1

I am trying to run some tests on Jacobie iterative method for solving linear systems. However, I have a problem with finding such matrix $A$, which:

  • isn't diagonally dominant
  • when we take two matrices $D$ and $R$, such that $D$ is diagonal from $A$ (so $d_{ii} = a_{ii}$ and rest of the fields are zeros); and $R$ which is the rest (so, $R = A - D$), then, spectral radius of matrix $D^{-1}R < 1$.

Can someone have an example of such matrix? Or is there any hints to create this kind of matrix?

Świstak35
  • 225
  • What form of diagonal dominance do you want? – AlexR Jan 24 '14 at 16:35
  • I don't want it to be strictly or irreducibly diagonally dominant. – Świstak35 Jan 24 '14 at 16:39
  • Maybe you should write down the definition of what you want... I figured that the toeplitz Matrix with $2$ on the diagonal and $-1$ on the first subdiagonals (symmetric), which is the finite difference approximation of the Laplacian with zero boundary conditions can be solved this way (convergence is slow, though) – AlexR Jan 24 '14 at 16:53

1 Answers1

1

"These are the matrices I was looking for".

$$ A = \begin{pmatrix} 1 & 2 & -2 \\ 1 & 1 & 1 \\ 2 & 2 & 1 \end{pmatrix} $$

It's not diagonally dominant, and it also satisfies second requirement.

Świstak35
  • 225