Question 9.3.2: Apply both the Power method and the Symmetric Power method t...
Apply both the Power method and the Symmetric Power method to the matrix
A=\left[\begin{array}{rrr}4 & -1 & 1 \\-1 & 3 & -2 \\1 & -2 & 3\end{array}\right],
using Aitken’s Δ² method to accelerate the convergence.
This matrix has eigenvalues \lambda_{1}=6, \lambda_{2}=3, \text { and } \lambda_{3}=1 . An eigenvector for the eigenvalue 6 is (1,-1,1)^{t} . Applying the Power method to this matrix with initial vector (1,0,0)^{t} gives the values in Table 9.2.
Table 9.2
\begin{array}{rllll}\hline m & {\left( y ^{(m)}\right)^{t}} & \mu^{(m)} & \hat{\mu}^{(m)} & {\left( x ^{(m)}\right)^{t} \text { with }\left\| x ^{(m)}\right\|_{\infty}=1} \\\hline 0 & & & & (1,0,0)\\1 & (4,-1,1) & 4 & & (1,-0.25,0.25)\\2 & (4.5,-2.25,2.25) & 4.5 & 7 &(1,-0.5,0.5) \\3 & (5,-3.5,3.5) & 5 & 6.2 & (1,-0.7,0.7) \\4 & (5.4,-4.5,4.5) & 5.4 & 6.047617 & (1,-0.833 \overline{3}, 0.833 \overline{3}) \\5 & (5.66 \overline{6},-5.166 \overline{6}, 5.166 \overline{6}) & 5.66 \overline{6} & 6.047617 & (1,-0.911765,0.911765) \\6 & (5.823529,-5.558824,5.558824) & 5.823529 & 6.011767 & (1,-0.954545,0.954545) \\7 & (5.909091,-5.772727,5.772727) & 5.909091 & 6.000733 & (1,-0.976923,0.976923) \\8 & (5.953846,-5.884615,5.884615) & 5.953846 & 6.000184 & (1,-0.988372,0.988372) \\9 & (5.976744,-5.941861,5.941861) & 5.976744 & & (1,-0.994163,0.994163) \\10 & (5.988327,-5.970817,5.970817) & 5.988327 & & (1,-0.997076,0.997076) \\\hline\end{array}