NAG C Library Function Document

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

k

in one variable,

\bar{x}

, and degree

l

in the other,

\bar{y}

. The range of both variables is

- 1

+ 1

. 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

x

and

y

by the transformations

\bar{x} = \frac{2 x - (x_{\max} + x_{\min})}{(x_{\max} - x_{\min})} and \bar{y} = \frac{2 y - (y_{\max} + y_{\min})}{(y_{\max} - y_{\min})} .

(Here

x_{\min}

and

x_{\max}

are the ends of the range of

x

which has been transformed to the range

- 1

+ 1

\bar{x}

y_{\min}

and

y_{\max}

are correspondingly for

y

. See Section 9). For this reason, the function has been designed to accept values of

x

and

y

rather than

\bar{x}

and

\bar{y}

, and so requires values of

x_{\min}

, 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

x

, all associated with the same value of

y

The double Chebyshev series can be written as

\sum_{i = 0}^{k} \sum_{j = 0}^{l} a_{i j} T_{i} (\bar{x}) T_{j} (\bar{y}),

where

T_{i} (\bar{x})

is the Chebyshev polynomial of the first kind of degree

i

and argument

\bar{x}

, and

T_{j} (\bar{y})

is similarly defined. However the standard convention, followed in this function, is that coefficients in the above expression which have either

i

j

zero are written

\frac{1}{2} a_{i j}

, instead of simply

a_{i j}

, and the coefficient with both

i

and

j

zero is written

\frac{1}{4} a_{0, 0}

The function first forms

c_{i} = \sum_{j = 0}^{l} a_{i j} T_{j} (\bar{y})

, with

a_{i, 0}

replaced by

\frac{1}{2} a_{i, 0}

, for each of

i = 0, 1, \dots, k

. The value of the double series is then obtained for each value of

x

, by summing

c_{i} \times T_{i} (\bar{x})

, with

c_{0}

replaced by

\frac{1}{2} c_{0}

, over

i = 0, 1, \dots, k

. 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 $x$ value in the array $x$ at which the evaluation is required respectively (see Section 9).

Constraint: $mlast \geq mfirst$ .
3: $k$ – IntegerInput
4: $l$ – IntegerInput: On entry: the degree $k$ of $x$ and $l$ of $y$ , respectively, in the polynomial.

Constraint: $k \geq 0$ and $l \geq 0$ .
5: $x [mlast]$ – const doubleInput: On entry: $x [i - 1]$ , for $i = mfirst, \dots, mlast$ , must contain the $x$ values at which the evaluation is required.

Constraint: $xmin \leq x [i - 1] \leq xmax$ , for all $i$ .
6: $xmin$ – doubleInput
7: $xmax$ – doubleInput: On entry: the lower and upper ends, $x_{\min}$ and $x_{\max}$ , of the range of the variable $x$ (see Section 3).
The values of xmin and xmax may depend on the value of $y$ (e.g., when the polynomial has been derived using nag_2d_cheb_fit_lines (e02cac)).

Constraint: $xmax > xmin$ .
8: $y$ – doubleInput: On entry: the value of the $y$ coordinate of all the points at which the evaluation is required.

Constraint: $ymin \leq y \leq ymax$ .
9: $ymin$ – doubleInput
10: $ymax$ – doubleInput: On entry: the lower and upper ends, $y_{\min}$ and $y_{\max}$ , of the range of the variable $y$ (see Section 3).

Constraint: $ymax > ymin$ .
11: $ff [mlast]$ – doubleOutput: On exit: $ff [i - 1]$ gives the value of the polynomial at the point $(x_{i}, y)$ , for $i = mfirst, \dots, mlast$ .
12: $a [\dim]$ – const doubleInput: Note: the dimension, dim, of the array a must be at least $((k + 1) \times (l + 1))$ .

On entry: the Chebyshev coefficients of the polynomial. The coefficient $a_{i j}$ defined according to the standard convention (see Section 3) must be in $a [i \times (l + 1) + j]$ .
13: $fail$ – 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_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT_2: On entry, $k = 〈value〉$ and $l = 〈value〉$ .
Constraint: $k \geq 0$ and $l \geq 0$ .

On entry, $mfirst = 〈value〉$ and $mlast = 〈value〉$ .
Constraint: $mfirst \leq mlast$ .
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 2.7.6 in How to Use the NAG Library and its Documentation 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 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_2: On entry, $xmin = 〈value〉$ and $xmax = 〈value〉$ .
Constraint: $xmin < xmax$ .

On entry, $y = 〈value〉$ and $ymax = 〈value〉$ .
Constraint: $y \leq ymax$ .

On entry, $y = 〈value〉$ and $ymin = 〈value〉$ .
Constraint: $y \geq ymin$ .

On entry, $ymin = 〈value〉$ and $ymax = 〈value〉$ .
Constraint: $ymin < ymax$ .
NE_REAL_ARRAY: On entry, $I = 〈value〉$ , $x [I - 1] = 〈value〉$ and $xmax = 〈value〉$ .
Constraint: $x [I - 1] \leq xmax$ .

On entry, $I = 〈value〉$ , $x [I - 1] = 〈value〉$ and $xmin = 〈value〉$ .
Constraint: $x [I - 1] \geq xmin$ .

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 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.

9

Further Comments

The time taken is approximately proportional to

(k + 1) \times (m + l + 1)

, where

m = mlast - mfirst + 1

, 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

y_{\min}

and

y_{\max}

supplied to the present function must be the same as those supplied to nag_2d_cheb_fit_lines (e02cac). The same applies to

x_{\min}

and

x_{\max}

if they are independent of

y

. If they vary with

y

, 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

y

, when, for example, other segments of x are associated with other values of

y

. 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,

y (S)

1 + \sum_{i = 1}^{s - 1} m (i)

and

\sum_{i = 1}^{s} m (i)

, the function will compute values of the polynomial surface at all data points which have

y [S - 1]

as their

y

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:

\begin{array}{l} N k l \\ a [i - 1], & for ​ i = 1, 2, \dots, (k + 1) \times (l + 1) \\ ymin ymax \\ y [i - 1] M (i - 1) xmin [i - 1] xmax [i - 1] X1 (i) XM (i), & for ​ i = 1, 2, \dots, N . \end{array}

For each line

y = y [i - 1]

the polynomial is evaluated at

M (i)

equispaced points between

X1 (i)

and

XM (i)

inclusive.

NAG C Library Function Document

nag_2d_cheb_eval (e02cbc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose