Roots of numerical equations

Methods for finding roots of numerical nonlinear equations first involve making estimates and subsequently improving them by some systematic procedure, ideal problems for implementation on computers. Many scientific calculators also have built-in capability for finding roots of single equations, for instance the HP-15C or HP-32SII. 

The averages (rs), ( C ), and (s) are found as numerical roots of these equations. If the statement that gradient corrections favor density inhomogeneity were the whole story, we would expect that gradient corrections would favor any physical process in which... 

Since the value of [Brd was small, the equation was easily solved by the method of successive approximations. In a preliminary calculation [Brj] was neglected and an approximate value of [Brsl was obtained. The cube of the value thus obtained was then substituted in equation 39 and the resulting quadratic equation again solved for [Brzl. After repeating this process several times, the numerical value of [Brsl approached a constant which was the correct root of the equation. 

Consider first a single equation /(X) = 0, in which f (X) is a function of the single variable X. Our purpose is to find a root of this equation, i.e., the value of X for which the function is zero. A simple function is illustrated inFig. I.l it exliibits a single root at the point where the curve crosses the X-axis. When it is not possible to solve directly for the root, a numerical procedure, such as Newton s method, is employed. 

COMMENT. If numerical methods are to be used to locate the roots of the equation which locates the extrema, then graphical/numerical methods might as well be used to locate the maxima directly. That is, the student may simply have a spreadsheet compute and examine or manipulate the spreadsheet to locate the maxima,... 

If an equation is written in the form f(x) = 0, where / is some function and jc is a variable, solving the equation means to find those constant values of x such that the equation is satisfied. These values are called solutions or roots of the equation. We discuss both algebraic and numerical methods for finding roots to algebraic equations. If there are two variables in the equation, such as F(jc, y) — 0, then the equation can be solved for y as a function of or x as a function of y, but in order to solve for constant values of both variables, a second equation, such as G(x,y) = 0, is required, and the two equations must be solved simultaneously. In general, if there are n variables, n independent and consistent equations are required. 

It can be easily shown that a single root of this equation = n(p) always exists in the interval 1 < D < Q[, and, with this root calculated numerically, one can calculate the quantities... 

In the delta function f (without) with an argument denotes (a specified numerical value) the function s value at the point x. Since the delta function vanishes except at the roots of the equation... 

We first discuss the graphical method to obtain a numerical approximation to the root or roots of an equation. This method is useful because you can see what you are doing and you can usually be sure that you do not obtain a different root than the one you want to find. The equation to be solved is written in the form... 

This result shows that the square root of the amount by which the ratio M /M exceeds unity equals the standard deviation of the distribution relative to the number average molecular weight. Thus if a distribution is characterized by M = 10,000 and a = 3000, then M /M = 1.09. Alternatively, if M / n then the standard deviation is 71% of the value of M. This shows that reporting the mean and standard deviation of a distribution or the values of and Mw/Mn gives equivalent information about the distribution. We shall see in a moment that the second alternative is more easily accomplished for samples of polymers. First, however, consider the following example in which we apply some of the equations of this section to some numerical data. 

If /I > 4, there is no formula which gives the roots of the general equation. For fourth and higher order (even third order), the roots can be found numerically (see Numerical Analysis and Approximate Methods ). However, there are some general theorems that may prove useful. 

Equation (13-14) is solved iteratively for V/F, followed by the calculation of values ofx, andy from Eqs. (13-12) and (13-13) andL from the total mole balance. Any one of a number of numerical root-finding... 

The basic theory of mass transfer to a RHSE is similar to that of a RDE. In laminar flow, the limiting current densities on both electrodes are proportional to the square-root of rotational speed they differ only in the numerical values of a proportional constant in the mass transfer equations. Thus, the methods of application of a RHSE for electrochemical studies are identical to those of the RDE. The basic procedure involves a potential sweep measurement to determine a series of current density vs. electrode potential curves at various rotational speeds. The portion of the curves in the limiting current regime where the current is independent of the potential, may be used to determine the diffusivity or concentration of a diffusing ion in the electrolyte. The current-potential curves below the limiting current potentials are used for evaluating kinetic information of the electrode reaction. 

Let us discuss some of the terms in equation 70-20. The simplest way to think about the covariance is to compare the third term of equation 70-20 with the numerator of the expression for the correlation coefficient. In fact, if we divide the last term on the RHS of equation 70-20 by the standard deviations (the square root of the variances) of X and Y in order to scale the cross-product by the magnitudes of the X and Y variables and make the result dimensionless, we obtain... 

A set of linear equations can be solved by a variety of procedures. In principle the method of determinants is applicable to any number of equations but for large systems other methods require much less numerical effort. The method of Gauss illustrated here eliminates one variable at a time, ends up with a single variable and finds all the roots by a reverse procedure. 

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. 

Each of the partial derivatives when equated to zero may well yield a nonlinear equation. Hence, the minimization of /(x) is converted into a problem of solving a set of nonlinear equations in n variables, a problem that can be just as difficult to solve as the original problem. Thus, most engineers prefer to attack the minimization problem directly by one of the numerical methods described in Chapter 6, rather than to use an indirect method. Even when minimizing a function of one variable by an indirect method, using the necessary conditions can lead to having to find the real roots of a nonlinear equation. 

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