Convergence of iteration method - Relation between norm and eigenvalue

In summary, the convergence of an iteration method depends on the spectral radius of the iteration matrix being less than 1, regardless of the norms of the matrix. For a symmetric matrix, the spectral radius is equal to the norm, so if the norm is 0.01, there will be an eigenvalue with the same magnitude.
  • #1
mathmari
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Hey! :eek:

Let $G$ be the iteration matrix of an iteration method. So that the iteration method converges is the only condition that the spectral radius id less than $1$, $\rho (G)<1$, no matter what holds for the norms of $G$ ?
I mean if it holds that $\|G\|_{\infty}=3$ and $\rho (G)=0.3<1$ or $\|G\|_{1}=1$ and $\rho (G)=0.1<1$ in these both cases the iteration method converges, or not? (Wondering)

I have also an other question. Let $A$ be a symmetric matrix. If $\|A\|=0.01$ then there is an eigenvalue $\lambda$ with $|\lambda|\leq 0.01$, isn't it? We get that using the spectral radius: $\rho (A)=\max |\lambda_i|\leq \|A\|_1$, right? (Wondering)
 
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  • #2
mathmari said:
Let $G$ be the iteration matrix of an iteration method. So that the iteration method converges is the only condition that the spectral radius id less than $1$, $\rho (G)<1$, no matter what holds for the norms of $G$ ?
I mean if it holds that $\|G\|_{\infty}=3$ and $\rho (G)=0.3<1$ or $\|G\|_{1}=1$ and $\rho (G)=0.1<1$ in these both cases the iteration method converges, or not? (Wondering)
For an iteration procedure, you are going to be interested in powers of $G$. So what matters is that $\|G^n\| < 1$ when $n$ is large. It doesn't matter whether $G$ itself has norm larger than $1$, so long as the powers of $G$ have smaller norms.

The spectral radius has the property that \(\displaystyle \rho(G) = \lim_{n\to\infty}\|G^n\|^{1/n}.\) So if $\rho(G)<1$ then $\|G^n\|<1$ whenever $n$ is large enough, and that is sufficient to ensure that the iteration method converges.

mathmari said:
I have also an other question. Let $A$ be a symmetric matrix. If $\|A\|=0.01$ then there is an eigenvalue $\lambda$ with $|\lambda|\leq 0.01$, isn't it? We get that using the spectral radius: $\rho (A)=\max |\lambda_i|\leq \|A\|_1$, right? (Wondering)
For a symmetric matrix, the spectral radius is equal to the norm. So if $\|A\|=0.01$ then there is actually an eigenvalue $\lambda$ with $|\lambda| = 0.01$.
 
  • #3
Opalg said:
For an iteration procedure, you are going to be interested in powers of $G$. So what matters is that $\|G^n\| < 1$ when $n$ is large. It doesn't matter whether $G$ itself has norm larger than $1$, so long as the powers of $G$ have smaller norms.

The spectral radius has the property that \(\displaystyle \rho(G) = \lim_{n\to\infty}\|G^n\|^{1/n}.\) So if $\rho(G)<1$ then $\|G^n\|<1$ whenever $n$ is large enough, and that is sufficient to ensure that the iteration method converges.For a symmetric matrix, the spectral radius is equal to the norm. So if $\|A\|=0.01$ then there is actually an eigenvalue $\lambda$ with $|\lambda| = 0.01$.
I see! Thank you very much! (Smile)
 

FAQ: Convergence of iteration method - Relation between norm and eigenvalue

1. What is the definition of convergence in the context of iteration methods?

Convergence in the context of iteration methods refers to the process of approaching a desired solution through repeated iterations or calculations. It is a measure of how quickly the method is able to find a solution that is close enough to the true solution.

2. How is the norm of a matrix related to its eigenvalues?

The norm of a matrix is a measure of its size or magnitude, while the eigenvalues of a matrix are the values that satisfy the equation Ax = λx, where A is the matrix and x is the eigenvector. The norm and eigenvalues of a matrix are related in that the norm can provide an upper bound for the magnitude of the eigenvalues.

3. Why is the convergence rate of an iteration method important?

The convergence rate of an iteration method is important because it determines how quickly the method is able to find a solution. A faster convergence rate means that fewer iterations are needed to reach a desired level of accuracy, which can save time and computational resources.

4. How can the convergence rate of an iteration method be improved?

The convergence rate of an iteration method can be improved by choosing an appropriate initial guess, using a more efficient iterative formula, or by preconditioning the matrix. Additionally, using a combination of different iteration methods or implementing parallel computing techniques can also improve the convergence rate.

5. Can the convergence rate of an iteration method be guaranteed?

No, the convergence rate of an iteration method cannot be guaranteed in all cases. It depends on the properties of the matrix and the chosen method. Some methods may converge quickly for certain matrices, but not for others. It is important to carefully select a method and analyze its convergence properties for a particular problem.

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