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Old   June 19, 2011, 06:30
Default about convergence of a linear system
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Wu Jian
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Hi,

I get a linear system, named A P = Q with A(n*n) , P(n*1), Q (n*1)

I will use iterative solver to solve it

before solving, i want to have some premiliary judgments about 'whether i can get convergent solution or not'

on this problem, my understanding is

1. I select an iterative method, like SOR or SIP;

2. I get the iterative matrix M using A; A P = Q --> MP = NP +Q

3. I use Matlab software to get the spectral radius of the matrix M-1N;

if the spectral radius is small than 1, then i can say ,'with this iterative method, i can obtain convergent solutions'


my question is,

it seems that the conclusion should be directly linked to what kind of iterative method i use;

so , can i directly compute the spectral radius of the coefficient matrix A, then i could get some general conclusions?
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Old   June 20, 2011, 03:16
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Arjun
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Just check if your matrix has diagonal dominance or not. If it has it would converge with most of the method.

You need at least one equation for which diagonal value is larger than sum of its off diagonal terms. (Compare absolute value only).
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Old   June 22, 2011, 06:22
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Wu Jian
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I am thinking the same way like you to check the diagonal dominance, and the condition is satisfied...

but I have another question,

diagonal dominance can guarantee convergence, but it said nothing about underrelaxation factor...

for some tough tests, i noticed that i have to introduce some underrelaxation factor, otherwise, i can not obtain convergent results....

so i guess,

1. diagonal dominance can guarantee convergence but we need also to find an appropriate linear system solver;

2. to improve the diagonal dominance , we can introduce under-relaxation technique

am I right ?

In fact, i am not confident with above claims, so does anyone meet the same situation?

that is, your coefficient matrix is of diagonal dominance, but you still need to introduce some underrelaxation factor ...

thank you for your help...
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