As asked in the title, how to transform a $3 × 3$ matrix into the fixed point form $x=Tx+b$?
How to transform a matrix into a fixed point form
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matrices
numerical-methods
numerical-linear-algebra
fixed-point-theorems
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1Could you be more specific? – 2017-02-02
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0"Let A be the matrix whose rows are (2 0 1), (1 2 0), (1 1 3). When Ax=b is put into the fixed-point form x = Tx+v, calculate the (usual) norm of the matrix T for the Jacobi method and for the Gauss-Seidel method to see that the latter can indeed be strictly smaller than the former. " This is the original question, but I am not sure what it means by putting that equation into the fixed point form. – 2017-02-02
1 Answers
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Hint: Observe \begin{align} Ax=b \ \ \Rightarrow \ \ (D+N)x = b \end{align} where $D$ is the diagonal of $A$ and $N$ is everything else. Then it follows \begin{align} x = D^{-1}b - D^{-1}Nx. \end{align} This is called the Jacobi fixed-point form.
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0Oh I got it. So T is equal to $D^{-1}N$ and b is equal to $D^{-1}b$? – 2017-02-02
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0Be careful. $T = -D^{-1}N$ – 2017-02-02
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0What if some element of the diagonal is $0$? – 2017-02-02
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0@Fimpellizieri Then you can't use Jacobi method. – 2017-02-02
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0@JackyChong Thanks! – 2017-02-02