 Convergence of an adjacent-term averaging recurrence

ID: convergence-of-an-adjacent-term-averaging-recurrence

For

0 < α < 1

, the successive differences satisfy

d_{n + 1} = - (1 - α) d_{n}

. Summing the resulting geometric series gives

x_{n} = x_{0} + \frac{x _{1} - x _{0}}{2 - α} [1 - (α - 1)^{n}], lim_{n \to \infty} x_{n} = \frac{( 1 - α ) x _{0} + x _{1}}{2 - α} .

Equivalently, the two-dimensional update matrix has eigenvalues

1

and

α - 1

, with only the second mode decaying. At

α = 0

, generic initial values instead alternate and do not converge.

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