Nonlinear algebraic systems

Many of the algorithms used to solve a BVP lead to the solution of a nonlinear algebraic system in the unknown v. 

The case of flash drum separator is considered (Figure A.1) it is a nonlinear algebraic system and the class BzzNonL InearSystem is required to solve it (Buzzi-Ferraris and Manenti, 2014). 

Another important topic is the solution of nonlinear algebraic systems, which demands very robust algorithms. Chapter 7 illustrates the numerical methods for square systems in their sequential and parallel implementation. In addition to this, methods and techniques are proposed by separating the small and medium dimension problems, which are considered dense for large-scale systems, where the management of matrix sparsity is crucial. Many practical examples are provided. 

After substitution (20) in (19) it is possible to derive the system of ordinary differential equations for Fourier coefficients hk X) and functions c(A), qo X), 5(A), 5(A). At any value of A the solution can be corrected by Newton s method applied to nonlinear algebraic system following from (19). 

Then nonlinear algebraic system for Fourier coefficients in paper Sh had been derived. That nonlinear system in the followin form could be represented... 

We have stated that we do not in general know the number or even the existence of solutions to a nonlinear algebraic system. This is true however, it is possible to identify points at which the existence properties of the system change through locating bifurcation points i.e., choices of parameters at which the Jacobian, evaluated at the solution, is singular. 

The main routine for solving nonlinear algebraic systems f(x) 0 is fsolve,... 

Armed with techniques for solving linear and nonlinear algebraic systems (Chapters 1 and 2) and the tools of eigenvalue analysis (Chapter 3), we are now ready to treat more complex problems of greater relevance to chemical engineering practice. We begin with the study of initial value problems (IVPs) of ordinary differential equations (ODEs), in which we compute the trajectory in time of a set of N variables Xj(t) governed by the set of first-order ODEs... 

Because fix 0) generally is nonlinear, (4.115) often cannot be rearranged to provide a direct expression for x. Then, (4.115) is said to generate an implicit integration method that requires a nonlinear algebraic system to be solved at each time step. 

Figure 4.16 Arc length continuation of a nonlinear algebraic system, showing solution path passing...

---