91Ó°ÊÓ

Find the LU factorization of the given matrices. Check by matrix multiplication. (a) \(\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]\) (b) \(\left[\begin{array}{ll}1 & 3 \\ 2 & 2\end{array}\right]\) (c) \(\left[\begin{array}{rr}3 & -4 \\ -5 & 2\end{array}\right]\)

Rearrange the equations to form a strictly diagonally dominant system. Apply two steps of the Jacobi and Gauss-Seidel Methods from starting vector \([0, \ldots, 0]\). (a) \(u+3 v=-1\) \(5 u+4 v=6\) (b) \(u+v+5 w=4\) \(u-8 v-2 w=1\) \(3 u-v+w=-2\) \(\begin{aligned} u+4 v &=5 \\ \text { (c) } \quad v+2 w &=2 \\ 4 u+3 w &=0 \end{aligned}\)

Find the PA \(=\) LU factorization (using partial pivoting) of the following matrices: (a) \(\left[\begin{array}{rrr}1 & 1 & 0 \\ 2 & 1 & -1 \\ -1 & 1 & -1\end{array}\right]\) (b) \(\left[\begin{array}{rrr}0 & 1 & 3 \\ 2 & 1 & 1 \\ -1 & -1 & 2\end{array}\right]\) (c) \(\left[\begin{array}{rrr}1 & 2 & -3 \\ 2 & 4 & 2 \\ -1 & 0 & 3\end{array}\right]\) (d) \(\left[\begin{array}{rrr}0 & 1 & 0 \\ 1 & 0 & 2 \\ -2 & 1 & 0\end{array}\right]\)

Find the LU factorization of the given matrices. Check by matrix multiplication. (a) \(\left[\begin{array}{lll}3 & 1 & 2 \\ 6 & 3 & 4 \\ 3 & 1 & 5\end{array}\right]\) (b) \(\left[\begin{array}{ccc}4 & 2 & 0 \\ 4 & 4 & 2 \\ 2 & 2 & 3\end{array}\right]\) (c) \(\left[\begin{array}{rrrr}1 & -1 & 1 & 2 \\ 0 & 2 & 1 & 0 \\ 1 & 3 & 4 & 4 \\ 0 & 2 & 1 & -1\end{array}\right]\)

Solve the system by finding the \(\mathrm{PA}=\mathrm{LU}\) factorization and then carrying out the two-step back substitution. (a) \(\left[\begin{array}{lll}4 & 2 & 0 \\ 4 & 4 & 2 \\ 2 & 2 & 3\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2} \\\ x_{3}\end{array}\right]=\left[\begin{array}{l}2 \\ 4 \\ 6\end{array}\right]\) (b) \(\left[\begin{array}{rrr}-1 & 0 & 1 \\ 2 & 1 & 1 \\ -1 & 2 & 0\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2} \\\ x_{3}\end{array}\right]=\left[\begin{array}{r}-2 \\ 17 \\\ 3\end{array}\right]\)

Use the approximate operation count \(2 n^{3} / 3\) for Gaussian elimination to estimate how much longer it takes to solve \(n\) equations in \(n\) unknowns if \(n\) is tripled.

Find the Cholesky factorization \(A=R^{T} R\) of each matrix. (a) \(\left[\begin{array}{rrr}4 & -2 & 0 \\ -2 & 2 & -3 \\ 0 & -3 & 10\end{array}\right]\) (b) \(\left[\begin{array}{lll}1 & 2 & 0 \\ 2 & 5 & 2 \\\ 0 & 2 & 5\end{array}\right]\) (c) \(\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3\end{array}\right](d)\left[\begin{array}{rrr}1 & -1 & -1 \\\ -1 & 2 & 1 \\ -1 & 1 & 2\end{array}\right]\)

(a) Write down the \(4 \times 4\) matrix \(P\) such that multiplying a matrix on the left by \(P\) causes the second and fourth rows of the matrix to be exchanged. (b) What is the effect of multiplying on the right by \(P\) ? Demonstrate with an example.

Solve the system of equations by finding the Cholesky factorization of \(A\) followed by two back substitutions. (a) \(\left[\begin{array}{rr}1 & -1 \\ -1 & 5\end{array}\right]\left[\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right]=\left[\begin{array}{r}3 \\ -7\end{array}\right]\) (b) \(\left[\begin{array}{rr}4 & -2 \\ -2 & 10\end{array}\right]\left[\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right]=\left[\begin{array}{r}10 \\ 4\end{array}\right]\)

Assume that your computer can solve 1000 problems of type \(U x=c\), where \(U\) is an upper-triangular \(500 \times 500\) matrix, per second. Estimate how long it will take to solve a full \(5000 \times 5000\) matrix problem \(A x=b\). Answer in minutes and seconds.