void	nag_1d_cheb_fit (Integer m, Integer kplus1, Integer tda, const double x[], const double y[], const double w[], double a[], double s[], NagError *fail)

3

Description

nag_1d_cheb_fit (e02adc) determines least squares polynomial approximations of degrees

0, 1, \dots, k

to the set of data points

(x_{r}, y_{r})

with weights

w_{r}

, for

r = 1, 2, \dots, m

The approximation of degree

i

has the property that it minimizes

σ_{i}

the sum of squares of the weighted residuals

ε_{r}

, where

ε_{r} = w_{r} (y_{r} - f_{r})

and

f_{r}

is the value of the polynomial of degree

i

at the

r

th data point.

Each polynomial is represented in Chebyshev series form with normalized argument

\bar{x}

. This argument lies in the range

- 1

+ 1

and is related to the original variable

x

by the linear transformation

\bar{x} = \frac{(2 x - x_{\max} - x_{\min})}{(x_{\max} - x_{\min})} .

Here

x_{\max}

and

x_{\min}

are respectively the largest and smallest values of

x_{r}

. The polynomial approximation of degree

i

is represented as

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

where

T_{j} (\bar{x})

is the Chebyshev polynomial of the first kind of degree

j

with argument

\bar{x}

For

i = 0, 1, \dots, k

, the function produces the values of

a_{i + 1, j + 1}

, for

j = 0, 1, \dots, i

, together with the value of the root mean square residual

s_{i} = \sqrt{σ_{i} / (m - i - 1)}

. In the case

m = i + 1

the function sets the value of

s_{i}

to zero.

The method employed is due to Forsythe (1957) and is based upon the generation of a set of polynomials orthogonal with respect to summation over the normalized dataset. The extensions due to Clenshaw (1960) to represent these polynomials as well as the approximating polynomials in their Chebyshev series forms are incorporated. The modifications suggested by Reinsch and Gentleman (Gentleman (1969)) to the method originally employed by Clenshaw for evaluating the orthogonal polynomials from their Chebyshev series representations 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 Chebyshev series representations of the polynomial approximations should be carried out using nag_1d_cheb_eval (e02aec).

4

References

Clenshaw C W (1960) Curve fitting with a digital computer Comput. J. 2 170–173

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

Forsythe G E (1957) Generation and use of orthogonal polynomials for data fitting with a digital computer J. Soc. Indust. Appl. Math. 5 74–88

Gentleman W M (1969) An error analysis of Goertzel's (Watt's) method for computing Fourier coefficients Comput. J. 12 160–165

Hayes J G (ed.) (1970) Numerical Approximation to Functions and Data Athlone Press, London

5

Arguments

1: $m$ – IntegerInput: On entry: the number $m$ of data points.

Constraint: $m \geq mdist \geq 2$ , where $mdist$ is the number of distinct $x$ values in the data.
2: $kplus1$ – IntegerInput: On entry: $k + 1$ , where $k$ is the maximum degree required.

Constraint: $0 < kplus1 \leq mdist$ , where $mdist$ is the number of distinct $x$ values in the data.
3: $tda$ – IntegerInput: On entry: the stride separating matrix column elements in the array a.

Constraint: $tda \geq kplus1$ .
4: $x [m]$ – const doubleInput: On entry: the values $x_{r}$ of the independent variable, for $r = 1, 2, \dots, m$ .

Constraint: the values must be supplied in nondecreasing order with $x [m - 1] > x [0]$ .
5: $y [m]$ – const doubleInput: On entry: the values $y_{r}$ of the dependent variable, for $r = 1, 2, \dots, m$ .
6: $w [m]$ – const doubleInput: On entry: the set of weights, $w_{r}$ , for $r = 1, 2, \dots, m$ . For advice on the choice of weights, see the e02 Chapter Introduction.

Constraint: $w [r] > 0.0$ , for $r = 0, 1, \dots, m - 1$ .
7: $a [kplus1 \times tda]$ – doubleOutput: On exit: the coefficients of $T_{j} (\bar{x})$ in the approximating polynomial of degree $i$ . $a [(i) \times tda + j]$ contains the coefficient $a_{i + 1, j + 1}$ , for $i = 0, 1, \dots, k$ and $j = 0, 1, \dots, i$ .
8: $s [kplus1]$ – doubleOutput: On exit: $s [i]$ contains the root mean square residual $s_{i}$ , for $i = 0, 1, \dots, k$ , as described in Section 3. For the interpretation of the values of the $s_{i}$ and their use in selecting an appropriate degree, see the e02 Chapter Introduction.
9: $fail$ – 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_2_INT_ARG_GT: On entry, $kplus1 = 〈value〉$ while the number of distinct $x$ values, $mdist = 〈value〉$ . These arguments must satisfy $kplus1 \leq mdist$ .
NE_2_INT_ARG_LT: On entry, $tda = 〈value〉$ while $kplus1 = 〈value〉$ .
The arguments must satisfy $tda \geq kplus1$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LT: On entry, kplus1 must not be less than 1: $kplus1 = 〈value〉$ .
NE_NO_NORMALISATION: On entry, all the $x [r]$ in the sequence $x [r]$ , $r = 0, 1, \dots, m - 1$ are the same.
NE_NOT_NON_DECREASING: On entry, the sequence $x [r]$ , $r = 0, 1, \dots, m - 1$ is not in nondecreasing order.
NE_WEIGHTS_NOT_POSITIVE: On entry, the weights are not strictly positive: $w [〈value〉] = 〈value〉$ .

7

Accuracy

No error analysis for the method has been published. Practical experience with the method, however, is generally extremely satisfactory.

8

Parallelism and Performance

nag_1d_cheb_fit (e02adc) is not threaded in any implementation.

9

Further Comments

The time taken by nag_1d_cheb_fit (e02adc) is approximately proportional to

m (k + 1) (k + 11)

The approximating polynomials may exhibit undesirable oscillations (particularly near the ends of the range) if the maximum degree

k

exceeds a critical value which depends on the number of data points

m

and their relative positions. As a rough guide, for equally spaced data, this critical value is about

2 \times \sqrt{m}

. For further details see page 60 of Hayes (1970).

10

Example

Determine weighted least squares polynomial approximations of degrees 0, 1, 2 and 3 to a set of 11 prescribed data points. For the approximation of degree 3, tabulate the data and the corresponding values of the approximating polynomial, together with the residual errors, and also the values of the approximating polynomial at points half-way between each pair of adjacent data points.

The example program supplied is written in a general form that will enable polynomial approximations of degrees

0, 1, \dots, k

to be obtained to

m

data points, with arbitrary positive weights, and the approximation of degree

k

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

NAG Library Function Document

nag_1d_cheb_fit (e02adc)

▸▿ 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

2

Specification