91Ó°ÊÓ

a. Show that the Gauss-Jordan method requires $$ \frac{n^{3}}{2}+n^{2}-\frac{n}{2} \quad \text { multiplications/divisions } $$ and $$ \frac{n^{3}}{2}-\frac{n}{2} \quad \text { additions/subtractions. } $$ b. Make a table comparing the required operations for the Gauss-Jordan and Gaussian elimination methods for \(n=3,10,50,100 .\) Which method requires less computation?

A Fredholm integral equation of the second kind is an equation of the form $$ u(x)=f(x)+\int_{a}^{b} K(x, t) u(t) d t $$ where \(a\) and \(b\) and the functions \(f\) and \(K\) are given. To approximate the function \(u\) on the interval \([a, b]\), a partition \(x_{0}=a

Let $$ A=\left[\begin{array}{rrr} 1 & 0 & -1 \\ 0 & 1 & 1 \\ -1 & 1 & \alpha \end{array}\right] $$ Find all values of \(\alpha\) for which a. \(\quad A\) is singular. b. \(\quad A\) is strictly diagonally dominant. c. \(A\) is symmetric. d. \(A\) is positive definite.

Let $$ A=\left[\begin{array}{lll} \alpha & 1 & 0 \\ \beta & 2 & 1 \\ 0 & 1 & 2 \end{array}\right] $$ Find all values of \(\alpha\) and \(\beta\) for which a. \(A\) is singular. b. \(\quad A\) is strictly diagonally dominant. c. \(A\) is symmetric. d. \(A\) is positive definite.

Show that Gaussian elimination can be performed on \(A\) without row interchanges if and only if all leading principal submatrices of \(A\) are nonsingular. [Hint: Partition each matrix in the equation $$ A^{(k)}=M^{(k-1)} M^{(k-2)} \cdots M^{(1)} A $$ vertically between the \(k\) th and \((k+1)\) st columns and horizontally between the \(k\) th and \((k+1)\) st rows (see Exercise 14 of Section 6.3). Show that the nonsingularity of the leading principal submatrix of \(A\) is equivalent to \(a_{k, k}^{(k)} \neq 0 .\) ]

Suppose that $$\begin{gathered} 2 x_{1}+x_{2}+3 x_{3}=1 \\ 4 x_{1}+6 x_{2}+8 x_{3}=5 \\ 6 x_{1}+\alpha x_{2}+10 x_{3}=5 \end{gathered}$$ with \(|\alpha|<10\). For which of the following values of \(\alpha\) will there be no row interchange required when solving this system using scaled partial pivoting? a. \(\alpha=6\) b. \(\quad \alpha=9\) c. \(\quad \alpha=-3\)