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Convergence in the symmetric positive definite case

In case that the system matrix A {\displaystyle A} is of symmetric positive-definite type one can show convergence.

Let C = C ω = I − ω D − 1 A {\displaystyle C=C_{\omega }=I-\omega D^{-1}A} be the iteration matrix.Then, convergence is guaranteed for

where λ max {\displaystyle \lambda _{\text{max}}} is the maximal eigenvalue.

The spectral radius can be minimized for a particular choice of ω = ω opt {\displaystyle \omega =\omega _{\text{opt}}} as follows

where κ {\displaystyle \kappa } is the matrix condition number.