Numerical Solution of Algebraic Equations

Strictly speaking, one does not solve an equation numerically. One obtains an approximation to a root. However, with a computer or with a hand calculator, it is easy to obtain enough significant digits for almost any purpose. 

EXAMPLE 3.8 Use the method of trial and error to find the positive root of the equation 

SOLUTION We let f(x) = 2sin(x) — x, which vanishes at the root. It is convenient to use a spreadsheet to carry out the evaluation of the function f x). We put the formula in the B eolumn, filling it down to enough rows for the number of times we think we will have to evaluate the function. We then put trial values into column A and inspect the value of the function in column B. We find quickly that /(I) = 0.68294, and that /(2) = —0.1814, so that there must be a root between x = 1 and x = 2. We find that /(1.5) = 0.49499, so the root lies between 1.5 and 2. We find that /(1.75) = 0.21798, so the root is larger than 1.75. However, /(1.9) = —0.00740, so the root is smaller than 1.9. We find that /(1.89) = 0.00897, so the root is between 1.89 and 1.90. We find that /(1.895) = 0.000809 and that /(1.896) = —0.000829. To five significant digits, the root is X = 1.8955. 

This is a systematic variation of the method of trial and error. You start with two values of x for which the fimction /(x) has opposite signs, and then evaluate the function for the midpoint of the interval and determine which half of the interval 

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Newton s method is a procedure for the numerical solution of algebraic equations, applicable to any number M of such equations expressed as functions of M variables. 

Section 3.3 Numerical Solution of Algebraic Equations TABLE 3.2 > Mathematical Functions in Mathematics... 

The Numerical Solution of Algebraic Equations". John Wiley Sons, 1979. 

The mathematical model developed in the preceding section consists of six coupled, three-dimensional, nonlinear partial differential equations along with nonlinear algebraic boundary conditions, which must be solved to obtain the temperature profiles in the gas, catalyst, and thermal well the concentration profiles and the velocity profile. Numerical solution of these equations is required. 

The model balance equation for each metal and ligand (e.g., Eqs. 2.49 and 2.52) is augmented to include formally the concentration of each possible solid phase. By choosing an appropriate linear combination of these equations, it is always possible to eliminate the concentrations of the solid phases from the set of equations to be solved numerically. Moreover, some of the free ionic concentrations of the metals and ligands also can be eliminated from the equations because of the constraints imposed by on their activities (combine Eqs. 3.2 and 3.3), which holds for each solid phase formed. The final set of nonlinear algebraic equations obtained from this elimination process will involve only independent free ionic concentrations, as well as conditional stability and solubility product constants. The numerical solution of these equations then proceeds much like the iteration scheme outlined in Section 2.4 for the case where only complexation reactions were considered, with the exception of an added requirement of self-consistency, that the calculated concentration of each solid formed be a positive number and that IAP not be greater than Kso (see Fig. 

As became apparently clear from the discussion above, during parameter estimation, analytical derivatives cannot be calculated, therefore, there is a need to solve algebraic and differential equations by numerical methods. In this section we briefly present strategies for numerical solution of algebraic and differential equations. 

For general case, the IAS equations must be solved numerically and this is quite effectively done with standard numerical tools, such as the Newton-Raphson method for the solution of algebraic equations and the quadrature method for the evaluation of integral. We shall develop below a procedure and then an algorithm for solving the equilibria problem when the gas phase conditions (P, yj are given... 

This is a highly nonlinear, imphcit system of algebraic equations. The integral in Eq. (28.63) contains a function of an unknown variable usually, Eq. (28.62) cannot be rearranged to exphcit equations in terms of the unknown overpotentials and Eq. (28.61) has a spatially lumped quantity on the left-hand side, but spatially distributed quantities on the right-hand side. This makes the numerical solution of this equation system difficult. This can also be seen in many pubUcations... 

Many engineering problems consist not just of systems of differential equations, but many times coupled systems of some differential equations and some algebraic equations. In such cases, some of the equations involve one or more derivatives of the solution variables and some of the equations simply involve an algebraic relationship between the variables with no derivative involved in the equation. It would be very nice if the code so far developed for the numerical solution of differential equations could accommodate such systems of differential-algebraic equations. As will be seen much of the work in this chapter is directly applicable to such physical problems. 

Families of finite elements and their corresponding shape functions, schemes for derivation of the elemental stiffness equations (i.e. the working equations) and updating of non-linear physical parameters in polymer processing flow simulations have been discussed in previous chapters. However, except for a brief explanation in the worked examples in Chapter 2, any detailed discussion of the numerical solution of the global set of algebraic equations has, so far, been avoided. We now turn our attention to this important topic. 

NUMERICAL SOLUTION OF THE GLOBAL SYSTEMS OF ALGEBRAIC EQUATIONS... 

As mentioned in Chapter 2, the numerical solution of the systems of algebraic equations is based on the general categories of direct or iterative procedures. In the finite element modelling of polymer processing problems the most frequently used methods are the direet methods. 

As mentioned earlier, overall accuracy of finite element computations is directly detennined by the accuracy of the method employed to obtain the numerical solution of the global system of algebraic equations. In practical simulations, therefore, computational errors which are liable to affect the solution of global stiffness equations should be carefully analysed. 

Following these procedures, we are led to a system of algebraic equations, thereby reducing numerical solution of an initial (linear) differential equation to solving an algebraic system. 

Usually the finite difference method or the grid method is aimed at numerical solution of various problems in mathematical physics. Under such an approach the solution of partial differential equations amounts to solving systems of algebraic equations. 

Thus Y1 is obtained not as the result of the numerical integration of a differential equation, but as the solution of an algebraic equation, which now requires an iterative procedure to determine the equilibrium value, Xj. The solution of algebraic balance equations in combination with an equilibrium relation has again resulted in an implicit algebraic loop. Simplification of such problems, however, is always possible, when Xj is simply related to Yi, as for example... 

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