Equations, finding roots

Now suppose that B is a diagonal matrix (all off-diagonal elements equal to zero) then the roots of its characteristic equation (eigenvalues) are identical with its diagonal elements. If A is not a diagonal matrix but is related to B by a similarity transformation, it follows that it has the same characteristic equation and roots as B. The problem of finding the eigenvalues... 

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. 

Note how much simpler this problem is to solve in the CSTR than in the PFTR (Figure 4—6), where we had to solve simultaneous differential equations. The CSTR involves only simultaneous algebraic equations so we just need to find roots of polynomials.]... 

Flowever, for more than a few reactions and with reversible reactions, we rapidly reach a situation where only numerical solutions are possible. While numerical solutions can easily be found with differential equation solvers or computer programs that find roots of simultaneous polynomials, note that these are only simulations for a given set of ks and... 

An Algebraic Method for Finding Roots of Polynomial Equations... 

Equations solving quadratic and higher order equations finding the factors and roots of simple polynomial equations using either algebraic or graphical procedures. 

Brent s method (Press et al. 1992) is used to find the root of Eq. 3.38 (Vr). Some terms in the latter equation have Vr raised to the fifth power, which means that there could be as many as five roots to such an equation. This root-finding method does not require a derivative as the following Newton-Raphson method does, but it does require a lower and upper bracket within which the true solution must he. In this particular case with... 

Both mass transfer resistances are interconnected in a rather complex manner. The structure of these interconnections was made visible by the presentation of analytical solutions which could be easily handled on a spreadsheet calculation tool, even for much more complicated VLE equations with variable relative volatilities and chemical equilibrium equations of any complexity because no integration procedure is necessary, but just finding roots. 

In this and the next subsection, we outline algorithms for finding roots of single-variable equations of the form f(x) = 0. The first procedure, termed the regula-falsi method, is appropriately used when an analytical expression for the derivative of / with respect to x is not available—as. for example, when f x) is obtained as the output of a computer program for an input value of x. The algorithm is as follows ... 

Equation 5-17 is solved for y by finding roots of the cubic equation using the following substitutions. 

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. 

The quantities x, and yj are the iterants, whereas gi and g2 are formed exactly the way Equation 9.5 was developed. Two common methods for finding roots to nonlinear systems are (1) Newton-Raphson and (2) the modified Newton-Raphson. Both approaches are briefly discussed in the subsections below. 

Chapter 11 we shall discuss numerical methods of finding roots to polynomial equations using a computer. 

To find the roots of Cg t) = 0 we divide Eq. (23) by the first term shown and transfer the unity to the left-hand side to obtain an equation of the form... 

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 solution of these equations requires a root-finding algorithm which iterates on assumed values of T i. At each value of T i, solve Eq. (26-95) (with T]i replacing Ra) foi Find R2 ffom Eq. (26-101), subject also to ... 

A simple way of finding the roots of an equation, other than by divine inspiration, symmetry or guesswork is afforded by the Newton method. We start at some point denoted x l along the x-axis, and calculate the tangent to the curve at... 

To compute the variance, we first find the mean concentration for that component over all of the samples. We then subtract this mean value from the concentration value of this component for each sample and square this difference. We then sum all of these squares and divide by the degrees of freedom (number of samples minus 1). The square root of the variance is the standard deviation. We adjust the variance to unity by dividing the concentration value of this component for each sample by the standard deviation. Finally, if we do not wish mean-centered data, we add back the mean concentrations that were initially subtracted. Equations [Cl] and [C2] show this procedure algebraically for component, k, held in a column-wise data matrix. 

Before considering how to proceed to find the remaining roots, another, closely related method will be described. The method has several variants, but basically it is due to Wielandt. Suppose p0 is closer (in the complex plane) to some particular root A than to any other. Then, for a given initial v0, form the iterates vltv2, by solving the equations... 

There are several ways to find the solutions of these simultaneous equations. One approach is to find the two roots of the determinantal equation known as the secular equation ... 

This important result is used to find the root mean square speeds of the gas-phase molecules at any temperature (Fig. 4.25). We can rewrite this equation to emphasize that, for a gas, the temperature is a measure of mean molecular speed. From... 

For some reactions, the equation for x in terms of K ma> be a higher-order polynomial. If an approximation is not valid, one approach to solving the equation is to use a graphing calculator or mathematical software to find the roots of the equation. 

If as an approximation we assume 6 = l( 3 = /3 = ), we find as the pertinent root of this equation z = 2.3062, corresponding to Wi,2 = Q + 1.8062a. A similar treatment applied to 1,4-dihydrobenzene (which involves no interaction between the benzene ring and the double bond) leads to Wi.4 = ( +3.6055a-2j8, or, in this case, Q+ 1.6055a. (In this paper as in the previous one... 

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