Spectral Radius

We wrote the program radius.c to compute the largest eigenvalue of a matrix using the power method. For the data file A.dat the output of this program is

n=4
(1 3) (6 1) (0 1) (1 0) 
(0 4) (-1 2) (4 0.2) (0 1) 
(1 1) (2 2) (3 3) (4 4) 
(6 0) (-1.2 2.3) (3.1 4.5) (-1 0) 
lambda=4.28214 6.06714
lambda=6.51923 6.26749
lambda=7.16173 7.29069
lambda=7.1359 7.13804
lambda=7.44994 7.27844
lambda=7.33535 7.34779
lambda=7.38929 7.29725
lambda=7.38333 7.34983
lambda=7.36579 7.32334
lambda=7.38217 7.32954

This indicates the largest eigenvalue is approximately 7.38+7.33i and therefore the spectral radius

ρ(A) ≈ sqrt(7.38²+7.33²) ≈ 10.4.

This can be verified by using the linear algebra program GNU Octave with the script check.m, which has output

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A =

 Columns 1 through 3:

   1.00000 + 3.00000i   6.00000 + 1.00000i   0.00000 + 1.00000i
   0.00000 + 4.00000i  -1.00000 + 2.00000i   4.00000 + 0.20000i
   1.00000 + 1.00000i   2.00000 + 2.00000i   3.00000 + 3.00000i
   6.00000 + 0.00000i  -1.20000 + 2.30000i   3.10000 + 4.50000i

 Column 4:

   1.00000 + 0.00000i
   0.00000 + 1.00000i
   4.00000 + 4.00000i
  -1.00000 + 0.00000i

lambda =

   7.3757 + 7.3276i
   1.6028 + 5.7916i
  -0.7272 - 4.1148i
  -6.2513 - 1.0044i

Last Updated: Wed Sep 17 11:22:30 PDT 2014