Multivalue method

As we will see later, certain algorithms (multivalue methods) can bypass this limitation since they allow the separation of the integration step effectively used from the points requested by the user. 

In this case, it is sufficient to replace each derivative with the corresponding component of the Nordsieck vector to have a nonlinear system in the single unknown b. For example, in the case of implicit first-order multivalue methods, the following system must be solved ... 

Each multivalue method is characterized by the same local error as the corresponding multistep method. 

The local error of a multivalue method is therefore obtained by means of... 

For certain reasons that we will explain soon, the error of a multivalue method should now be calculated in a slightly different way than the traditional... 

To estimate the local error of a p-order multivalue method, it is necessary to have an approximate value for the (p + l)-order derivative. It is trivial since we collected the derivatives m the Nordsieck vector up to the order p. 

This feature of the multivalue methods is also exploited in another way and for a completely different objective to the change of the integration step. In fact, it is... 

It is worth illustrating the reason why it can be useful to change the order of the method. Let us consider a problem with a single dependent variable for the sake of simplicity and let us suppose we assign the maximum admitted error e. Using a p-ovdev multivalue method, it is necessary to have... 

If we use the previous considerations, it should be easy to see how to initialize a multivalue method without using (although it is possible) a Runge-Kutta method with an opportune order. 

In the equivalent situation of the multivalue methods, the vector v is used as predictor and the vector b as corrector obtained iteratively from the relation... 

This strength can also become a weakness If functions have certain discontinuities, the memory of past calculations plays against the multivalue methods. 

The objects of these classes not only remember their history but also exploit the property of multivalue methods to maintain the optimal integration step distinct from the user requests. 

The idea of using a multivalue method to solve DAE problems was introduced by Gear (1971). As demonstrated in Chapter 2, given a system in the following implicit form ... 

While in the case of ODE with initial conditions the multivalue methods are very promising also for stiff systems, in the special case of BVP here considered the imphcit Runge-Kutta methods are the most performing ones. 

Differential cross section Deflection function. First we describe methods which take advantage of the close relationship between semi-classical cross sections and deflection function as outlined in Section III. A procedure which uses nearly all measurable quantities has been proposed and applied by Buck (1971). In order to unfold the multivalued character of 6(9), the deflection function is separated into monotonic functions g/[b) such that 0(6) = . gj(6)and6 = g (9). The g are represented by the usual functional approximations made in the semiclassical scattering theory ... 

Many of these multistep methods (and also their modern multivalue version described later) that were of great interest in the past are now considered obsolete due to their small stability region. 

The multistep methods are only of historical interest since their corresponding multivalue versions are used nowadays. 

Each p-oidei multivalue is characterized by the vector r used to correct the prediction V. It corresponds to the coefficients ao, i,. .., f>-i, bq, , bk used in the multistep methods and is selected to make the algorithm stable, accurate, and exact for the / -degree polynomial solutions. 

As we can see, the Adams-Moulton method is implicit even in its multivalue form. 

The alternative of solving the nonlinear system iteratively with a substitution method leads to a multivalue version of the traditional predictor-corrector method of the multistep algorithms. 

For example, the method applied to the fourth-order multivalue Adams-Moulton algorithm consists of iterating the equation ... 

The multivalue algorithms have their strength in the collection of the previous history and, therefore, they allow a better approximation with respect to the one-step methods. 

By monitoring T using a photomultiplier, R can be determined. Note that the sign of R is unknown since T varies with the sine squared of R hence, one must determine the sign by another method. Also T is a multivalued function of so that the order of the retardation must be established via another route. 

A simple scheme. Now suppose that a numerical solution for the pressure field is available, for example, the finite difference solutions presented later in Chapter 7. The solution, for instance, may contain the effects of arbitrary aquifier and solid wall no-flow boundary conditions we also suppose that this pressure solution contains the effects of multiple production and injection wells. How do we pose the streamline tracing problem using T without dealing with multivalued functions The solution is obvious subtract out multivalued effects and treat the remaining single-valued formulation using standard methods. Let us assume that there exist N wells located at the coordinates (Xn,Yn), having... 

The alternative way to deal with unknown or other such states is to catch the good ones and throw out the bad. This approach should be used wiffi more caution as not all synthesis tools support the required constructs. The advantage of this method is that the VHDL can be constructed so that the simulation will accurately reflect the operation of the synthesized circuit design. Wiih the previous method, tfds would only occur after the initialization signal had been issued. This topic is discussed in more detail in Chapter 8 when multivalued logic systems are introduced. 

The method lies on a set of binary coded numerical values. This coding that we call multivalue coding is considered, according to the case, either as a numerical value or as a set of numerical values. In the first case we apply to it mathematical operations (addition, substraction...) and logical operations (and, or...) in the second case. 

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