Polynomial third-degree

This third-degree polynomial has the value zero with two eigenvalues ... 

The correlation for C° of the ideal gas at low pressure is a third degree polynomial, which is a function of temperature. 

Equation 5-197 is a polynomial of the third degree, and by employing either a numerieal method or a spreadsheet paekage sueh as Mierosoft Exeel, the roots (C ) of the equation ean be determined. A developed eomputer program PROGS 1 using the Newton-Raphson method to determine was used. The Newton-Raphson method for the roots of Equation 5-197 is... 

The fraction undissolved data until the critical time can be least-square fitted to a third degree polynomial in time as dictated by Eq. (29). The moments of distribution ij, p2, and p3 can be evaluated from Eqs. (30) through (32), with three equations used to solve for three unknowns. These values may be used as first estimates in a nonlinear least-squares fit program, and the curve will, hence, reveal the best values of both shape factor, size distribution, and A -value. 

Equation 56-27 contains scaled coefficients for the zeroth through third derivative convolution functions, using a third degree polynomial fitting function. The first row of equation 56-27 contains the coefficients for smoothing, the second row contains the coefficients for the first derivative, and so forth. 

Final substitution of v will give the sought for result, which will be a ratio of fourth and third degree polynomials in r. 

Cubic interpolation to find the minimum of f(x) is based on approximating the objective function by a third-degree polynomial within the interval of interest and then determining the associated stationary point of the polynomial... 

A polynomial is represented by a sum of symbols raised to different powers, each with a different coefficient. For example, 3X3—2x+l involves a sum of x raised to the third, first and zeroth powers (remember that jc° = 1) with coefficients 3, —2, and 1, respectively. The highest power indicates the degree of the polynomial and so, for this example, the expression is a polynomial of the third degree. 

MATLAB has a built-in root finder for scalar equations f(x) = 0 in one real variable x that are in standard form. The built-in MATLAB function is fzero. The use of fzero hinges on a user-defined function, such as the function f inside the following fzero tester, called fzerotryl, that we apply to our previously studied third degree polynomial. 

Figure 3.9 3,n -lattices a) for a second-degree polynomial, b) for an incomplete-third-degree polynomial, c) for a third-degree polynomial, d) for a fourth-degree polynomial... 

For incomplete-third, third- and fourth-degree polynomials the relationships are derived in much the same way. So, for the incomplete-third-degree polynomial ... 

Table 3.38 tabulates the D-optimal design for the derivation of a ternary system third-degree polynomial. Following this design the coefficients are obtained for a third-degree polynomial having the same form as that from a conventional simplex lattice ... 

Table 3.38 D-optimal design for a ternary system third-degree polynomial 3,3 ...

A nonlinear increase of Ss upon increasing 0 is evident for all the adsorbate/adsorbent systems (Figure 3a) the curvatures reflect many-body interactions between a single xenon atom with benzene molecules within the zeolite supercage. Regressional fitting by a third-degree polynomial to the experimental values produces ... 

Figure 5-1 Examples of Savitsky Golay smoothing. A second-degree polynomial was used for the left column and a third-degree polynomial was used for the right column with a 5-, 15-, and 25-point smooth.

Curves of pressure dependence of the separation temperature of (1-x) water+(x)methyl-di(n-amyl)phosphinoxide for six mole fractions have been measured. The results of the measurements are given in Fig 2. Separation lines have been approximated by a third-degree polynomial... 

Before data can be interpreted, whether by analytic or numerical means, it is sometimes necessary to smooth the data so that differences from point to point do not show random fluctuations. This is the equivalent of drawing a smooth curve through the data points on a graph, but it can be done more objectively by numerical methods, several of which are described elsewhere. We shall give here one method, based on fitting a third-degree polynomial to 2n + 1 adjacent points. For n= 2, the smoothed value of y, is... 

The shape described by a spline between two adjacent points, or knots, is a cubic, third-degree polynomial. For the six pairs of data points representing our magnesium study, we would consider the curve connecting the data to comprise five cubic polynomials. Each of these take the form... 

The numerator of the second differential is a third-degree polynomial which may have 1 or 3 real roots and, thus may have 0,1,2, or 3 positive roots and as many inflection points. The quadratic isotherm is the first equation to which the experimental data should be fitted when an inflection point is observed. 

Table 1.2 The accuracy of first, second and third degree polynomial approximations to the function f(x)=e ...

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