Fitting orthogonal polynomials

The least squares estimate of the parameters s0, slf. .., sn in (3.100) are simply obtained by 

Rearranging the polynomial (3.100) to the canonical form (3.99) gives the estimates for the coefficients aQ, a, . .., a The following module based an 

1 SPECIFIED ND IS TOO LARGE ( IN THIS CASE A FURTHER OUTPUT IS 

Example 3.9 Polynomial regression through Forsythe orthogonalization 

The DATA statements of the following program include 12 data pairs 

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Try to fit polynomials of degree 3 through 7 to the data using the module M42. Discuss the advantages of orthogonal polynomials in view of your experiences. 

Forsythe, Generation and use of orthogonal polynomials for data-fitting with a digital computer. J. SIAM, 5 (1957) 74-88. 

D.B. Marsland, Data fitting by orthogonal polynomials, in. CACHE Thermodynamics (ed. R.V. Jelinek), Sterling Swift, Manchaca TX, 1971. 

From a computational point of view it is important to ensure that the problem is not ill-conditioned so as to maintain numerical stability. Therefore, the use of rational functions as proposed by Bergmann et a/.81 is combined with the established practice of fitting polynomials to given data using orthogonal polynomial bases.83 A system of linear equations is solved at each iteration.81 Hence, the condition number83 of this system may be used to monitor numerical stability. 

This is one of the variants of the finite element methods. The essence of orthogonal collocation (OC) is that a set of orthogonal polynomials is fitted to the unknown function, such that at every node point, there is an exact fit. The points are called collocation points, and the set of polynomials is chosen suitably, usually as Jacobi polynomials. The optimal choice of collocation points is to make them the roots of the polynomials. There are tables of such roots, and thus point placements, in Appendix A. The notable things here are the small number of points used (normally, about 10 or so will do), their... 

A computationally efficient method of function fitting using an orthogonal polynomial expansion is presented for approximating continuous wall temperature profiles. 

As a check, the low temperature experimental heat capacity values are smoothed by fitting the data with orthogonal polynomials over selected overlapping temperature intervals. This fitting procedure gave an S (298.15 K) identical to that reported by Osborne and Flotow (2 ). The ice calorimetric data of Satoh (7) are not extensive enough to reliably define the entire heat capacity dependence from 350 to 1086 K, however the three data prints at 373 K, 578 K, and 773 K are sufficient to define a reasonable temperature dependence. 

The low temperature heat capacities of Mo(cr) have been measured by Clusius and Franzosini ( ) between 16 and 256 K (63 measured Cp points). The reported Cp values are smoothed by fitting the data with orthogonal polynomials over selected overlapping temperature intervals. This fitting procedure also includes the smoothed Cp values (275-335 K) reported in the critical evaluation by Ditmars et al. (2 ) so as to provide smoothly varying heat capacity values in the range 256-275 K. The data of Simon and Zeidler (3 ), 15-238 K, are as much as 2,5% high below 78 K and as much as 1% low above 78 K. 

In the same paper, Pankratz et al. ( ) reported measurements of the high temperature enthalpies (401-1794 K) for VN g which were obtained in a copper-block drop calorimeter. The subnitride sample was the same as that used in their combustion work and was contained in Pt-Rh alloy capsules during the "drop" experiments. Temperature measurements were based on the IPTS-68 scale. A technique employing orthogonal polynomials is used to fit their experimental enthalpies by computer. The curve is constrained to join smoothly with the low temperature C data near 298.15 K. The average deviation of the smoothed enthalpies from the experimental values is +0.46% the maximum deviation is +0.73% at 702 K. C data above 1800 K are obtained by graphical extrapolation. No anomalies are observed in either the low temperature C data or the high temperature enthalpies. 

Low temperature heat capacities of ZnS0 (cr, o) have been measured by Weller ( ) from 51.7 - 296.5 K. A small heat capacity maximum was observed at 124.37 K. Our adopted value of S°(298.15 K) = 26.42+0.3 cal K mol obtained from C is based on S (51 K) = 2.27 cal K mol obtained by Weller (1 ) by extrapolation of the measured heat capacity with a combination of Debye and Einstein functions. We have smoothed the data of Weller (H)) by fitting the data with orthogonal polynomials over selected overlapping temperature intervals. 

The adopted heat capacities are derived from the heat capacity (, and enthalpy (2) data by fitting the experimental data with orthogonal polynomials over selected overlapping temperature intervals. The temperature region of 1-9 K is described by the equation C 3.281 x... 

The Chebyshev polynomials of the first kind have long been recognized as an effective basis for fitting non-periodic ftinctions.[7] In many aspects they resemble the Fourier basis for periodic systems. This special type of classical orthogonal polynomials can be generated by the following three-term recurrence relationship [8]... 

Orthogonal polynomials are particularly useful when the order of the equation is not known beforehand. The problem of finding the lowest-order polynomial to represent the data adequately can be achieved by first fitting a straight line, then a quadratic curve, then a cubic, and so on. At each stage it is only necessary to determine one additional parameter and apply the f-test to estimate the significance of each additional term. 

What coordinate systems-one for polynomials of the nth degree and one for polynomials of the n + 1st degree-will make the study of this difference simplest The answer will depend on the rule by which we fit. For least squares, the rule with which we are all fortunately or unfortunately most familiar, the answer is a suitable set of orthogonal polynomials... 

The estimation was of the parameters of a second-degree polynomial fitted to 20 equally spaced points. Thus three coefficients were calculated corresponding to the constant term and the linear and quadratic orthogonal polynomials. The coefficients, obviously uncorrelated, were rescaled to have variance 1 under least squares and the Gaussian distribution. The discussion will be in terms of the sum of these three variances. 

As mentioned in Chapter 1 of Vol. 2 (Buzzi-Ferraris and Manenti, 2010b), the orthogonal polynomial that best fits the selection of the P support points used to build the interpolating polynomial is the P-order Chebyshev polynomial. 

Besides the aforementioned time-domain approaches, many frequency-domain methods have also been developed and widely used. Examples are the complex curve fitting method [153], the maximum entropy method [4,263], the pole/zero assignment technique [271], the simultaneous frequency-domain approach [62], the rational fraction polynomial approach [219], the orthogonal polynomial approach [264], the polyreference frequency-domain approach [73], the multi-reference simultaneous frequency-domain approach [64] and the best-fit reciprocal vectors method [173]. 

In the treatment of data which exhibit curvilinear relationships such as those of Bard (1951) the curves may often be fitted through use of orthogonal polynomials. Where the values of the independent variable are... 

As for efficiency, this appears to be high (Magno et al s 1983 result to the contrary is doubtful), although one has yet to see - as far as I am aware - a definite error analysis, for example the question of how well the P-derivatives match those of C. In principle, one can imagine a polynomial P that precisely fits C at all but swings wildly everywhere else. By choosing the set as the roots of orthogonal polynomials,... 

Equation (29) may be extrapolated outside the experimental range-limitation of the value of x is required only for the fitting procedure. The fitting of experimental results by orthogonal polynomials and their transformation to the Chebyshev form by computer is straightforward the experimental points should be evenly spaced throughout the temperature range, but in practice little difficulty arises if this condition is not fully met. 

In a general case parameters re, XdP and y must be determined by self-consistent two-parameter fitting. Owing to the property of orthogonality of Laguerre polynomials, one has for the spectral band shapes... 

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