Underdimensioned Nonlinear System Solution

In the BzzMath library, the class used to solve underdimensioned nonlinear systems with dense matrices is 

The BzzNonLinearUnderdimensionedSystem class is designed to solve underdimensioned nonlinear systems by means of the quasi-Newton method as the main algorithm and by availing of all the devices described in Chapter 7. In the following, we will use 

To initialize the object nlus, the operator () is adopted. Two alternatives are possible  

In the first case, we must insert the indices of the variables for which we want to solve the system. The indices are inserted using a Bz zVector Int dependent-variables dimensioned to suit the number of the equations. 

In the second case, the following arguments must be assigned to the operator ()  

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Note that many problems of different kinds (i.e., the solution of differential-algebraic equation systems or constrained optimization problems) lead to the numerical solution of an underdimensioned nonlinear system. 

Since these underdimensioned nonlinear systems will also be solved using a modified Newton method, it is essential to tackle the subproblem of the solution of underdimensioned linear systems. 

If the linearized system together with the linear equations has some linear dependencies, the program switches to the underdimensioned nonlinear system of the active constraints solution. A BzzNonLinearUnderdimen-sionedSystem class object is adopted to solve this underdimensioned... 

If the nonlinear system does not satisfactorily fulfill the constraints, a Newton method is iteratively used to solve the underdimensioned system. The iterations proceed using the same Jacobian J for a limited number of iterations and only if the solution of the nonlinear system is improved. Otherwise, a BzzNonLinearUnderdimensionedSystem class object is adopted to solve this underdimensioned nonlinear system properly. 

Chapter 8 deals with underdimensioned nonlinear systems. It proposes a stable Gauss factorization for their solution and compares the novel method to the most common factorizations. Sparse underdimensioned nonlinear systems also have a dedicated class of algorithms. 

The problem of selecting the most reasonable pivot to detect real linearly dependent equations occurs not only for underdimensioned linear systems but also in the solution of singular square systems. It happens, for instance, when a Newton s method is adopted to solve a square nonlinear system and the resulting... 

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