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Can you please give me criteria for convergence of these 2 iterative methods(Gauss–Seidel and Jacoby methods)? What is necessary and sufficient condition and what just a sufficient from these 3?

1.The standard convergence condition (for any iterative method) is when the spectral radius of the iteration matrix is less than 1.

2.The method is guaranteed to converge if the matrix A is strictly or irreducibly diagonally dominant.

3.Norm of the iteration matrix is less than 1.

Thanks a lot!

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1 Answer 1

up vote 2 down vote accepted

Obviously all these three conditions are not necessary, as both iteration methods can be used to solve the matrix equation $A0=0$ regardless of $A$ (as long as the iteration matrices exist).

Apparently, you copied the first two conditions from the Wikipedia entries on Gauss-Seidal method and Jacobi method. If you read the two articles careful enough, you should know that both conditions 1 and 2 are sufficient conditions for convergence. As to condition 3, the answer depends on the norm. If the matrix norm in question is a consistent norm (which is true for virtually all matrix norms we encounter in practice) and the iteration matrix is $E$, then by the spectral radius formula, $\rho(E)\le\|E^k\|^{1/k}$ for any $k\in\mathbb{N}$. In particular, $\rho(E)\le\|E\|$. Since $\|E\|<1$, we conclude that the spectral radius $\rho(E)$ is also smaller than $1$ and hence both iteration methods converge.

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