NAG CL Interface
e02cbc (dim2_cheb_eval)
1
Purpose
e02cbc evaluates a bivariate polynomial from the rectangular array of coefficients in its double Chebyshev series representation.
2
Specification
void |
e02cbc (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) |
|
The function may be called by the names: e02cbc, nag_fit_dim2_cheb_eval or nag_2d_cheb_eval.
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
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:
– Integer
Input
-
2:
– Integer
Input
-
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:
– Integer
Input
-
4:
– Integer
Input
-
On entry: the degree of and of , respectively, in the polynomial.
Constraint:
and .
-
5:
– const double
Input
-
On entry: , for , must contain the values at which the evaluation is required.
Constraint:
, for all .
-
6:
– double
Input
-
7:
– double
Input
-
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
e02cac).
Constraint:
.
-
8:
– double
Input
-
On entry: the value of the coordinate of all the points at which the evaluation is required.
Constraint:
.
-
9:
– double
Input
-
10:
– double
Input
-
On entry: the lower and upper ends,
and
, of the range of the variable
(see
Section 3).
Constraint:
.
-
11:
– double
Output
-
On exit: gives the value of the polynomial at the point , for .
-
12:
– const double
Input
-
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:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- 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.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Unexpected internal failure when evaluating the polynomial.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- 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
e02cbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also 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
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
e02cac. The same applies to
and
if they are independent of
. If they vary with
, their values must be consistent with those supplied to
e02cac (see
Section 9 in
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
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
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
10.2
Program Data
10.3
Program Results