91Ó°ÊÓ

Use Householder's method to place the following matrices in tridiagonal form. a. \(\left[\begin{array}{rrrr}4 & -1 & -1 & 0 \\ -1 & 4 & 0 & -1 \\ -1 & 0 & 4 & -1 \\ 0 & -1 & -1 & 4\end{array}\right]\) c. \(\left[\begin{array}{lllll}8 & 0.25 & 0.5 & 2 & -1 \\ 0.25 & -4 & 0 & 1 & 2 \\ 0.5 & 0 & 5 & 0.75 & -1 \\ 2 & 1 & 0.75 & 5 & -0.5 \\ -1 & 2 & -1 & -0.5 & 6\end{array}\right]\) d. \(\left[\begin{array}{rrrrr}2 & -1 & -1 & 0 & 0 \\ -1 & 3 & 0 & -2 & 0 \\\ -1 & 0 & 4 & 2 & 1 \\ 0 & -2 & 2 & 8 & 3 \\ 0 & 0 & 1 & 3 & 9\end{array}\right]\)

Apply two iterations of the QR method without shifting to the following matrices. a. \(\left[\begin{array}{rrr}2 & -1 & 0 \\ -1 & -1 & -2 \\ 0 & -2 & 3\end{array}\right]\) b. \(\left[\begin{array}{lll}3 & 1 & 0 \\ 1 & 4 & 2 \\ 0 & 2 & 3\end{array}\right]\) c. \(\left[\begin{array}{lllll}4 & 2 & 0 & 0 & 0 \\ 2 & 4 & 2 & 0 & 0 \\ 0 & 2 & 4 & 2 & 0 \\ 0 & 0 & 2 & 4 & 2 \\ 0 & 0 & 0 & 2 & 4\end{array}\right]\) d. \(\left[\begin{array}{rrccc}5 & -1 & 0 & 0 & 0 \\ -1 & 4.5 & 0.2 & 0 & 0 \\\ 0 & 0.2 & 1 & -0.4 & 0 \\ 0 & 0 & -0.4 & 3 & 1 \\ 0 & 0 & 0 & 1 & 3\end{array}\right]\)