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Convergence of recursive simultaneous equations
Posted:
Dec 3, 2013 12:30 PM


I have a set of four recursive equations:
A[n] = A[0] + 1/3 B[n1] + 1/3 C[n1] + 1/3 D[n1]
B[n] = B[0] + 1/3 A[n1] + 1/3 C[n1] + 1/3 D[n1]
C[n] = C[0] + 1/3 A[n1] + 1/3 B[n1] + 1/3 D[n1]
D[n] = D[0] + 1/3 A[n1] + 1/3 B[n1] + 1/3 C[n1]
where A[0], B[0], C[0], and D[0] are constants.
Is there a way other than brute force recursion to determine if the values converge (and, if so, to what) as n approaches positive infinity?
This can be expressed in matrix form as:
[A(n)] [ 0 1/3 1/3 1/3 A(0)] [A(n1)] [B(n)] [1/3 0 1/3 1/3 B(0)] [B(n1)] [C(n)] = [1/3 1/3 0 1/3 C(0)] [C(n1)] [D(n)] [1/3 1/3 1/3 0 D(0)] [D(n1)] [ 1 ] [ 0 0 0 0 1 ] [ 1 ]



