nag_fit_2dcheb_eval (e02cb) evaluates a bivariate polynomial from the rectangular array of coefficients in its double Chebyshev series representation.

Syntax

[ff, ifail] = e02cb(mfirst, k, l, x, xmin, xmax, y, ymin, ymax, a, 'mlast', mlast)

[ff, ifail] = nag_fit_2dcheb_eval(mfirst, k, l, x, xmin, xmax, y, ymin, ymax, a, 'mlast', mlast)

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_fit_2dcheb_lines (e02ca), 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 Further Comments). 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.

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

Parameters

Compulsory Input Parameters

1: $mfirst$ – int64int32nag_int scalar: The index of the first and last $x$ value in the array $x$ at which the evaluation is required respectively (see Further Comments).

Constraint: $mlast \geq mfirst$ .
2: $k$ – int64int32nag_int scalar
3: $l$ – int64int32nag_int scalar: The degree $k$ of $x$ and $l$ of $y$ , respectively, in the polynomial.

Constraint: $k \geq 0$ and $l \geq 0$ .
4: $x (mlast)$ – double array: $x (i)$ , for $i = mfirst, \dots, mlast$ , must contain the $x$ values at which the evaluation is required.

Constraint: $xmin \leq x (i) \leq xmax$ , for all $i$ .
5: $xmin$ – double scalar
6: $xmax$ – double scalar: The lower and upper ends, $x_{\min}$ and $x_{\max}$ , of the range of the variable $x$ (see Description).
The values of xmin and xmax may depend on the value of $y$ (e.g., when the polynomial has been derived using nag_fit_2dcheb_lines (e02ca)).

Constraint: $xmax > xmin$ .
7: $y$ – double scalar: The value of the $y$ coordinate of all the points at which the evaluation is required.

Constraint: $ymin \leq y \leq ymax$ .
8: $ymin$ – double scalar
9: $ymax$ – double scalar: The lower and upper ends, $y_{\min}$ and $y_{\max}$ , of the range of the variable $y$ (see Description).

Constraint: $ymax > ymin$ .
10: $a (na)$ – double array: na, the dimension of the array, must satisfy the constraint $na \geq (k + 1) \times (l + 1)$ , the number of coefficients in a polynomial of the specified degree.
The Chebyshev coefficients of the polynomial. The coefficient $a_{i j}$ defined according to the standard convention (see Description) must be in $a (i \times (l + 1) + j + 1)$ .

Optional Input Parameters

1: $mlast$ – int64int32nag_int scalar: Default: For mlast, the dimension of the array x.
The index of the first and last $x$ value in the array $x$ at which the evaluation is required respectively (see Further Comments).

Constraint: $mlast \geq mfirst$ .

Output Parameters

1: $ff (mlast)$ – double array: $ff (i)$ gives the value of the polynomial at the point $(x_{i}, y)$ , for $i = mfirst, \dots, mlast$ .
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$mfirst > mlast$ ,
or	$k < 0$ ,
or	$l < 0$ ,
or	$na < (k + 1) \times (l + 1)$ ,
or	$nwork < k + 1$ .

$ifail = 2$

On entry,	$ymin \geq ymax$ ,
or	$y < ymin$ ,
or	$y > ymax$ .

$ifail = 3$

On entry,	$xmin \geq xmax$ ,
or	$x (i) < xmin$ , or $x (i) > xmax$ , for some $i = mfirst, mfirst + 1, \dots, mlast$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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.

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_fit_2dcheb_lines (e02ca), which provides the double 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_fit_2dcheb_lines (e02ca). 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_fit_2dcheb_lines (e02ca) (see Further Comments in nag_fit_2dcheb_lines (e02ca)).

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_fit_2dcheb_lines (e02ca) 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_fit_2dcheb_lines (e02ca)) 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)

as their

y

coordinate (from which values the residuals of the fit may be derived).

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

For each line

y = y (i)

the polynomial is evaluated at

M (i)

equispaced points between

X1 (i)

and

XM (i)

inclusive.

Open in the MATLAB editor: e02cb_example

function e02cb_example


fprintf('e02cb example results\n\n');


% domain
dx   = 4/19;
x    = [0.5:dx:4.5];
y    = [0:dx:4];
xmin = 0.1; xmax = 4.5;
ymin = 0;   ymax = 4;

% Fit characteristics and coefficients
k = int64(3);
l = int64(2);
a = [15.3482   5.15073  -2.20140   1.14719  -0.64419   0.30464 ...
     -0.4901  -0.00314  -6.69912   0.00153   3.00033  -0.00022];

% Evaluations on mesh
mfirst = int64(1);
mlast  = int64(20);

% Evaluate fit 
for i = 1:mlast
  [fit(:,i), ifail] = e02cb( ...
                             mfirst, k, l, x, xmin, xmax, y(i), ...
                             ymin, ymax, a, 'mlast', mlast);
  sol = [x; fit(:,i)'];
end

fprintf('\nThe bivariate polynomial fit values for y = %5.1f are:\n',y(mlast));
sol = [x; fit(:,mlast)'];
fprintf('      x       p(x,y=4)\n', sol);
fprintf('%11.4f%11.4f\n', sol);

fig1 = figure;
meshc(y,x,fit);
title('Least-squares bi-variate polynomial fit');
xlabel('x');
ylabel('y');
zlabel('p(x,y)');

e02cb example results


The bivariate polynomial fit values for y =   4.0 are:
      x       p(x,y=4)
     0.5000     3.5575
     0.7105     6.8145
     0.9211     9.3405
     1.1316    11.1986
     1.3421    12.4520
     1.5526    13.1637
     1.7632    13.3970
     1.9737    13.2148
     2.1842    12.6803
     2.3947    11.8566
     2.6053    10.8069
     2.8158     9.5942
     3.0263     8.2816
     3.2368     6.9323
     3.4474     5.6094
     3.6579     4.3760
     3.8684     3.2951
     4.0789     2.4300
     4.2895     1.8437
     4.5000     1.5993

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