help on a proof of convergence (an exercise problem)
- From: "Yecloud" <yecloud@xxxxxxxxx>
- Date: 19 Feb 2007 17:19:27 -0800
Hi, all,
I have a question on how to prove the convergence of the following
algorithm:
For iterations
V(t+1)=(1-a(t))^2*V(t) + a(t)^2*\sigma
where \sigma >0 is a constant, a(t) >0 is the step size at iteration
t. Start from V(1)>0.
If we require
\sum_{t=1}^{\infty}{a(t)}=\infty; (*)
and
\sum_{t=1}^{\infty}{a(t)^2} < \infty, (**)
how to prove that V(t) converges to zero?
It is an exercise problem in the book of "Parallel and Distributed
Numerical Algorithms"
(Prob.8.1, pp.568). It gives a hint
(1) with (**), to prove V(t) is bounded above and obtain V(t
+1)<=V(t)-2a(t)V(t)+e(t), where
\sum_{t=1}^{\infty}{e(t)} < \infty;
(2) then with (*) to prove the convergence of V(t).
I figured out the first step and with (*) I can show that liminf V(t)
= 0. But I don't know how
to proceed further to get lim V(t) = 0?
Can anyone give me a help or hint?
Thanks,
Cloud
.
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