Course Code: BMTE-144

Assignment Code: BMTE-144/TMA/2023

Maximum Marks: 100

PART – A (40 marks)

1. a) Find the approximate root of the equation \(2 x^3=3 x+6\) using Newton-Raphson method. Perform only 3 iterations with \(\mathrm{x}_0=2\).

b) The roots of the quadratic equation \(x^2+a x+b=0\) are given by \(\alpha\) and \(\beta\). Show that the iteration \(\mathrm{x}_{\mathrm{k}+1}=\frac{-\left(\mathrm{ax} \mathrm{x}_{\mathrm{k}}+\mathrm{b}\right)}{\mathrm{x}_{\mathrm{k}}}\) will converge near \(\mathrm{x}=\alpha\) when \(|\alpha|>|\beta|\).

c) If \(\delta^2 \mathrm{f}\left(\mathrm{x}_0\right)=\mathrm{C}_1 \mathrm{~h}^2 \mathrm{f}^{\prime \prime}\left(\mathrm{x}_0\right)+\mathrm{C}_2 \mathrm{~h}^4 \mathrm{f}^{(4)}\left(\mathrm{x}_0\right)+\cdots\), find the values of \(\mathrm{C}_1\) and \(\mathrm{C}_2\).

2. a) The Gauss-Seidel method is used to solve the system of equations
\[
\left[\begin{array}{ccc}
4 & 0 & 2 \\
0 & 5 & 2 \\
5 & 4 & 10
\end{array}\right]\left[\begin{array}{l}
x_1 \\
x_2 \\
x_3
\end{array}\right]=\left[\begin{array}{c}
4 \\
-3 \\
2
\end{array}\right]
\]
Determine the rate of convergence of the method.

b) Find the interpolating polynomial by Newton’s divided difference formula for the following data:
\begin{tabular}{|c|c|c|c|c|}
\hline \(\mathrm{x}\) & 0 & 1 & 2 & 4 \\
\hline \(\mathrm{y}\) & 1 & 1 & 2 & 5 \\
\hline
\end{tabular}

c) Using synthetic division method, show that 2 is a simple root of the equation
\[
p(x)=x^4-2 x^3+x^2-x-2=0 .
\]

3. a) Using Gauss-Jordan method, find the inverse of the matrix
\[
\left[\begin{array}{ccc}
1 & 1 & 3 \\
1 & 3 & -3 \\
-2 & -4 & -4
\end{array}\right] .
\]

b) Find the largest step length that can be used for constructing a table of values for the function
\[
f(x)=\frac{4}{3} x^3+5 \ln x, 10 \leq x \leq 20,
\]
so that a quadratic interpolation can be used with an accuracy of \(5 \times 10^{-6}\).

4. a) Find the missing values in the following table:
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline \(\mathrm{x}\) & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline \(\mathrm{y}\) & 0 & 2 & – & 18 & – & 90 \\
\hline
\end{tabular}

b) Using Classical Runge-Kutta fourth order method, find an approximate value of \(\mathrm{y}(1.2)\) for the IVP \(\frac{d y}{d x}=x y, y(1)=2\) with \(h=0.2\).

PART – B (40 marks)

5. a) For the following data, use Gauss backward difference method to obtain the interpolating polynomial \(\mathrm{f}(\mathrm{x})\) :
\begin{tabular}{|c|c|c|c|c|c|}
\hline \(\mathrm{x}\) & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 \\
\hline \(\mathrm{f}(\mathrm{x})\) & 1.40 & 1.56 & 1.76 & 2.00 & 2.28 \\
\hline
\end{tabular}
Hence, find the value of \(\mathrm{f}(0.45)\).

b) The velocity of a vehicle beginning from rest is given in the following table for part of the first four. Using Simpson’s \(\frac{1}{3}\) rule, find the distance travelled by the vehicle in this hour:
\begin{tabular}{|c|l|l|l|l|l|l|}
\hline \(\mathrm{t}=\) time in \(\mathrm{min}\). & 10 & 20 & 30 & 40 & 50 & 60 \\
\hline \(\mathrm{v}=\) velocity in \(\mathrm{km} / \mathrm{hr}\). & 80 & 60 & 70 & 75 & 70 & 80 \\
\hline
\end{tabular}

6. a) Evaluate \(\int_0^1 \frac{1}{1+\mathrm{x}^2} \mathrm{dx}\) by using trapezoidal rule with \(\mathrm{h}=0.5\) and \(\mathrm{h}=0.25\). Use Romber’s method to find the best value of \(\pi\).

b) Estimate the eigenvalues of the matrix
\[
\left[\begin{array}{ccc}
1 & -1 & 2 \\
-1 & 1 & 2 \\
2 & 2 & -2
\end{array}\right]
\]
using the Gerschgorin bounds.

7. a) Determine the largest eigenvalue in magnitude and the corresponding eigenvector of the matrix \(\left(\begin{array}{lll}1 & 6 & 1 \\ 1 & 2 & 0 \\ 0 & 0 & 3\end{array}\right)\) using the power method. Take \((1,0,0)^{\mathrm{T}}\) as the initial approximation and perform 4 iterations.

b) The method
\[
\mathrm{x}_{\mathrm{n}+1}=\frac{1}{9}\left[5 \mathrm{x}_{\mathrm{n}}+\frac{5 \mathrm{~N}}{\mathrm{x}_{\mathrm{n}}^2}-\frac{\mathrm{N}^2}{\mathrm{x}_{\mathrm{n}}^5}\right], \mathrm{n}=0,1,2, \ldots
\]
where \(\mathrm{N}\) is a positive constant, converges to \(\mathrm{N}^{1 / 3}\). Find the rate of convergence of the method.

8. a) Find the inverse of the matrix \(\left[\begin{array}{lll}2 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 1 & 2\end{array}\right]\) using Gauss-Jordan method.

b) Divide the polynomial
\[
x^5-6 x^4+8 x^3+8 x^2+4 x-40
\]
by \((x-3)\) by the synthetic division method and find the remainder.

c) Determine a unique polynomial \(\mathrm{f}(\mathrm{x})\) of degree \(\leq 3\) such that \(\mathrm{f}\left(\mathrm{x}_0\right)=1, \mathrm{f}^{\prime}\left(\mathrm{x}_0\right)=2\), \(\mathrm{f}\left(\mathrm{x}_1\right)=2, \mathrm{f}^{\prime}\left(\mathrm{x}_1\right)=3\), where \(\mathrm{x}_1-\mathrm{x}_0=\mathrm{h}\).

PART – C (20 marks)

9. a) Obtain the interpolating polynomial in simplest form which fits the following data:
\begin{tabular}{|c|c|c|c|c|}
\hline \(\mathrm{x}\) & -1 & 0 & 1 & 2 \\
\hline \(\mathrm{f}(\mathrm{x})\) & 3 & -4 & 5 & -6 \\
\hline
\end{tabular}

b) Prove that \(\mu^2=1+\frac{\delta^2}{4}\).

c) Determine the order of convergence of the iterative method
\[
x_{n+1}=\frac{x_{n-1} f\left(x_n\right)-x_n f\left(x_{n-1}\right)}{f\left(x_n\right)-f\left(x_{n-1}\right)}
\]
for finding a simple root of the equation \(f(x)=0\).

10. a) Solve the initial value problem using Euler method
\[
y^{\prime}=\frac{1}{x^2-3 y}, y(3)=2 .
\]
Find \(\mathrm{y}(3.1)\) taking \(\mathrm{h}=0.1\).

b) Set up the Gauss-Seidel iteration scheme in matrix form for solving the system of equations
\[
\left[\begin{array}{ccc}
1 & 1 & 1 \\
4 & 3 & -1 \\
3 & 5 & 3
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
1 \\
6 \\
4
\end{array}\right] .
\]
Show that the method is convergent and hence find its rate of convergence.

c) Write the error in linear interpolation. Hence, show that
\[
\mid \text { error }\left|\leq \frac{h^2}{8} \max \right| f^{\prime \prime}(x) \mid
\]
where \(\mathrm{h}=\mathrm{x}_1-\mathrm{x}_0, \mathrm{x} \in\left[\mathrm{x}_0, \mathrm{x}_1\right]\).