The SpectralRadius command computes the maximum of the absolute values of the eigenvalues of the matrix A.

The output from this procedure will be as symbolic as computationally feasible. If the exact spectral radius of A is too time-consuming to compute, it may be computed numerically.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{NumericalAnalysis}\right]\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[2\,-1\,1\right]\,\left[2\,2\,2\right]\,\left[-1\,-1\,2\right]\right]\right)$

$A≔\left[\begin{array}{ccc}2& \mathrm{-1}& 1\\ 2& 2& 2\\ \mathrm{-1}& \mathrm{-1}& 2\end{array}\right]$

$b≔\mathrm{Vector}\left(\left[-1\,4\,-5\right]\right)$

$b≔\left[\begin{array}{c}\mathrm{-1}\\ 4\\ \mathrm{-5}\end{array}\right]$

$\mathrm{T\_j}≔\mathrm{IterativeFormula}\left(A\,b\,\mathrm{method}=\mathrm{jacobi}\,\mathrm{output}='T'\right)$

$\mathrm{T\_j}≔\left[\begin{array}{ccc}0& \frac{1}{2}& -\frac{1}{2}\\ \mathrm{-1}& 0& \mathrm{-1}\\ \frac{1}{2}& \frac{1}{2}& 0\end{array}\right]$

$\mathrm{SpectralRadius}\left(\mathrm{T\_j}\right)$

$\frac{\sqrt{5}}{2}$