NAG Library Routine Document

E02AFF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

E02AFF computes the coefficients of a polynomial, in its Chebyshev series form, which interpolates (passes exactly through) data at a special set of points. Least squares polynomial approximations can also be obtained.

2 Specification

SUBROUTINE E02AFF (

NPLUS1, F, A, IFAIL)

INTEGER	NPLUS1, IFAIL
REAL (KIND=nag_wp)	F(NPLUS1), A(NPLUS1)

3 Description

E02AFF computes the coefficients

a_{j}

, for

j = 1, 2, \dots, n + 1

, in the Chebyshev series

\frac{1}{2} a_{1} T_{0} (\bar{x}) + a_{2} T_{1} (\bar{x}) + a_{3} T_{2} (\bar{x}) + \dots + a_{n + 1} T_{n} (\bar{x}),

which interpolates the data

f_{r}

at the points

{\bar{x}}_{r} = \cos ((r - 1) π / n), r = 1, 2, \dots, n + 1 .

Here

T_{j} (\bar{x})

denotes the Chebyshev polynomial of the first kind of degree

j

with argument

\bar{x}

. The use of these points minimizes the risk of unwanted fluctuations in the polynomial and is recommended when the data abscissae can be chosen by you, e.g., when the data is given as a graph. For further advantages of this choice of points, see Clenshaw (1962).

In terms of your original variables,

x

say, the values of

x

at which the data

f_{r}

are to be provided are

x_{r} = \frac{1}{2} (x_{\max} - x_{\min}) \cos (π (r - 1) / n) + \frac{1}{2} (x_{\max} + x_{\min}), r = 1, 2, \dots, n + 1

where

x_{\max}

and

x_{\min}

are respectively the upper and lower ends of the range of

x

over which you wish to interpolate.

Truncation of the resulting series after the term involving

a_{i + 1}

, say, yields a least squares approximation to the data. This approximation,

p (\bar{x})

, say, is the polynomial of degree

i

which minimizes

\frac{1}{2} ε_{1}^{2} + ε_{2}^{2} + ε_{3}^{2} + \dots + ε_{n}^{2} + \frac{1}{2} ε_{n + 1}^{2},

where the residual

ε_{r} = p ({\bar{x}}_{r}) - f_{r}

, for

r = 1, 2, \dots, n + 1

The method employed is based on the application of the three-term recurrence relation due to Clenshaw (1955) for the evaluation of the defining expression for the Chebyshev coefficients (see, for example, Clenshaw (1962)). The modifications to this recurrence relation suggested by Reinsch and Gentleman (see Gentleman (1969)) are used to give greater numerical stability.

For further details of the algorithm and its use see Cox (1974) and Cox and Hayes (1973).

Subsequent evaluation of the computed polynomial, perhaps truncated after an appropriate number of terms, should be carried out using E02AEF.

4 References

Clenshaw C W (1955) A note on the summation of Chebyshev series Math. Tables Aids Comput. 9 118–120

Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO

Cox M G (1974) A data-fitting package for the non-specialist user Software for Numerical Mathematics (ed D J Evans) Academic Press

Cox M G and Hayes J G (1973) Curve fitting: a guide and suite of algorithms for the non-specialist user NPL Report NAC26 National Physical Laboratory

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: NPLUS1 – INTEGERInput

On entry: the number

n + 1

of data points (one greater than the degree

n

of the interpolating polynomial).

Constraint:

NPLUS1 \geq 2

2: F(NPLUS1) – REAL (KIND=nag_wp) arrayInput

On entry: for

r = 1, 2, \dots, n + 1

F (r)

must contain

f_{r}

the value of the dependent variable (ordinate) corresponding to the value

{\bar{x}}_{r} = \cos (π (r - 1) / n)

of the independent variable (abscissa)

\bar{x}

, or equivalently to the value

x (r) = \frac{1}{2} (x_{\max} - x_{\min}) \cos (π (r - 1) / n) + \frac{1}{2} (x_{\max} + x_{\min})

of your original variable

x

. Here

x_{\max}

and

x_{\min}

are respectively the upper and lower ends of the range over which you wish to interpolate.

3: A(NPLUS1) – REAL (KIND=nag_wp) arrayOutput

On exit:

A (j)

is the coefficient

a_{j}

in the interpolating polynomial, for

j = 1, 2, \dots, n + 1

4: 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,

NPLUS1 < 2

7 Accuracy

The rounding errors committed are such that the computed coefficients are exact for a slightly perturbed set of ordinates

f_{r} + δ f_{r}

. The ratio of the sum of the absolute values of the

δ f_{r}

to the sum of the absolute values of the

f_{r}

is less than a small multiple of

(n + 1) ε

, where

ε

is the machine precision.

8 Further Comments

The time taken is approximately proportional to

{(n + 1)}^{2} + 30

For choice of degree when using the routine for least squares approximation, see Section 3.2 in the E02 Chapter Introduction.

9 Example

Determine the Chebyshev coefficients of the polynomial which interpolates the data

{\bar{x}}_{r}, f_{r}

, for

r = 1, 2, \dots, 11

, where

{\bar{x}}_{r} = \cos (π \times (r - 1) / 10)

and

f_{r} = e^{{\bar{x}}_{r}}

. Evaluate, for comparison with the values of

f_{r}

, the resulting Chebyshev series at

{\bar{x}}_{r}

, for

r = 1, 2, \dots, 11

The example program supplied is written in a general form that will enable polynomial interpolations of arbitrary data at the cosine points

\cos (π \times (r - 1) / n)

, for

r = 1, 2, \dots, n + 1

, to be obtained for any

n

(

= NPLUS1 - 1

). Note that E02AEF is used to evaluate the interpolating polynomial. The program is self-starting in that any number of datasets can be supplied.

NAG Library Routine DocumentE02AFF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

E02AFF