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NAG Toolbox: nag_fit_1dcheb_eval (e02ae)

Purpose

nag_fit_1dcheb_eval (e02ae) evaluates a polynomial from its Chebyshev series representation.

Syntax

[p, ifail] = e02ae(a, xcap, 'nplus1', nplus1)
[p, ifail] = nag_fit_1dcheb_eval(a, xcap, 'nplus1', nplus1)

Description

nag_fit_1dcheb_eval (e02ae) evaluates the polynomial
(1/2)a1T0(x) + a2T1(x) + a3T2(x) + + an + 1Tn(x)
12a1T0(x-)+a2T1(x-)+a3T2(x-)++an+1Tn(x-)
for any value of xx- satisfying 1x1-1x-1. Here Tj(x)Tj(x-) denotes the Chebyshev polynomial of the first kind of degree jj with argument xx-. The value of nn is prescribed by you.
In practice, the variable xx- will usually have been obtained from an original variable xx, where xminxxmaxxminxxmax and
x = (((xxmin)(xmaxx)))/((xmaxxmin))
x-=((x-xmin)-(xmax-x)) (xmax-xmin)
Note that this form of the transformation should be used computationally rather than the mathematical equivalent
x = ((2xxminxmax))/((xmaxxmin))
x-= (2x-xmin-xmax) (xmax-xmin)
since the former guarantees that the computed value of xx- differs from its true value by at most 4ε4ε, where εε is the machine precision, whereas the latter has no such guarantee.
The method employed is based on the three-term recurrence relation due to Clenshaw (1955), with modifications to give greater numerical stability due to Reinsch and Gentleman (see Gentleman (1969)).
For further details of the algorithm and its use see Cox (1974) and Cox and Hayes (1973).

References

Clenshaw C W (1955) A note on the summation of Chebyshev series Math. Tables Aids Comput. 9 118–120
Cox M G (1974) A data-fitting package for the non-specialist user Software for Numerical Mathematics (ed D J Evans) Academic Press
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

Parameters

Compulsory Input Parameters

1:     a(nplus1) – double array
nplus1, the dimension of the array, must satisfy the constraint nplus11nplus11.
a(i)ai must be set to the value of the iith coefficient in the series, for i = 1,2,,n + 1i=1,2,,n+1.
2:     xcap – double scalar
xx-, the argument at which the polynomial is to be evaluated. It should lie in the range 1-1 to + 1+1, but a value just outside this range is permitted (see Section [Error Indicators and Warnings]) to allow for possible rounding errors committed in the transformation from xx to xx- discussed in Section [Description]. Provided the recommended form of the transformation is used, a successful exit is thus assured whenever the value of xx lies in the range xminxmin to xmaxxmax.

Optional Input Parameters

1:     nplus1 – int64int32nag_int scalar
Default: The dimension of the array a.
The number n + 1n+1 of terms in the series (i.e., one greater than the degree of the polynomial).
Constraint: nplus11nplus11.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     p – double scalar
The value of the polynomial.
2:     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
ABS(xcap) > 1.0 + 4εABS(xcap)>1.0+4ε, where εε is the machine precision. In this case the value of p is set arbitrarily to zero.
  ifail = 2ifail=2
On entry,nplus1 < 1nplus1<1.

Accuracy

The rounding errors committed are such that the computed value of the polynomial is exact for a slightly perturbed set of coefficients ai + δaiai+δai. The ratio of the sum of the absolute values of the δaiδai to the sum of the absolute values of the aiai is less than a small multiple of (n + 1) × machine precision(n+1)×machine precision.

Further Comments

The time taken is approximately proportional to n + 1n+1.
It is expected that a common use of nag_fit_1dcheb_eval (e02ae) will be the evaluation of the polynomial approximations produced by nag_fit_1dcheb_arb (e02ad) and nag_fit_1dcheb_glp (e02af).

Example

function nag_fit_1dcheb_eval_example
a = [2;
     0.5;
     0.25;
     0.125;
     0.0625];
xcap = -1;
[p, ifail] = nag_fit_1dcheb_eval(a, xcap)
 

p =

    0.6875


ifail =

                    0


function e02ae_example
a = [2;
     0.5;
     0.25;
     0.125;
     0.0625];
xcap = -1;
[p, ifail] = e02ae(a, xcap)
 

p =

    0.6875


ifail =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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