Computing x₁x₂ and x1/x2 for x₁=2.48, x₂=3.4415, evaluate the relative and absolute error. Further comment on the accuracy of the result.

Why the calculations through calculator are approximate?

Using Power method, obtain the largest eigen value and corresponding eigen vector correct upto 2d of the following matrices:

1. \(\left[\begin{array}{ccc} 1&2&-1 \\ 2&1&2 \\ -1&2&1\end{array}\right]\)   
2. \(\left[\begin{array}{ccc} 4&1&1&1 \\ 3&-1&1&1\\ -1&1&2&0\\1&1&0&2\end{array}\right]\)

       3. \(\left[\begin{array}{ccc} 0&11&-5\\ -2&17&-7 \\ -4&26&-10\end{array}\right]\)

Find the smallest eigen value and corresponding eigen vector of the following matrices using Power method:

1. \(\left[\begin{array}{ccc} 3&2&1 \\ 2&3&2 \\ 1&2&2\end{array}\right]\)
2. \(\left[\begin{array}{ccc} 2&2&-1 \\ -5&9&-3 \\ -4&4&1\end{array}\right]\)
3. \(\left[\begin{array}{ccc} 2&-1&0 \\ -1&2&-1 \\ 0&-1&2\end{array}\right]\)

Find all the eigen values and corresponding eigen vectors using Power method:

1. \(\left[\begin{array}{ccc} 1&2&3 \\ 3&1&0 \\ -2&0&1\end{array}\right]\)
2. \(\left[\begin{array}{ccc} 4&1&0\\ 1&20&1 \\ 0&1&4\end{array}\right]\)
3. \(\left[\begin{array}{ccc} 2&1&1&0 \\ 1&1&0&1 \\ 1&0&0&1\\1&1&1&2\end{array}\right]\)

If the largest eigen value of the matrix\(\left[\begin{array}{ccc} 5&-2&0 \\ 1&2&-3 \\ 1&-2&4\end{array}\right]\)is 6 and corresponding eigen vector is\(\left[\begin{array}{ccc} 1 \\ -1/2 \\ 1\end{array}\right]\),then find the subdominant eigen value.

Find the eigen values of following matrix:

\(\left[\begin{array}{ccc} 1&-1&0&0&0&0 \\ 1&1&-1&0&0&0 \\ 0&1&1&-1&0&0\\0&0&1&1&-1&0\\0&0&0&1&1&-1\\0&0&0&0&1&1\end{array}\right]\)

Let y(x) be a continuous solution of the initial value problem

y' + 2y = f(x), y(0) = 0, where \( f(x) = \begin{cases} 1, & \quad \text{if } 0 \le x\le1,\\ 0, & \quad \text{if } x>1 \end{cases} \)

Then y\(( \frac{3}{2})\) is equal to

(a)  \(( \frac{sinh(1)}{e^3})\)  (b)   \(( \frac{cosh(1)}{e^3})\)   (c)  \(( \frac{sinh(1)}{e^2})\)  (d)   \(( \frac{cosh(1)}{e^2})\)

The singular integral of the ODE 

\((xy'-y)^2 =x^2(x^2-y^2)\)

is

(a) \(y=xsinx\)       (b)  \(y=xsin(x+\frac{\pi}{4})\)   (c)  \(y=x\)     (d)  \(y=x+ \frac{\pi}{4}\)

The initial value problem

\(y'=2\sqrt{y}, y(0)=a,\)

has

(a) a unique solution if a < 0,

(b) no solution if a > 0,

(c) infinitely many solutions if a = 0

(d) a unique solution if a ≥ 0

For the initial value problem

\(\begin {cases} \frac{dy}{dx}=y^2+cos^2x, x>0,\\ y(0)=0 \end{cases}\)

The largest interval of existence of the solution predicted by Picard’s theorem is:

(a) [0, 1] (b) [0, \(\frac{1}{2}\) ] (c) [0, \(\frac{1}{3}\) ] (d) [0, \(\frac{1}{4} \) ]