Root-finding problem

We start with a short history of the polynomial-root finding problem that will explain the eminent role of matrices for numerical computations by example. 

Since ordinary zero finders fail us often in root-finding problems with multiple roots, we now set out to develop a more reliable graphical level-set method for finding all... 

It is interesting to estimate the order of magnitude of d for the root-finding problem. Let us suppose that the function y(t) can be expanded in Taylor series in... 

The calculation of the volume of a nonideal gas is a root-finding problem. Let us consider, for example, Rice s problem (Rice, 1993) where the Beattie-Bridgeman equation of state joins the temperature T(K), the pressure P(atm), and the molar volume V(l/mol) through the formula... 

Before ending this chapter, we should emphasize several points crucial to the function root-finding problem. 

The stationary point of the parabola (2.37) is obtained by solving the root-finding problem with its first derivative ... 

In these cases, it is opportune to transform the root-finding problem into a onedimensional minimization, by taking the absolute value of the function and adopting the techniques described in Chapter 5. 

Chapter 1 discusses methods for handling the function root-finding problem. Algorithms are proposed in the renewed forms to exploit the multiprocessor machines. The use of parallel computing is also common to the chapters that follow. 

The steady-state approximation effectively turns the microkinetic model from a set of coupled nonlinear differential equations in time into a time-independent algebraic root-finding problem, which is simpler to solve and which can sometimes even be solved analytically. The rate corresponding to the steady-state solution is also continuous when the rate constants are varied continuously, which we... 

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). 

The next phase of the problem is to find those values for T and V that will give the lowest product cost. This is a problem in optimization rather than root-finding. Numerical methods for optimization are described in Appendix 6. The present example of consecutive, mildly endothermic reactions provides exercises for these optimization methods, but the example reaction sequence is... 

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. 

Water from a reservoir passes over a dam through a turbine and discharges from a 70-cm ID pipe at a point 65 m below the reservoir surface. The turbine delivers 0.80 MW. Calculate the required flow rate of water in m /min if friction is neglected. (See Example 7.7-3.) If friction were included, would a higher or lower flow rate be required Note The equation you will solve in this problem has multiple roots. Find a solution less than 2 m /s.)... 

In principle, we could use any root-finding method for solving problems involving the equations Ki = 1 for combinations of species representing different tuples, such as all binary combinations, all ternary combinations, quaternary. 

It would not be practical or even desirable, however, to carry out a classical calculated in the above framework. The practical difficulty would be related to finding the roots of (117), the usual multi-dimensional root-search problem, and the result would be undesirable because zeros in the Jacobian determinant cause singularities, classical rainbows , in the classical probability distribution in (116), To remedy both of these features one averages the classical expression over a quantum number increment about n2 and over some increment about 2 ... 

The zero-crossing problem consists of function root finding during the integration of a differential system. In other words, during the integration of ODE/DAE systems, there may be a need to calculate the value of the independent variable t at which a certain fimction of the dependent variables y is zeroed. 

Intrinsically similar problems (e.g., root-finding for different functions) are not solved by the same calculation procedure. 

This section illustrates a set of case studies in which root-finding plays an important role in chemical engineering including the calculation of the volume of a nonideal gas, bubble point, and zero-crossing. However, these scenarios also crop up in several other areas. For instance, the calculation of the volume of a nonideal gas is a typical problem in fiuid dynamics, whereas the zero-crossing problem is very common in all disciplines involving differential and differential-algebraic systems as convolutions models, such as the optimal control for electrical and electronic purposes. 

Stop criteria for one-dimensional optimization problems are very similar to function root-finding and are discussed at exhaustive length in Chapter 1. 

The numerical integration of the model equations often requires the determination of the time instant ofthe discrete events and a reinitialisation. Hence, the numerical computation of a hybrid system model may be viewed as the solution of a sequence of initial value problems (IVPs). Modern numerical solvers for DAE systems such as IDA [2] from the SUNDIALS suite or DASRT [3] provide a root finding feature such that the time instances of mode switches can be located. 

There are two obvious advantages in Eq. (4.59) over Eq. (4.58) The root-search problem to find pairs of (g/, g ) is avoided, since one can let a path start with a natural condition (qi,Pi). The other one is the singularities inherent in Eq. (4.58) due to the zero of dqf/dpi are all removed. This is called the Initial Value Representation (IVR). 

Newton s method is a classic iterative scheme for solving a nonlinear system /(x) = 0 or for minimizing the multivariate function /(x). These root-finding and minimization problems are closely related since obtaining the minimum of a function f(x) can be formulated as solving for the zeros of / (x) for which f (x) > 0. (See historical note below on the method s name.)... 

Appendix A. Mathematical appendix contains a survey of selected mathematical-physical methods that are often used for solution of practical problems. These include numerical processing and uncertainty calculation, dimensional analysis, linear regression, iterative root finding, etc. 

Thevalueof is 12(h) x 1.05(1/mol-h) = 12.6(1/mol).ThesolutionofEqn.(4-12)isa trial-and-error problem that can be solved in several ways. For example, the value of the left-hand side of Eqn. (4-12) could be calculated for various values of Ca over a range that bracketed the value of —12.6 (1/mol). The desired value of Ca dien could be found by interpolation, either graphical or numerical. A simpler approach is to use the GOALSEEK function that is a Tool in the Microsoft Excel spreadsheet application. GOALSEEK is a root-finding technique, i.e., it finds the value of a variable (Ca in this case) that makes a specified function of that variable equal to zero. In this case, we want to make the function... 

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