Multiple roots

The modified Newton method [12] offers one way of dealing with multiple roots. If a new function is defined... 

But in order for a matrix to have a multiple root, it is necessary that its elements satisfy a certain algebraic relation to have a triple root they must satisfy two relations, and so forth for roots of higher order. Thus, if a matrix is considered as a point in 2-space, only those matrices that lie on a certain algebraic variety have multiple roots. Clearly, if the elements of a matrix are selected at random from any reasonable distribution, the probability that the matrix selected will have multiple roots is zero. Moreover, even if the matrix itself should have, the occurrence of any rounding errors would almost certainly throw the matrix off the variety and displace the roots away from one... 

This is possible since P has maximal rank p. Now all roots of M are roots of A. Assume A to be nonsingular, and suppose that if A has any multiple roots occurring among the roots of M, then it has the same multiplicity as a root of M. Then it can be shown that... 

Multiple roots of matrices, 68 Mutual information, 205 average, 206... 

If A has repeated eigenvalues (multiple roots of the characteristic polynomial), the result, again from introductory linear algebra, is the Jordan canonical form. Briefly, the transformation matrix P now needs a set of generalized eigenvectors, and the transformed matrix J = P 1 AP is made of Jordan blocks for each of the repeated eigenvalues. For example, if matrix A has three repealed eigenvalues A,j, the transformed matrix should appear as... 

The solution to Equation 9-34 for GTA is done by an iterative or trial-and-error technique. Equation 9-34 is likely to produce multiple roots. In this case the correct solution is the minimum mass flux GT. For the special case of no overpressure, Tm = Ts, and Equation 9-34 reduces to... 

Equation (65) can have many roots. For each of these values of (x/m) there exists a value of (z/x) and a value of (y/x), as determined by Eqs. (66) and (67). But since Eqs. (68)-(70) contain only positive constants, these equations provide criteria that the roots must satisfy in order that they apply to the model. Thus, according to Eq. (68), (x/u) > 1 according to Eqs. (69) and (70), both (z/x) and (y/x) must be positive. Since (x/u), (z/x), and (y/x) are linked, rejection of one root means that the two associated roots must also be rejected. In the 1-butene case we were thus able to remove the multiple-root problem from Eq. (65), although we have not proved that such removal can always be accomplished by using these two criteria. 

This chapter addresses methods and tools used successfully to identify multiple root causes. Process safety incidents are usually the result of more than one root cause. This chapter provides a structured approach for determining root causes. It details some powerful, widely used tools and techniques available to incident investigation teams including timelines, logic trees, predefined trees, checklists, and fact/hypothesis. Examples are included to demonstrate how they apply to the types of incidents readers are likely to encounter. 

In recognition that most incidents have multiple root causes, the team is generally required to identify a minimum of three factors one from each of the following categories organizational, human, and material factors. 

MULTIPLE CAUSES - Most current methods recognize the concept of multiple root causes. 

This chapter addresses typical data gathering needs of major investigations. A team may need to augment the activities in this chapter for the unique circumstances of the incident. Performing the activities oudined in this chapter plus special activities provides the incident investigation team with the data needed to complete the next step—systematic determination of the multiple root causes of the incident. However, data gathering and analysis typically involve much iteration as shown in Figure 8-1. 

To give further insight into the role of management systems and the distinction between multiple root causes and non-root causes, consider the following actual case histories. 

The causal factors need to be examined further to determine why those factors existed. The investigation team may use a predefined tree to examine each causal factor individually. The first causal factor is analyzed starting at the top of the tree, and then working down all of the branches as far as the facts permit. When an appropriate subcategory on one of the branches is identified, it is recorded as a root cause. The remaining branches are checked as one causal factor may have multiple root causes. The procedure is then repeated for each causal factor in turn. 

Analysis of Esso Longford as well as analysis in the UK Health and Safety Executive (HSE) investigation report into petrochemical complex major incidents all show that common underlying causes are often repeated. The Longford incident clearly illustrates the multiple root cause concept. A number of PSM system failures occurred either in... 

The following case study describes the investigation work process for a hypothetical occurrence using a logic tree based multiple root-cause systems approach. An example incident investigation report follows the work process description. The example is intended for instructive purposes only descriptions of process equipment and conditions are not intended to reflect actual operating conditions. 

So the smallest positive value of cr corresponds to the ground state. The second smallest similarly provides the energy of the first excited state and so on. The possible occurrence and significance of multiple roots are discussed in Refs. [1,2]. 

We took the 4- sign on the square root term for second-order kinetics because the other root would give a negative concentration, which is physically unreasonable. This is true for any reaction with nth-order kinetics in an isothermal reactor There is only one real root of the isothermal CSTR mass-balance polynomial in the physically reasonable range of compositions. We will later find solutions of similar equations where multiple roots are found in physically possible compositions. These are true multiple steady states that have important consequences, especially for stirred reactors. However, for the nth-order reaction in an isothermal CSTR there is only one physically significant root (0 < Ca < Cao) to the CSTR equation for a given T. ... 

These steady states are within the physically possible range of T 0 < T < oo) and X(0 < X < 1). This is in contrast to many situations in the physical sciences where equations have multiple roots but only one root is physically acceptable because the other solutions are either outside the bounds of parameters (such as negative concentrations or temperatures) or occur as imaginary or complex numbers and can therefore be ignored. 

This is the example from the previous chapter, and k( T) was determined previously. X(T) is shown in Figure 6-9 for T = 1 mm. As T increases so that the curves move into the situation where multiple roots emerge, the system does not jump to them because it is already in a stable steady state. Only when Tq is so high that the heat removal line becomes tangent to the heat generation curve does the lower intersection disappear. The system then has no alternative but to jump to the upper intersection. A similar argument holds in the decrease in Tq from the upper steady state. 

Detailed description of the domains of convergence of hyper geometric series in terms of amoeba of the discriminant of the polynomial has been given recently in Passare and Tsikh (2004). The discriminant A(a) is an irreducible polynomial with integer coefficients in terms of the coefficients , of polynomial (54) that vanishes if this polynomial has multiple roots. For instance, for cubic polynomial the discriminant is... 

Assume for simplicity that all roots A/- of (4.2.9) are different (the case of multiple roots is handled in a standard fashion). Then according to (4.2.8)... 

From the discussion above, we need only look for multiple roots of eq. (39) to find the maximum value of n for our problem. For v = V2/2 and y = 12, w = 12 is a triple root of eq. (39), the most degenerate root obtained for any values of v and y. Therefore, we would expect that the value of n is 6 (star... 

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... 

In Figure 1.2 we note that for x within around 3% of the multiple root x = 2 of p, the values of p(x) and p (x) in extended polynomial form (both plotted as dots in Figure 1.2) are small random numbers of magnitudes up to around 5 10 u. Hence... 

The following experiments validate our assessment of troubles with Newton or bisection root finders for multiple roots. First we use the bisection method based MATLAB root finder f zero, followed by a simple Newton iteration code, both times using the chosen polynomial p x) of degree 9 in its extended form (1.6). 

Unfortunately, in chemical and biological engineering problems multiple roots or near multiple roots occur often for example, look at Figure 3.2 on p. 71 and the slanted intersections marked by (1), (2), and (3). These force us to devise specific better numerical methods for such problems in Sections 3.1 and 3.2. 

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