NAG Library Function Document
nag_2d_cheb_eval (e02cbc)
1 Purpose
nag_2d_cheb_eval (e02cbc) evaluates a bivariate polynomial from the rectangular array of coefficients in its double Chebyshev series representation.
2 Specification
#include <nag.h> |
#include <nage02.h> |
void |
nag_2d_cheb_eval (Integer mfirst,
Integer mlast,
Integer k,
Integer l,
const double x[],
double xmin,
double xmax,
double y,
double ymin,
double ymax,
double ff[],
const double a[],
NagError *fail) |
|
3 Description
This function evaluates a bivariate polynomial (represented in double Chebyshev form) of degree
in one variable,
, and degree
in the other,
. The range of both variables is
to
. However, these normalized variables will usually have been derived (as when the polynomial has been computed by
nag_2d_cheb_fit_lines (e02cac), for example) from your original variables
and
by the transformations
(Here
and
are the ends of the range of
which has been transformed to the range
to
of
.
and
are correspondingly for
. See
Section 9). For this reason, the function has been designed to accept values of
and
rather than
and
, and so requires values of
, etc. to be supplied by you. In fact, for the sake of efficiency in appropriate cases, the function evaluates the polynomial for a sequence of values of
, all associated with the same value of
.
The double Chebyshev series can be written as
where
is the Chebyshev polynomial of the first kind of degree
and argument
, and
is similarly defined. However the standard convention, followed in this function, is that coefficients in the above expression which have either
or
zero are written
, instead of simply
, and the coefficient with both
and
zero is written
.
The function first forms
, with
replaced by
, for each of
. The value of the double series is then obtained for each value of
, by summing
, with
replaced by
, over
. The Clenshaw three term recurrence (see
Clenshaw (1955)) with modifications due to Reinsch and
Gentleman (1969) is used to form the sums.
4 References
Clenshaw C W (1955) A note on the summation of Chebyshev series Math. Tables Aids Comput. 9 118–120
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:
mfirst – IntegerInput
- 2:
mlast – IntegerInput
On entry: the index of the first and last
value in the array
at which the evaluation is required respectively (see
Section 9).
Constraint:
.
- 3:
k – IntegerInput
- 4:
l – IntegerInput
On entry: the degree of and of , respectively, in the polynomial.
Constraint:
and .
- 5:
x[mlast] – const doubleInput
On entry: , for , must contain the values at which the evaluation is required.
Constraint:
, for all .
- 6:
xmin – doubleInput
- 7:
xmax – doubleInput
On entry: the lower and upper ends,
and
, of the range of the variable
(see
Section 3).
The values of
xmin and
xmax may depend on the value of
(e.g., when the polynomial has been derived using
nag_2d_cheb_fit_lines (e02cac)).
Constraint:
.
- 8:
y – doubleInput
On entry: the value of the coordinate of all the points at which the evaluation is required.
Constraint:
.
- 9:
ymin – doubleInput
- 10:
ymax – doubleInput
On entry: the lower and upper ends,
and
, of the range of the variable
(see
Section 3).
Constraint:
.
- 11:
ff[mlast] – doubleOutput
On exit: gives the value of the polynomial at the point , for .
- 12:
a[] – const doubleInput
-
Note: the dimension,
dim, of the array
a
must be at least
.
On entry: the Chebyshev coefficients of the polynomial. The coefficient
defined according to the standard convention (see
Section 3) must be in
.
- 13:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT_2
-
On entry, and .
Constraint: and .
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
Unexpected failure in internal call to
nag_1d_cheb_eval (e02aec).
- NE_REAL_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_REAL_ARRAY
-
On entry, , and .
Constraint: .
On entry, , and .
Constraint: .
7 Accuracy
The method is numerically stable in the sense that the computed values of the polynomial are exact for a set of coefficients which differ from those supplied by only a modest multiple of machine precision.
8 Parallelism and Performance
nag_2d_cheb_eval (e02cbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the
Users' Note for your implementation for any additional implementation-specific information.
The time taken is approximately proportional to , where , the number of points at which the evaluation is required.
This function is suitable for evaluating the polynomial surface fits produced by the function
nag_2d_cheb_fit_lines (e02cac), which provides the
array
a in the required form. For this use, the values of
and
supplied to the present function must be the same as those supplied to
nag_2d_cheb_fit_lines (e02cac). The same applies to
and
if they are independent of
. If they vary with
, their values must be consistent with those supplied to
nag_2d_cheb_fit_lines (e02cac) (see
Section 9 in nag_2d_cheb_fit_lines (e02cac)).
The arguments
mfirst and
mlast are intended to permit the selection of a segment of the array
x which is to be associated with a particular value of
, when, for example, other segments of
x are associated with other values of
. Such a case arises when, after using
nag_2d_cheb_fit_lines (e02cac) to fit a set of data, you wish to evaluate the resulting polynomial at all the data values. In this case, if the arguments
x,
y,
mfirst and
mlast of the present function are set respectively (in terms of arguments of
nag_2d_cheb_fit_lines (e02cac)) to
x,
,
and
, the function will compute values of the polynomial surface at all data points which have
as their
coordinate (from which values the residuals of the fit may be derived).
10 Example
This example reads data in the following order, using the notation of the argument list above:
For each line
the polynomial is evaluated at
equispaced points between
and
inclusive.
10.1 Program Text
Program Text (e02cbce.c)
10.2 Program Data
Program Data (e02cbce.d)
10.3 Program Results
Program Results (e02cbce.r)