Two nonlinear equations

Fig. 4.2. Newton-Raphson iteration for solving two nonlinear equations containing the unknown variables x and y. Planes are drawn tangent to the residual functions R and R2 at an initial estimate (r, > (o)) to the value of the root. The improved guess (v(l y(l)) is the point at which the tangent planes intersect each other and the plane R = 0.

The multidimensional counterpart to Newton s method is Newton-Raphson iteration. A mathematics professor once complained to me, with apparent sincerity, that he could visualize surfaces in no more than twelve dimensions. My perspective on hyperspace is less incisive, as perhaps is the reader s, so we will consider first a system of two nonlinear equations / = a and g = b with unknowns, v and y. 

If there is no added salt, c = 0, then we define simply c (cg = 0) = c. Above the cmc, the quantities Cj, C2, c, Cj +, and x are found from Equations 11 through 14 and the three mass balances. By eliminating Cl, C2, and c from these equations we obtain a system of two nonlinear equations with unknowns x and C] +. 

When the flow regime is not known, the analytical calculation of ut from Equation (12) requires choosing an equation for Cd, such as, for example, Equation (15), and solving a non-trivial system of two nonlinear equations. To avoid the analytical solution, an iterative procedure can be applied by guessing an initial value of CD or Rep ... 

Similarly, for the case of the reaction A -y B carried out in a CSTR. we could use Polymath or. MATLAB to solve two nonlinear equations in X and T. These two equations are combined mole balance... 

These are two nonlinear equations for the determination of the temperatures 1 and 2 with known starting temperatures lo and 20 and the known temperatures +00 and oo as functions of time. Thereby the problem of the transport of energy through radiation in principle is solved. The consideration of convection in this system is simple ... 

Calculation involves the solution of two nonlinear equations. We may solve for component A in terms of B and then substitute the expression into the CSTR equation for B. This results in... 

In the first instance, when the results were analyzed by simple mean and standard deviation analysis, Amico et al. [16-18] got large relative standard deviatiOTi, indicating limitatimi of this method for the proper characterizatiOTi of the diameter. Then, they used Weibull probability density and cumulative distribution functions [20,56,58] to estimate two parameters, the characteristic life and a dimensionless positive pure number, which were supposed to determine the shape and scale of the distribution curve. For this, they adopted two methods, the maximum likelihood technique, which requires the solution of two nonlinear equations, and the analytical method using the probability plot as mentioned earlier for coir fibers. 

Equations (4.6) and (4.7) form a system of two nonlinear equations for two unknowns v and p. Flow velocity is typically much less than the speed of sound. In the next section, this inequality is used to simplify and solve this system. 

The implemented algorithm is verified with a test problem (Floudas and Maranas, 1995) comprising two nonlinear equations subject to bounds on the two variables. 

After successfully solving a simple system of two nonlinear equations, the algorithm is next applied to a simple distillation problem that does not involve rigorous thermodynamic calculations. 

The principle of the method will be described on the basis of a set of two nonlinear equations... 

This equation can be solved numerically in three different ways. The first is by direct integration of equation (8.21) and is described in Section 8.5.2. The second method is to treat the objective function as an unconstrained optimization, which can be solved using any direct search technique such as Powell s method (1964). This approach is computationally faster than the direct integration but requires good initial estimates for the volumes. The third method is to differentiate equation (8.21) to give two nonlinear equations ... 

The method just described for two nonlinear equations is readily expandable to the case of k simultaneous nonlinear equations in k unknowns ... 

Consider a simple case of two nonlinear equations whose solution(s) depend upon some parameter vector 0. 

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