Underdimensioned equations, nonlinear

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. 

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... 

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... 

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See also in sourсe #XX -- [ ]

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