Polynomials high-degree

Let us lump together the m observations y (i = 1,..., m) into a vector y, the polynomial coefficients otj (/=0,...,n — 1) into a vector jc of unknowns, and define the (/—l)th power of the ith observable (uj> 1 as the current term atJ of the matrix Am, . We now apply the usual method. Polynomials of high degrees tend to generate nearly singular matrices A which result in excessive fluctuations. 

Therefore, irrespective of the beautiful algebraic form of Cioslowski s UDA one must conclude that its predictions are not in complete agreement with empirical findings [115, 116]. Furthermore, it was shown [117] that a hidden assumption behind Eq. (78) is that the coefficients b(B, k) of the characteristic polynomial are not mutually correlated. Empirical testing [118] revealed that, on the contrary, a high degree of correlation exists between these coefficients. 

In carrying ont polynomial fits, one must be alert to possible pathological behavior. If the observed quantities 7 vary in a smooth but rapid and complex way with x one will need a high-degree polynomial to fit the data. This leads to a potentially serious problem. Two high-degree polynomials of different order that each fit a limited data set very well may yield different interpolated values and derivatives. Another problem situation arises when the measured physical quantity represents a singular function for example the heat capacity of a pure fluid at its critical density tends smoothly to infinity as Tapproaches the critical temperature y. In such a case, a polynomial fit will improperly round off the sharp physical feature. 

Many problems can be found in large systems in which most of the degrees of freedom are artificially frozen. Then the potential energy, as a function of the coordinates over a wide range, can become very complex. An alternative procedure is to use interpolation techniques. By the common polynomial techniques all the values of either the potential or its derivatives can be exactly fitted whenever it is necessary but this leads to polynomials of a very high degree (hundreds of mesh points may have to be considered). Consequently, instabilities can appear, especially on the sides of the region studied (see Fig. 1 for a one dimensional case). 

Fig. 1. An example of a possible behavior of an interpolation polynomial of high degree, as compared with the exact function. This is what we call an instability on the sides of the range of interpolation...

Specific Volume of Water. The data for the specific volume of water are available over a wide range of temperatures and pressures. Normally high degree polynomials (ustially at least eight degrees) are used as the state equations of both steam and liquid water (10. 11). 

If P is sampled from a polynomial of degree d, then the right hand end of the series is truncated because a sufficiently high power of <5 (greater than d) annihilates the term. If sufficiently many a are zero, the left hand end of the series is also zero, so that p(z) = P(z), which is the interpolation condition. 

The method described here includes no approximation at the data treatment step, so it can be used generally. In addition, the required level of mathematical knowledge is not high, only a formula for polynomials of degree 2, therefore the logical basis can be easily understood. Moreover, statistical treatment of the obtained data is understandable with primary statistics. These are the merits to use this method at first in order to understand the way of determination of binding constants. 

For this reason and because the fourth order is usually a good compromise between accuracy and the issues arising from the high-degree polynomial... 

Table I and Fig. I indicate how various asymptotic functions grow with n. As is evident, the function 2" grows very rapidly with n. In fact, if a program needs 2" steps for execution, then when n = 40 the number of steps needed is I.I X 10. On a computer performing 1 billion steps per second, this would require 18.3 min (Table II). If n = 50, the same program would run for I3 days on this computer. When n = 60, 310.56 years will be required to execute the program, and when n = I 00 4x1013 years will be needed. So we may conclude that the utility of programs with exponential complexity is limited to small n (typically n 40). Programs that have a complexity that is a polynomial of high degree are also of limited utility.

For problems of small and medium dimensions, it is not important to repeat blocks of operations, carry out operations between constants, insert elaborated expressions useless to obtaining the solution, calculate high-degree polynomials with the standard form rather than the Horner technique (Buzzi-Ferraris and Manenti, 2010b), and so on. However, these factors become relevant when the systems are large. It is, therefore, useful that users write their portion of code efficiently. 

Chebyshev polynomials of rather high degree were necessary and additional techniques were required to suppress the Gibbs phenomena. In contrast, the polynomials used in our approach are of relatively low degree (say <20). They exploit the fast growth property of Chebyshev polynomials outside the interval [—1,1] to filter out undesired eigencomponents. 

---