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Considering the following iteration:

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For any initial value of $x^{(0)}$, find the value of $\alpha$ for which the iteration converges.

Hayden
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user146402
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  • This is a 1-step discrete dynamical system. You should evaluate the eigenvalues and find for which $\alpha$ those are between $-1$ and $1$ – 7raiden7 Apr 28 '14 at 16:09
  • Yes, your solution sounds really good. Can you please give me a hint of how I could evaluate the eigenvalues or a partial solution? Because I've worked only with very simple dynamical systems. Which method can I use for finding α? Thank you in advance. – user146402 Apr 28 '14 at 16:37

1 Answers1

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Suppose that $x^{(0)}=b$. You can write you equation as:

$$x^{(N)}=\left(\sum\limits_{k=0}^{N}\left(\alpha A\right)^{k}\right)b.$$

For this to converge, the eigenvalues of the matrix $\alpha A$ must satisfy: $|\lambda_i|<1$.

If I did not make mistakes, we have: $\lambda_1=\alpha$ and $\lambda_2=3\alpha$.

Hence, $\alpha$ must satisfy the following:

$$|\alpha|<1/3.$$

Jika
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