Root of equation

The roots of equation (8.95) are the open-loop poles or eigenvalues. For the closed-loop system described by equation (8.94), the characteristic equation is... 

The roots of equation (8.96) are the closed-loop poles or eigenvalues. 

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... 

Expansion in Series 60. Finite Difference Calculus 60. Interpolation 64. Roots of Equations 69. 

The time dependence of the various species concentrations will depend on the relative magnitudes of the four rate constants. In some cases the curves will involve a simple exponential rise to an asymptote, as is the case for irreversible reactions. In other cases the possibility of overshoot (exists, as indicated in Figure 5.2. Whether or not this phenomenon will occur depends on the relative magnitudes of the rate constants and the initial conditions. However, the fact that both roots of equation 5.2.34 must be real requires that there be only one maximum in the curve for R(t) or S(t). 

Where we note that the limit of n/ n — 1) -> 1 as n becomes indefinitely large. Of course, the noise level we want will be the square root of equation 44-75a. 

Equation 21.3-10 provides the value of the bed diameter D for a given allowable pressure drop, (-AP), a value of W calculated as described in Section 21.5, and known values of the other quantities. Since a, j8, and are all positive, from the Descartes rule of signs (Section 14.3.3), there is only one positive real root of equation 21.3-10. If the equations are solved for L instead of D, a cubic equation results. 

A dichotomy arises in attempting to minimize function (h). You can either (1) minimize the cost function (h) directly or (2) find the roots of Equation (i). Which is the best procedure In general it is easier to minimize C directly by a numerical method rather than take the derivative of C, equate it to zero, and solve the resulting nonlinear equation. This guideline also applies to functions of several variables. 

APPLICATION OF THE GIBBS FUNCTION TO CHEMICAL CHANGES The only physically meaningful root of Equation (12.20) is... 

Therefore, given the initial conditions, first find z, then the mean reaction time as a function of can be calculated. According to Vieta s formulas, the two roots of Equation 2-18 satisfy... 

The quadratic equation (10) has two roots and the shelf life is obtained by computing the root of Equation (10) that is smaller than a reference point, which is defined as... 

Therefore, the shelf life is the root smaller than 28.90. A simple and practical tool to compute the roots of Equation (12) is perhaps solving the following equivalent problem. Find such that it minimizes the absolute value of /( ). This root is obtained by using the quasi-Newton line search (QNLS) algorithm [13]. The computer program requires an initial point and we recommend using the value... 

The approach to estimate the shelf life of a single batch can be applied to the pooled stability data from all batches. Thus, the shelf life is obtained by finding the minimum root of Equation (10), where a and b are the estimates of a and p from model (45), respectively, x is the average sampling time, and Sxx is defined by Equation (4) and considering for all sampling times. 

The two roots of equation (3.15) may be abstracted by routine application of the quadratic formula. Before we do that, let us examine the results from three simplified special... 

The constants D and a were evaluated from two sets of the experimental release data using Equation 8. Although Equation 8 is in the form of an infinite series, numerical evaluation readily shows that the first term predominates even for very short times. This is also obvious from the plot in Figure 5 where it can be seen that a straight line may be drawn through the points starting from t = 0. Thus, the slopes of the lines taken after a reasonably short time will yield the exponential power, —fi2Dt/l2, where f3 is the first positive root of Equation 9. 

The smallest root of Equation 15-14 is zL when AL and BL are used. The largest root of Equation 15-14 is zg when Ag and Bg are used. 

The energies of the two TUg) terms are then given by the roots of equation (41) ... 

For a particular solution, where R is the particular root of Equation 24 found by analysis, and the respective Pi /s are substituted from Equation 14... 

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