$$
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Iterative Methods
Consider the following iteration for solving \(Mx + q = 0\):
\[x_{k+1} = x_k - \inv{P}\block{Mx_k +q}\]
We look for conditions on \(P\) such that the iteration matrix \(B =
I - \inv{P}M\) is convergent, that is:
\[\rho(B) < 1\]
We start by splitting \(M = P - N\) so that \(B = \inv{P}N\), and let
\(\lambda\) be an eigenvalue of \(B\) with corresponding eigenvector
\(x\), that is:
\[\inv{P}Nx = \lambda x\]
Equivalently:
\[\block{P - M}x = \lambda P x\]
We now look for sufficient conditions for:
\[|\lambda| < 1\]
Case 1: \(M\) is positive definite
In this case, \(x^*Mx >0\) and we may normalize \(x\) so that \(x^*Mx
= 1\). \(x\) verifies:
\[x^*\block{P - M}x = \lambda x^*P x\]
We let \(x^*Px = a + ib\) and rewrite the above as:
\[a + ib - 1 = \lambda \block{a + ib}\]
Taking squared modulus on both sides gives:
\[(a - 1)^2 + b^2 = |\lambda|^2 \block{a^2 + b^2}\]
That is:
\[a^2 + b^2 + 1 - 2a = |\lambda|^2 \block{a^2 + b^2}\]
Our convergence condition reduces to \(1 - 2a < 0\), which we rewrite
as:
\[x^*Mx < x^*Px + \bar{x^*Px} = x^*\block{P + P^T} x\]
A sufficient condition is then:
\[P + P^T - M > 0\]
Jacobi Iteration
Given a diagonal preconditioner \(P\) and a positive definite matrix
\(M\), the above sufficient condition reduces to:
\[2P > M\]
Gauss-Seidel Iteration
Given a positive definite matrix \(M = L + D + L^T\), where \(D\) is
diagonal, the Gauss-Seidel preconditioner is \(P = L + D\) and the
sufficient convergence condition is:
\[L + 2D + L^T - M = D + M - M = D > 0\]
which is always true for \(M\) positive definite.
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