@MapleEnthusiast

This site is about **using Maple to solve problems** so first question: what **exactly** is Maple doing incorrectly?

If I go through your previous response more-or-less line by line, then you have to ask yourself some pretty basic questions, which I have itemised in the following

*The general solver -LinearSolve() confirms the existence of a solution. *

**Incorrect. statement**. LinearAlgebra:-LinearSolve demonstrate that there are an **infinite number of solutions**

*The -IterativeFormula() suggests that this solution can be reached using one of the standard iterative methods (such Gauss-Seidel, Jacobi, etc).*

**Incorrect statement**. Since there are an **inifinite** number of solutions, all the IterativeFormula() command does is find **one** of them

*I guess, the fact that these methods can deal with the case of spectral radius of M being 1 is not so surprizing, since the singular matrix (I-M) from equation (5) of my write-up can be inverted using the Moore-Penrose pseudoinverse and still generate an (up-to-scale) unique solution.*

**Overcomplicating the issue:** For a numerical process to generate one (of an infinite number of) solution(s), all it has to do is to **assign an arbitrary value to one of the variables**, throw away one of the redundant equations, and solve the remaining three **independent** (consistent) linear equations in three unknowns. Not exactly difficult

*Presumably, however, the -IterativeFormula() applies a different type of iteration then the one you've coded in Example1.mw and Example2.mw above (where we start with an initial, model-consistent guess for the unknowns (of 0's) on the right hand side (RHS), calculate the left-hand side (LHS), plug those values again on the RHS, etc.), for which we have a ***convergence** in Ex1 and a **divergence **in Ex2. (Presumably, the -IterativeFormula() plugs in the LHS values with some sort of 'damping' factor on the RHS to ensure the convergence to the existent solution. However, the **existence** of a solution that can be reached iteratively, e.g., using -IterativeFormula(), does not yet imply the **stability** of the equilibrium, right?).

**Complete misrepresentation**: the problem you posted in your very first worksheet was for a system of **non-linear** **equation**s - lots of products* of* y[1]*y[2] etc. I was **doubtful** about the success of an iterative process in solving this **non-linear** system*,* which is why my original answer was headlined "use with care"*.* At some point in the evolution of this thread, you morphed from a **system of non-linear equations** to a **system of linear equations*** -* I have no idea how/why this occurred*, *but if your original post had specified a** system of linear equations**, then no way would I have proposed the iterative solution in my original response. For systems of linear equations, there are several "better" methods of constructing an iterative solution - see for example the IterativeFormula() command

*Hence, I wonder whether the fact of a ***divergence **in** **Example2.mw (via a "mechanical" updating of the RHS using LHS) is some sort of indication that the equilibrium in Example2 is unstable and, if so, how can I formally tackle this question?

**More-or-less meaningless**:* *As stated above, I have little confidence that the iterative approach in my original response would work for non-linear system. For linear systems, far better iteration processes are avaliable. (See the documentation for the IterativeFormula() command.) I also have no idea what you mean by terms such as "equilbrium" or "unstable" in this context. All you have is two linear systems of four non-independent equations, each of which has an infinite number of solutions