NAG C Library Function Document
nag_1d_cheb_eval (e02aec)
1
Purpose
nag_1d_cheb_eval (e02aec) evaluates a polynomial from its Chebyshev series representation.
2
Specification
| #include <nag.h> |
| #include <nage02.h> |
| void |
nag_1d_cheb_eval (Integer nplus1,
const double a[],
double xcap,
double *p,
NagError *fail) |
|
3
Description
nag_1d_cheb_eval (e02aec) evaluates the polynomial
for any value of
satisfying
. Here
denotes the Chebyshev polynomial of the first kind of degree
with argument
. The value of
is prescribed by you.
In practice, the variable
will usually have been obtained from an original variable
, where
and
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
, where
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).
4
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
5
Arguments
- 1:
– IntegerInput
-
On entry: the number of terms in the series (i.e., one greater than the degree of the polynomial).
Constraint:
.
- 2:
– const doubleInput
-
On entry: must be set to the value of the th coefficient in the series, for .
- 3:
– doubleInput
-
On entry:
, the argument at which the polynomial is to be evaluated. It should lie in the range
to
, but a value just outside this range is permitted (see
Section 9) to allow for possible rounding errors committed in the transformation from
to
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
to
.
- 4:
– double *Output
-
On exit: the value of the polynomial.
- 5:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_INT_ARG_LT
-
On entry,
nplus1 must not be less than 1:
.
- NE_INVALID_XCAP
-
On entry,
, where
is the
machine precision.
In this case the value of
p is set arbitrarily to zero.
7
Accuracy
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
nag_1d_cheb_eval (e02aec) is not threaded in any implementation.
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).
10
Example
Evaluate at 11 equally-spaced points in the interval the polynomial of degree 4 with Chebyshev coefficients, , , , , 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)