NAG Library Routine Document

E02CBF

. However, these normalized variables will usually have been derived (as when the polynomial has been computed by E02CAF, 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 8). For this reason, the subroutine 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 routine 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 subroutine, 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 subroutine 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 Parameters

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 8).
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) – REAL (KIND=nag_wp) arrayInput: On entry: $X (i)$ , 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 $i$ .
6: XMIN – REAL (KIND=nag_wp)Input
7: XMAX – REAL (KIND=nag_wp)Input: 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 E02CAF).

Constraint: $XMAX > XMIN$ .
8: Y – REAL (KIND=nag_wp)Input: 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 – REAL (KIND=nag_wp)Input
10: YMAX – REAL (KIND=nag_wp)Input: 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) – REAL (KIND=nag_wp) arrayOutput: On exit: $FF (i)$ gives the value of the polynomial at the point $(x_{i}, y)$ , for $i = MFIRST, \dots, MLAST$ .
12: A(NA) – REAL (KIND=nag_wp) arrayInput: 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 $A (i \times (l + 1) + j + 1)$ .
13: NA – INTEGERInput: On entry: the dimension of the array A as declared in the (sub)program from which E02CBF is called.
Constraint: $NA \geq (K + 1) \times (L + 1)$ , the number of coefficients in a polynomial of the specified degree.
14: WORK(NWORK) – REAL (KIND=nag_wp) arrayWorkspace
15: NWORK – INTEGERInput: On entry: the dimension of the array WORK as declared in the (sub)program from which E02CBF is called.
Constraint: $NWORK \geq K + 1$ .
16: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

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 $i = MFIRST, MFIRST + 1, \dots, MLAST$ .

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 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 subroutine is suitable for evaluating the polynomial surface fits produced by the subroutine E02CAF, which provides the real array A in the required form. For this use, the values of

y_{\min}

and

y_{\max}

supplied to the present subroutine must be the same as those supplied to E02CAF. 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 E02CAF (see Section 8 in E02CAF).

The parameters 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 E02CAF to fit a set of data, you wish to evaluate the resulting polynomial at all the data values. In this case, if the parameters X, Y, MFIRST and MLAST of the present routine are set respectively (in terms of parameters of E02CAF) to X,

Y (S)

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

and

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

, the routine 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).

9 Example

This example reads data in the following order, using the notation of the parameter 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.

NAG Library Routine DocumentE02CBF

+− Contents

1 Purpose

2 Specification

3 Description