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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. 

Iterative_solutionQ.mw

Dear Maple community, 
I've been trying to calculate all eigenvalues of a (large) symbolic matrix M (see EigenvaluesQ.mw) using:

with(LinearAlgebra):
LinearAlgebra:-Eigenvalues(M):

or simply 

Eigenvalues(M)

but, unfortunately, my computer (64Gb RAM, Maple 2020) doesn't deliver a solution after 1 hour of computation. Am I doing something wrong? Thank you very much in advance!

Dear Maple community,

I'm trying to calculate a Steady State Vector of the attached matrix using this Maple routine, but, unfortunately, don't get any solution even after a 30 min calculation on a 64Gb machine. Am I doing something wrong and, if so, could you please give me a hand how to conduct this calculation? Thanks!

Maple_SteadyStateVector.mw

Dear Maple community,

I was wondering whether you could help me with the following problem: Consider the non-linear system of 4 equations in 4 unknowns: {y[2,1], y[2,2], y[4,1], y[4,2]}, see MapleQ.mw. Please also note that equations y[2,1] and y[2,2] in sys2 are homogenous of degree 1 (i.e., defined only up to scale). In principle, it should be possible to choose either y[2,1] or y[2,2] as a free and unconstrained variable (a scaling factor) and express all other unknowns in terms of it. Unfortunately, I couldn't yet implement it in the current setting and was wondering whether you give me a hand. Thank you very much in advance!

``

sys1 := {y[4, 1] = 33^(2/3)*50^(1/3)/(33*(1/(y[2, 1]^(3/2)*y[4, 1]^(3/2)))^(1/3))+2^(2/3)*5^(1/3)/(2*(1/(y[2, 2]^(3/2)*y[4, 2]^(3/2)))^(1/3)), y[4, 2] = 11^(2/3)*50^(1/3)/(11*(1/(y[2, 1]^(3/2)*y[4, 1]^(3/2)))^(1/3))+4^(2/3)*5^(1/3)/(4*(1/(y[2, 2]^(3/2)*y[4, 2]^(3/2)))^(1/3))}

sys2 := {y[2, 1] = 33/(100*sqrt(y[2, 1])*y[4, 1]^(3/2)*(33/(50*y[2, 1]^(3/2)*y[4, 1]^(3/2))+2/(5*y[2, 2]^(3/2)*y[4, 2]^(3/2))))+11/(40*sqrt(y[2, 1])*y[4, 1]^(3/2)*(11/(20*y[2, 1]^(3/2)*y[4, 1]^(3/2))+1/(2*y[2, 2]^(3/2)*y[4, 2]^(3/2))))+3*y[2, 2]/(20*y[2, 1]^(3/2)*y[4, 1]^(3/2)*(11/(50*y[2, 1]^(3/2)*y[4, 1]^(3/2))+4/(5*y[2, 2]^(3/2)*y[4, 2]^(3/2))))+3*y[2, 2]/(10*y[2, 1]^(3/2)*y[4, 1]^(3/2)*(11/(25*y[2, 1]^(3/2)*y[4, 1]^(3/2))+3/(5*y[2, 2]^(3/2)*y[4, 2]^(3/2)))), y[2, 2] = 11*y[2, 1]/(75*y[2, 2]^(3/2)*y[4, 2]^(3/2)*(33/(50*y[2, 1]^(3/2)*y[4, 1]^(3/2))+2/(5*y[2, 2]^(3/2)*y[4, 2]^(3/2))))+11*y[2, 1]/(60*y[2, 2]^(3/2)*y[4, 2]^(3/2)*(11/(20*y[2, 1]^(3/2)*y[4, 1]^(3/2))+1/(2*y[2, 2]^(3/2)*y[4, 2]^(3/2))))+2/(5*sqrt(y[2, 2])*y[4, 2]^(3/2)*(11/(50*y[2, 1]^(3/2)*y[4, 1]^(3/2))+4/(5*y[2, 2]^(3/2)*y[4, 2]^(3/2))))+3/(10*sqrt(y[2, 2])*y[4, 2]^(3/2)*(11/(25*y[2, 1]^(3/2)*y[4, 1]^(3/2))+3/(5*y[2, 2]^(3/2)*y[4, 2]^(3/2))))}

 

Download MapleQ.mw

Dear MaplePrimes community,

I'm trying to calculate the (Moore-Penrose) pseudoinverse of a matrix using the following code:

with(LinearAlgebra):
Identity := Matrix(2, 2, [[1, 0], [0, 1]]):
A := Matrix(2, 2, [[0.5661180126, 0.4338819876], [0.8316071431, 0.1683928571]]):
MatrixInverse(Identity - A, method = pseudo)

Unfortunately, the output appears to be meaningless (and it conincides with what I'm getting by calculating a simple matrix inverse using "MatrixInverse(Identity - A)"). Am I missing something? (And how do I calculate the (Moore-Penrose) pseudoinverse of the matrix (Identity-A)?)

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