Rate of Convergence for iterative method.

**sean.1986** · September 9th 2009, 04:37 AM

Let an iterative solution of a system of linear equations be defined by...

$x^{(k+1)} = Gx^{(k)} + c, k = 1, 2, ...$

$G = \left[ \begin{array}{cccc} 0.08 & 0 & 0.05 \\ 0.04 & 0.1 & 0.02 \\ 0 & 0 & 0.08 \end{array} \right]$

Determine the rate of convergence for the iterative method and explain your work.

Once again, finding it hard to find information that explains this sort of thing clearly, so any help greatly appreciated.

Thanks.

**BobP** · September 10th 2009, 01:52 AM

Have you tried Varga, R.S., Matrix Iterative Analysis ?

**sean.1986** · September 11th 2009, 10:40 AM

Okay, I think I have the answer but perhaps someone might check for me...

$G = \left[ \begin{array}{cccc} 0.08 & 0 & 0.05 \\ 0.04 & 0.1 & 0.02 \\ 0 & 0 & 0.08 \end{array} \right]$

$H = \left[ \begin{array}{cccc} 0.08 - \lambda & 0 & 0.05 \\ 0.04 & 0.1 - \lambda & 0.02 \\ 0 & 0 & 0.08 - \lambda \end{array} \right]$

$det(H) = (0.08 - \lambda)(0.1 - \lambda)(0.08 - \lambda)$

$+ (0)(0.02)(0) + (0.05)(0.04)(0)$

$- (0.08 - \lambda)(0.02)(0) - (0)(0.04)(0.08 - \lambda) - (0.05)(0.1 - \lambda)(0)$

$= (0.08 - \lambda)(0.1 - \lambda)(0.08 - \lambda)$

$max(|\lambda|) = 0.1 = p(G)$

$r(G) = -log_{10}p(G) = -log_{10}(0.1) = 1$

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