NAG Library Function Document

nag_1d_cheb_interp_fit (e02afc)

+− 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

nag_1d_cheb_interp_fit (e02afc) 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

#include <nag.h>

#include <nage02.h>

void	nag_1d_cheb_interp_fit (Integer nplus1, const double f[], double a[], NagError *fail)

3 Description

nag_1d_cheb_interp_fit (e02afc) 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 you can choose the data abscissae, 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 ((r - 1) π / 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), Cox and Hayes (1973).

Subsequent evaluation of the computed polynomial, perhaps truncated after an appropriate number of terms, should be carried out using nag_1d_cheb_eval (e02aec).

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 Arguments

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] – const doubleInput

On entry: for

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

f [r - 1]

must contain

f_{r}

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

{\bar{x}}_{r} = \cos (\frac{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] – doubleOutput

On exit:

a [j - 1]

is the coefficient

a_{j}

in the interpolating polynomial, for

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

4: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_INT_ARG_LT: On entry, nplus1 must not be less than 2: $nplus1 = ⟨value⟩$ .

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 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_1d_cheb_interp_fit (e02afc) is approximately proportional to

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

For choice of degree when using the function for least squares approximation, see the e02 Chapter Introduction.

10 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 ((r - 1) π / 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 ((r - 1) π / n)

, for

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

to be obtained for any

n

(

= nplus1 - 1

). Note that nag_1d_cheb_eval (e02aec) is used to evaluate the interpolating polynomial. The program is self-starting in that any number of datasets can be supplied.

NAG Library Function Documentnag_1d_cheb_interp_fit (e02afc)

+− 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

NAG Library Function Document

nag_1d_cheb_interp_fit (e02afc)