Integer, Intent (In)	::	m, kplus1, lda
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(m), y(m), w(m)
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,kplus1)
Real (Kind=nag_wp), Intent (Out)	::	work1(3m), work2(2kplus1), s(kplus1)

C Header Interface

#include <nag.h>

void	e02adf_ (const Integer m, const Integer kplus1, const Integer lda, const double x[], const double y[], const double w[], double work1[], double work2[], double a[], double s[], Integer ifail)

The routine may be called by the names e02adf or nagf_fit_dim1_cheb_arb.

3 Description

e02adf 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})

, for

j = 0, 1, \dots, i

, are the Chebyshev polynomials of the first kind of degree

j

with argument

(\bar{x})

For

i = 0, 1, \dots, k

, the routine 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 routine sets the value of

s_{i}

to zero.

The method employed is due to Forsythe (1957) and is based on 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 (see 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) and Cox and Hayes (1973).

Subsequent evaluation of the Chebyshev series representations of the polynomial approximations should be carried out using e02aef.

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: $m$ – Integer Input: 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: $kplus1$ – Integer Input: 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: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which e02adf is called.

Constraint: $lda \geq kplus1$ .
4: $x (m)$ – Real (Kind=nag_wp) array Input: 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 $x (m) > x (1)$ .
5: $y (m)$ – Real (Kind=nag_wp) array Input: On entry: the values $y_{r}$ of the dependent variable, for $r = 1, 2, \dots, m$ .
6: $w (m)$ – Real (Kind=nag_wp) array Input: On entry: the set of weights, $w_{r}$ , for $r = 1, 2, \dots, m$ . For advice on the choice of weights, see Section 2.1.2 in the E02 Chapter Introduction.

Constraint: $w (r) > 0.0$ , for $r = 1, 2, \dots, m$ .
7: $work1 (3 \times m)$ – Real (Kind=nag_wp) array Workspace
8: $work2 (2 \times kplus1)$ – Real (Kind=nag_wp) array Workspace
9: $a (lda, kplus1)$ – Real (Kind=nag_wp) array Output: On exit: the coefficients of $T_{j} (\bar{x})$ in the approximating polynomial of degree $i$ . $a (i + 1, j + 1)$ contains the coefficient $a_{i + 1, j + 1}$ , for $i = 0, 1, \dots, k$ and $j = 0, 1, \dots, i$ .
10: $s (kplus1)$ – Real (Kind=nag_wp) array Output: On exit: $s (i + 1)$ 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 Section 3.1 in the E02 Chapter Introduction.
11: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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, $i = ⟨ value ⟩$ and $w (i) = ⟨ value ⟩$ .
Constraint: $w (i) > 0.0$ .

$ifail = 2$: On entry, $i = ⟨ value ⟩$ , $x (i) = ⟨ value ⟩$ and $x (i - 1) = ⟨ value ⟩$ .
Constraint: $x (i) \geq x (i - 1)$ .

$ifail = 3$: On entry, all $x (I)$ have the same value: $x (1) = ⟨ value ⟩$ .

$ifail = 4$: On entry, $kplus1 = ⟨ value ⟩$ .
Constraint: $kplus1 \geq 1$ .

On entry, $kplus1 = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $kplus1 \leq m$ .

On entry, $kplus1 = ⟨ value ⟩$ and $mdist = ⟨ value ⟩$ .
Constraint: $kplus1 \leq mdist$ , where mdist is the number of distinct x values.

$ifail = 5$: On entry, $lda = ⟨ value ⟩$ and $kplus1 = ⟨ value ⟩$ .
Constraint: $lda \geq kplus1$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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

e02adf is not threaded in any implementation.

9 Further Comments

The time taken 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. e02aef is used to evaluate the approximating polynomial. The program is self-starting in that any number of datasets can be supplied.

e02ad: FL CL CPP AD

NAG FL Interfacee02adf (dim1_​cheb_​arb)

▸▿ 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 FL Interface
e02adf (dim1_cheb_arb)