Multivalue Algorithms

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. 

To better understand this new way of looking at the problem, we can take the example of a particular algorithm, such as the fourth-order Adams-Moulton algorithm in its multivalue version. 

For each multistep algorithm, there is a corresponding multivalue algorithm with the same features. 

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. 

For example, the multivalue fourth-order Adams-Moulton algorithm has the same local error as the corresponding multistep fourth-order Adams-Moulton algorithm ... 

For example, the fourth-order multivalue Adams-Moulton algorithm has a local error ... 

As mentioned on several occasions, each multistep algorithm has a corresponding multivalue algorithm. Thus, many multivalue algorithms and families of algorithms are available. 

All the multivalue algorithms that we consider are implicit since they are the only stable algorithms. 

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. 

To implement a multivalue algorithm to solve stiff problems, it is crucial to be careful about its robustness. In the following sections, the main causes that can make a program unstable are analyzed and their respective remedies are... 

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. 

To implement a multivalue algorithm for solving stiff problems, it is essential to take great care regarding efficiency. The key point with stiff problems is the solution of the nonlinear system (2.232) ... 

The following classes, which are based on multivalue algorithms, are implemented in the BzzMath library. 

For the nonstiff problems, based on multivalue algorithms of the Adams-Moulton family ... 

The root finding classes can be combined with classes for differential systems based on multivalue algorithms. 

If the function to be zeroed has a single root and changes sign during the integration of the system, an object from the BzzFunctionRoot class can be used. The root-finding classes can be combined with classes for differential systems based on multivalue algorithms (see Vol. 4 - Buzzi-Ferraris and Manenti, in press). 

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