif, ifail] = e02ah(n, xmin, xmax, a, ia1, iadif1)
 

patm1 =

    0.3679


adif =

    1.6881
    0.7535
    0.1810
    0.0295
    0.0036
    0.0003
         0


ifail =

                    0



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Chapter Introduction
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