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Topic: Convergence of recursive simultaneous equations
Replies: 4   Last Post: Dec 4, 2013 12:41 AM

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 Don Del Grande Posts: 2 Registered: 12/3/13
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[n-1] + 1/3 C[n-1] + 1/3 D[n-1]

B[n] = B[0] + 1/3 A[n-1] + 1/3 C[n-1] + 1/3 D[n-1]

C[n] = C[0] + 1/3 A[n-1] + 1/3 B[n-1] + 1/3 D[n-1]

D[n] = D[0] + 1/3 A[n-1] + 1/3 B[n-1] + 1/3 C[n-1]

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(n-1)]
[B(n)] [1/3 0 1/3 1/3 B(0)] [B(n-1)]
[C(n)] = [1/3 1/3 0 1/3 C(0)] [C(n-1)]
[D(n)] [1/3 1/3 1/3 0 D(0)] [D(n-1)]
[ 1 ] [ 0 0 0 0 1 ] [ 1 ]

Date Subject Author
12/3/13 Don Del Grande
12/3/13 James Van Buskirk
12/3/13 Don Del Grande
12/4/13 James Van Buskirk
12/3/13 Ken.Pledger@vuw.ac.nz