NAG FL Interface
e02adf (dim1_cheb_arb)
1
Purpose
e02adf computes weighted least squares polynomial approximations to an arbitrary set of data points.
2
Specification
Fortran Interface
Subroutine e02adf ( |
m, kplus1, lda, x, y, w, work1, work2, a, s, ifail) |
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(3*m), work2(2*kplus1), 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) |
|
C++ Header Interface
#include <nag.h> extern "C" {
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 to the set of data points with weights , for .
The approximation of degree
has the property that it minimizes
the sum of squares of the weighted residuals
, where
and
is the value of the polynomial of degree
at the
th data point.
Each polynomial is represented in Chebyshev series form with normalized argument
. This argument lies in the range
to
and is related to the original variable
by the linear transformation
Here
and
are respectively the largest and smallest values of
. The polynomial approximation of degree
is represented as
where
, for
, are the Chebyshev polynomials of the first kind of degree
with argument
.
For , the routine produces the values of , for , together with the value of the root-mean-square residual . In the case the routine sets the value of 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:
– Integer
Input
-
On entry: the number of data points.
Constraint:
, where is the number of distinct values in the data.
-
2:
– Integer
Input
-
On entry: , where is the maximum degree required.
Constraint:
, where is the number of distinct values in the data.
-
3:
– Integer
Input
-
On entry: the first dimension of the array
a as declared in the (sub)program from which
e02adf is called.
Constraint:
.
-
4:
– Real (Kind=nag_wp) array
Input
-
On entry: the values
of the independent variable, for .
Constraint:
the values must be supplied in nondecreasing order with .
-
5:
– Real (Kind=nag_wp) array
Input
-
On entry: the values
of the dependent variable, for .
-
6:
– Real (Kind=nag_wp) array
Input
-
On entry: the set of weights,
, for
. For advice on the choice of weights, see
Section 2.1.2 in the
E02 Chapter Introduction.
Constraint:
, for .
-
7:
– Real (Kind=nag_wp) array
Workspace
-
8:
– Real (Kind=nag_wp) array
Workspace
-
-
9:
– Real (Kind=nag_wp) array
Output
-
On exit: the coefficients of
in the approximating polynomial of degree . contains the coefficient , for and .
-
10:
– Real (Kind=nag_wp) array
Output
-
On exit:
contains the root-mean-square residual
, for
, as described in
Section 3. For the interpretation of the values of the
and their use in selecting an appropriate degree, see
Section 3.1 in the
E02 Chapter Introduction.
-
11:
– Integer
Input/Output
-
On entry:
ifail must be set to
,
. If you are unfamiliar with this argument you should refer to
Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
.
When the value is used it is essential to test the value of ifail on exit.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
or
, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
-
On entry, and .
Constraint: .
-
On entry, , and .
Constraint: .
-
On entry, all have the same value: .
-
On entry, .
Constraint: .
On entry, and .
Constraint: .
On entry,
and
.
Constraint:
, where
mdist is the number of distinct
x values.
-
On entry, and .
Constraint: .
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.
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.
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.
The time taken is approximately proportional to .
The approximating polynomials may exhibit undesirable oscillations (particularly near the ends of the range) if the maximum degree
exceeds a critical value which depends on the number of data points
and their relative positions. As a rough guide, for equally-spaced data, this critical value is about
. For further details see page 60 of
Hayes (1970).
10
Example
Determine weighted least squares polynomial approximations of degrees , , and to a set of prescribed data points. For the approximation of degree , 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
to be obtained to
data points, with arbitrary positive weights, and the approximation of degree
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.
10.1
Program Text
10.2
Program Data
10.3
Program Results