Exercise 2.1.5

For $b=(1,0,0)$, Let’s run the Arnoldi iteration with $A$ to produce an orthonormal basis $q_1,q_2,q_3$: $A=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 2\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 2\hfill \end{array}\right]$

• $k=1$, Let $q_1=\frac{b}{\left|b\right|}=(1,0,0)$.

• $v=A{q}_1=(2,-1,0)$

  • $j=1$: $h_{11}=q_1^{T}v=2$, $v=v-h_{11}{q}_1=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill -1\hfill \\ \hfill 0\hfill \end{array}\right]$
• $h_{2,1}=1$
• $q_2=\frac{v}{h_{2,1}}=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill -1\hfill \\ \hfill 0\hfill \end{array}\right]$

• $k=2$, we have $v=A{q}_2=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right]$

  • $j=1$: $h_{12}=q_1^{T}v=1$, $v=v-h_{12}q_1 =

    $\left[\begin{array}{c}\hfill 0\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \end{array}\right]$

    $

  • $j=2$: $h_{22}=q_2^{T}v=2$, $v=v-h_{22}q_2 =

    $\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \end{array}\right]$

    $

• $h_{3,2}=1$

• $q_3 =

  $\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \end{array}\right]$

  $

So from the calculations, we have $H=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill -2\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$. We can prove that $A{Q}_2=Q_{3}H$