Underdimensioned nonlinear systems

Examples of this chapter can be found in the Vol3 Chapter8 directory in the WileyVolS. zip file available atwww.chem.polimi.it/homes/gbuzzi. 

This chapter deals with the solution of the system of he nonlinear equations  

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. 

We also encounter this situation when we need to solve a nonlinear system using Newton s method when the Jacobian is singular at a certain iteration. This problem is relatively easy to solve when the variables for which the underdimensioned system has to be solved are known. In Chapter 7, we saw how to solve this problem using the objects from the BzzNonLinearSystem class, predisposed for square systems. Real problems are often in this fortunate position. For instance, we often know a priori which equations are algebraic and which other are differential in the case of differential-algebraic systems (Vol. 4 - Buzzi-Ferraris and Manenti, in press). If the differential equations are explicit and first order, the variables of the differential equations are known and, consequently, the variables to be used to solve the algebraic equations are known too. 

In the other circumstances, it is necessary to solve the underdimensioned algebraic system with respect to the variables that make the solution stable. 

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The BzzNonLinearUnderdimensionedSystem class specifically developed to solve underdimensioned nonlinear systems shall be described in Chapter 8. 

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. 

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... 

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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