Iterative methods for linear systems
- From: kingpin@xxxxxxxxxxx
- Date: 26 Mar 2007 01:42:18 -0700
Hi to all,
i have the following exercise. I have solved it. Unfortunately, it
seems that my solution is wrong, but i can't spot the error in it.
---
We have a linear system Ax=b, with A an NxN in R and with a unique
solution x*.
Consider the following iterative method
x_(k+1) = P*x_(k) + q, for k>=0
with
x_0 (starting vector) supposed given
P (a real NxN matrix) supposed given
q (an N real-valued vector) supposed given
If || e_(k+1) || > || e_(k) || (where || || is the norm-2, e_(k) is
defined as e_(k) = x_(k) - x*) )
for some k, then the method do not converge.
--
Discussion.
Let x_0 be a vector such that e_0 = x_0 - x* is the eigenvector
relative to the
eigenvalue of maximum module of P. Let "a" be that eigenvalue.
We know also that
e_(k) = (P^k)*e_0.
We have:
|| e_(k+1) || = || (P^(k+1)) * e_0|| = || a^(k+1) * e_0|| = |a^(k+1)|
||e_0||
and
|| e_(k) || = || (P^(k)) * e_0|| = || a^(k) * e_0|| = |a^(k)| ||
e_0|| .
If for some k || e_(k+1) || > || e_(k) ||, then we have, for that k:
|a^(k+1)| ||e_0|| > |a^(k)| ||e_0||
so must be |a| > 1 (since ||e_0|| is always > 0).
But iterative methods have convergence if and only if |a| < 1, so the
method do not converge.
Where i am wrong???
The right answer should be
"It is false that the method do not converge, because convergence do
not imply a monotonic error in norm-2".
.
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