Finding roots, Newton method

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... 

Computer programs (such as Mathcad) automatically implement the Newton-Raphson method of finding roots. Usually, such programs require the input of an... 

The iterative solution can be carried out by one of various algorithms, for example Newton s approximation to find roots, steepest descent to find a minimum quadratic error, rough search, successive substitution. Newton s method in four dimensions works reasonably well, although instability can set in if the incremental changes are allowed to be too large. Hence some deceleration is required to stabilise the algorithm. The method of successive substitution is more efficient, but also... 

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. 

Here is another application of the Taylor series method. Suppose that you want to find the values x = x that cause a function to equal zero, fix) = 0. Such values of x are called the roots of the equation. The Newton method is an iteratixe scheme that works best when you can make a reasonable first guess X, and when the function does not have an extremum between the guess and... 

The Newton method to find a root of F z) = 0 and with an initial guess, z , is given... 

In Figure 6.12 two different slopes have been guessed, resulting in two different solutions, neither of which match the boundary value Different root-finding algorithms may be used to find the missing initial condition, e.g. the secant method or the Newton method. When the missing initial condition has been found, the solution of the IVP y(b) will match... 

The message here is that the Newton-Raphson method will converge but probably to the nearest root if there is more than one root. Even so, this method is quite valuable for finding roots of complicated functions. Of course with a programmable calculator, the iteration can be automated and will converge rapidly ... 

A fairly thorough look has been taken at Newton s method for solving a single nonlinear (or linear) equation for a root (or roots) of a function. If a function has multiple roots, Newton s method must be started close to each root in order to find more than one solution. It is also seen that for some formulation of a physical problem, it is essentially impossible to get Newton s method to converge to a solution while a rearrangement of the basic equation can rapidly lead to valid solutions with Newton s method. The more one knows about a particular physical problem... 

The emphasis in this chapter has been on using the Newton method for finding roots of various functions. Newton s method is based upon a local Taylor series expansion of a function and approximating the function locally by the first two terms representing the value of the function and the first derivative as expressed by Eq. (3.17). But what about functions for which the first derivative is zero at the solution value that corresponds to zeros of the function From Eq. (3.19) one can see that the correction value beeomes ... 

The root-finding method used up to this point was chosen to illustrate iterative solution, not as an efficient method of solving the problem at hand. Actually, a more efficient method of root finding has been known for centuries and can be traced back to Isaac Newton (1642-1727) (Eig. 1-2). 

This example found the reactor throughput that would give the required annual capacity. For prescribed values of the design variables T and V, there is only one answer. The program uses a binary search to find that answer, but another root-finder could have been used instead. Newton s method (see Appendix 4) will save about a factor of 4 in computation time. 

The function given by dj(t)/dt (calculated case) has a maximum that can be identified as the carrier transit time, which can be derived by using Newton s method of finding the root for the maximum of corresponding equation (for details see Ref. [11]). 

Solving (1.4) for x and setting Xj+i = x as well as y = 0 gives rise to the iteration (1.3). Newton s method is the basis of many root-finding algorithms. It can be extended to several, i.e., n real variables... 

Another method to solve scalar equations in one real variable x uses inclusion and bisection. Assume that for a given one variable continuous function / R —> R we know of two points X < xup G R with f xi) f(xup) < 0, i.e., / has opposite signs at X and xup. Then by the intermediate value theorem for continuous functions, there must be at least one value x included in the open interval (x ,xup) with f(x ) = 0. The art of inclusion/bisection root finders is to make judicious choices for the location of the root x G (x , xup) from the previously evaluated / values and thereby to bisect the interval of inclusion [x , xup] to find closer values v < u e [x , xup] with v — u < x — xup and f(v) f u) < 0, thereby closing in on the actual root. Inclusion and bisection methods are very efficient if there is a clear intersection of the graph of / with the x axis, but for slanted, near-multiple root situations, both Newton s method and the inclusion/bisection... 

Looking at the three shallow intersections of the horizontal axis with the graph of / in Figure 3.4, we are reminded of the problems encountered in Chapter 1 on p. 30 and 31 with both the bisection and the Newton root finder for polynomials with repeated roots. The common wisdom is that the shallower these intersections become, the worse the roots will be computed by standard root-finding methods (see the exercises below), and multiple roots will easily be missed. 

Compared to solveadiabxy. m for the adiabatic CSTR case in Section 3.1, the above MATLAB function solveNadiabxy. m depends on the two extra parameters Kc and yc that were defined following equation (3.9). It uses MATLAB s built-in root finder fzero.m. As explained in Section 3.1, such root-finding algorithms are not very reliable for finding multiple steady states near the borders of the multiplicity region. The reason - as pointed out earlier in Section 1.2 - is geometric the points of intersection of the linear and exponential parts of equations such as (3.16) are very shallow, and their values are very hard to pin down via either a Newton or a bisection method, especially near the bifurcation points. 

In this section we will discuss the specific mathematical techniques used to estimate chemical equilibria using the sequential approach, which is the foundation for all versions of the FREZCHEM model, except for versions 2 and 10 (see above). The techniques used to solve (find the roots of) the equilibrium relations can be grouped into three classes simple one-dimensional (1-D) techniques, Brents method for more complex 1-D cases, and the Newton-Raphson technique that is used for both 1-D and multidimensional cases. 

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... 

The Newton-Raphson method (NRM) is used to find the roots of three chemical equilibria Eq. 3.29 for finding the H+ concentration of carbonate systems (one unknown), Eq. 3.32 for finding the H+ concentration of acidic systems where the partial pressure of acids controls equilibria (one unknown), and the equations governing dolomite equilibria where there are four unknowns [H+, Mg2+, MgCOg, and CaMg(CC>3)2(cr)]. In the dolomite case, Ca2+ is calculated indirectly. 

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