E02AGF computes constrained weighted least squares polynomial approximations in Chebyshev series form to an arbitrary set of data points. The values of the approximations and any number of their derivatives can be specified at selected points.
E02AGF determines least squares polynomial approximations of degrees up to to the set of data points with weights , for . The value of , the maximum degree required, is to be prescribed by you. At each of the values , for , of the independent variable , the approximations and their derivatives up to order are constrained to have one of the values , for , specified by you, where .
The approximation of degree has the property that, subject to the imposed constraints, it minimizes , the sum of the squares of the weighted residuals , for , where
and is the value of the polynomial approximation 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
where and , specified by you, are respectively the lower and upper end points of the interval of over which the polynomials are to be defined.
The polynomial approximation of degree can be written as
where is the Chebyshev polynomial of the first kind of degree with argument . For , the routine produces the values of the coefficients , for , together with the value of the root mean square residual,
where is the number of data points with nonzero weight.
Values of the approximations may subsequently be computed using E02AEF or E02AKF.
First E02AGF determines a polynomial , of degree , which satisfies the given constraints, and a polynomial , of degree , which has value (or derivative) zero wherever a constrained value (or derivative) is specified. It then fits , for , with polynomials of the required degree in each with factor . Finally the coefficients of are added to the coefficients of these fits to give the coefficients of the constrained polynomial approximations to the data points , for . The method employed is given in Hayes (1970): it is an extension of Forsythe's orthogonal polynomials method (see Forsythe (1957)) as modified by Clenshaw (see Clenshaw (1960)).
Clenshaw C W (1960) Curve fitting with a digital computer Comput. J.2 170–173
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
Hayes J G (ed.) (1970) Numerical Approximation to Functions and Data Athlone Press, London
1: M – INTEGERInput
On entry: , the number of data points to be fitted.
2: KPLUS1 – INTEGERInput
On entry: , where is the maximum degree required.
is the total number of constraints and is the number of data points with nonzero weights and distinct abscissae which do not coincide with any of the .
3: LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which E02AGF is called.
4: XMIN – REAL (KIND=nag_wp)Input
5: XMAX – REAL (KIND=nag_wp)Input
On entry: the lower and upper end points, respectively, of the interval . Unless there are specific reasons to the contrary, it is recommended that XMIN and XMAX be set respectively to the lowest and highest value among the and . This avoids the danger of extrapolation provided there is a constraint point or data point with nonzero weight at each end point.
On entry: must contain the weight to be applied to the data point , for . For advice on the choice of weights see the E02 Chapter Introduction. Negative weights are treated as positive. A zero weight causes the corresponding data point to be ignored. Zero weight should be given to any data point whose and values both coincide with those of a constraint (otherwise the denominators involved in the root mean square residuals will be slightly in error).
9: MF – INTEGERInput
On entry: , the number of values of the independent variable at which a constraint is specified.
On entry: the values which the approximating polynomials and their derivatives are required to take at the points specified in XF. For each value of
, YF contains in successive elements the required value of the approximation, its first derivative, second derivative, th derivative, for . Thus the value, , which the th derivative of each approximation ( referring to the approximation itself) is required to take at the point must be contained in , where
where and . The derivatives are with respect to the independent variable .
12: LYF – INTEGERInput
On entry: the dimension of the array YF as declared in the (sub)program from which E02AGF is called.
On exit: contains , for , the root mean square residual corresponding to the approximating polynomial of degree . In the case where the number of data points with nonzero weight is equal to , is indeterminate: the routine sets it to zero. For the interpretation of the values of and their use in selecting an appropriate degree, see Section 3.1 in the E02 Chapter Introduction.
16: NP1 – INTEGEROutput
On exit: , where is the total number of constraint conditions imposed: .
On entry: the dimension of the array IWRK as declared in the (sub)program from which E02AGF is called.
21: IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to , . 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 is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this parameter, 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).
, where is the number of data points with nonzero weight and distinct abscissae which do not coincide with any . Thus there is no unique solution.
The polynomials and/or cannot be determined. The problem supplied is too ill-conditioned. This may occur when the constraint points are very close together, or large in number, or when an attempt is made to constrain high-order derivatives.
No complete error analysis exists for either the interpolating algorithm or the approximating algorithm. However, considerable experience with the approximating algorithm shows that it is generally extremely satisfactory. Also the moderate number of constraints, of low-order, which are typical of data fitting applications, are unlikely to cause difficulty with the interpolating routine.
8 Further Comments
The time taken to form the interpolating polynomial is approximately proportional to , and that to form the approximating polynomials is very approximately proportional to .
To carry out a least squares polynomial fit without constraints, use E02ADF. To carry out polynomial interpolation only, use E01AEF.
This example reads data in the following order, using the notation of the parameter list above: