Question: How do I store and retrieve the iterates after each iteration of solving a (non-linear) system of equations?

Dear Maple community,

I'm trying to solve iteratively a system of (non-)linear equations and, to better understand the dynamics behind this process, would like to know the new/updated values of the unknowns after each iteration. Unfortunately, I wasn't able to implement this myself and was wondering whether you could give me hand?

Specifically, consider the following system of 4 non-linear equations in 4 unknowns: y[1], y[2], y[3], y[4]:

(EQ1): y[1]=f(y[1], y[2], y[3], y[4])

(EQ2): y[2]=f(y[1], y[2], y[3], y[4])

(EQ3): y[3]=f(y[1], y[2], y[3], y[4])

(EQ4): y[4]=f(y[1], y[2], y[3], y[4])

The goal is to set-up an iterative process along the following lines:

Step 1: Set the values of y[1] through y[4] on the right-hand side (RHS) equal to 1 and calculate the respective values of y[1] through y[4] on the left-hand side (LHS). Store the latter LHS values as {y[1]^0, y[2]^0, y[3]^0, y[4]^0}.

Step 2: Substitute {y[1]^0, y[2]^0, y[3]^0, y[4]^0} on the RHS, update the LHS values and store those as {y[1]^1, y[2]^1, y[3]^1, y[4]^1}.

Iterate Step 2 until convergence (which should yield the exact same result as with solving numerically the above system of 4 equations using the fsolve(.) command). Importantly, I would like to "retrieve" the values of each variable y[k]^i at each iteration i, where k=1,2,3,4, and i=0,1,... and use them for further analysis.

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