NAG Library Function Document
nag_1d_cheb_eval (e02aec) evaluates a polynomial from its Chebyshev series representation.
||nag_1d_cheb_eval (Integer nplus1,
const double a,
nag_1d_cheb_eval (e02aec) evaluates the polynomial
for any value of
denotes the Chebyshev polynomial of the first kind of degree
. The value of
is prescribed by you.
In practice, the variable
will usually have been obtained from an original variable
Note that this form of the transformation should be used computationally rather than the mathematical equivalent
since the former guarantees that the computed value of
differs from its true value by at most
is the machine precision
, whereas the latter has no such guarantee.
The method employed is based upon 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)
, Cox and Hayes (1973)
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
nplus1 – IntegerInput
On entry: the number of terms in the series (i.e., one greater than the degree of the polynomial).
a[nplus1] – const doubleInput
On entry: must be set to the value of the th coefficient in the series, for .
xcap – doubleInput
, the argument at which the polynomial is to be evaluated. It should lie in the range
, but a value just outside this range is permitted (see Section 9
) to allow for possible rounding errors committed in the transformation from
discussed in Section 3
. Provided the recommended form of the transformation is used, a successful exit is thus assured whenever the value of
lies in the range
p – double *Output
On exit: the value of the polynomial.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
On entry, nplus1
must not be less than 1:
is the machine precision
In this case the value of p
is set arbitrarily to zero.
The rounding errors committed are such that the computed value of the polynomial is exact for a slightly perturbed set of coefficients . The ratio of the sum of the absolute values of the to the sum of the absolute values of the is less than a small multiple of machine precision.
8 Parallelism and Performance
The time taken by nag_1d_cheb_eval (e02aec) is approximately proportional to .
It is expected that a common use of nag_1d_cheb_eval (e02aec) will be the evaluation of the polynomial approximations produced by nag_1d_cheb_fit (e02adc)
and nag_1d_cheb_interp_fit (e02afc)
Evaluate at 11 equally-spaced points in the interval the polynomial of degree 4 with Chebyshev coefficients, 2.0, 0.5, 0.25, 0.125, 0.0625.
The example program is written in a general form that will enable a polynomial of degree in its Chebyshev series form to be evaluated at equally-spaced points in the interval . The program is self-starting in that any number of datasets can be supplied.
10.1 Program Text
Program Text (e02aece.c)
10.2 Program Data
Program Data (e02aece.d)
10.3 Program Results
Program Results (e02aece.r)