Student[NumericalAnalysis]

 compute the spectral radius of a square matrix

Parameters

 A - Matrix; a square matrix

Description

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

Notes

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{NumericalAnalysis}\right]\right):$
 > $A≔\mathrm{Matrix}\left(\left[\left[2,-1,1\right],\left[2,2,2\right],\left[-1,-1,2\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{ccc}{2}& {-1}& {1}\\ {2}& {2}& {2}\\ {-1}& {-1}& {2}\end{array}\right]$ (1)
 > $b≔\mathrm{Vector}\left(\left[-1,4,-5\right]\right)$
 ${b}{≔}\left[\begin{array}{c}{-1}\\ {4}\\ {-5}\end{array}\right]$ (2)
 > $\mathrm{T_j}≔\mathrm{IterativeFormula}\left(A,b,\mathrm{method}=\mathrm{jacobi},\mathrm{output}='T'\right)$
 ${\mathrm{T_j}}{≔}\left[\begin{array}{ccc}{0}& \frac{{1}}{{2}}& {-}\frac{{1}}{{2}}\\ {-1}& {0}& {-1}\\ \frac{{1}}{{2}}& \frac{{1}}{{2}}& {0}\end{array}\right]$ (3)
 > $\mathrm{SpectralRadius}\left(\mathrm{T_j}\right)$
 $\frac{\sqrt{{5}}}{{2}}$ (4)