Course Code: MTE-10

Assignment Code: MTE-10/TMA/2023

Maximum Marks: 100

1. a) The equation \(x^{3}-x-1=0\) has a positive root in the interval \(] 1,2[\). Write a fixed point iteration method and show that it converges. Starting with initial approximation \(\mathrm{x}_{0}=1.5\) find the root of the equation correct to three decimal places.

b) Find an appropriate root of \(x^{3}+2 x^{2}-5=0\) in [1, 2] with \(10^{-5}\) accuracy by

i) Newton Raphson Method

ii) Secant Method

What conclusions can you draw from here about the two methods?

2. a) Using Maclaurin’s expansion for \(\sin x\), find the approximate value of \(\sin \frac{\pi}{4}\) with the error bound \(10^{-5}\)

b) Find an approximate value of the positive real root of \(\mathrm{xe}^{\mathrm{x}}=1\) using graphical method. Use this value to find the positive real root of \(\mathrm{xe}^{\mathrm{x}}=1\) correct to three decimal places by fixed point iteration method.

c) Using \(\mathrm{x}_{0}=0\) find an approximation to one of the zeros of \(\mathrm{x}^{3}-4 \mathrm{x}+1=0\) by using BirgeVieta Method. Perform two iterations.

3. a) Solve the system of equations

\[
\begin{aligned}
2 x_{1}+3 x_{2}+4 x_{3}+x_{4} & =3 \\
x_{1}+2 x_{2}+x_{4} & =2 \\
2 x_{1}+3 x_{2}+x_{3}-x_{4} & =1 \\
x_{1}-2 x_{2}-x_{3}+4 x_{4} & =5
\end{aligned}
\]

using Gauss elimination method with pivoting.

b) Find the inverse of the matrix \(\left[\begin{array}{ccc}3 & 1 & 2 \\ -2 & 3 & -5 \\ 1 & 2 & 4\end{array}\right]\) using Gauss Jordan Method.

c) Solve the following linear system \(A \mathrm{x}=\mathrm{b}\) of equations with partial pivoting

\[
\begin{gathered}
x_{1}-x_{2}+3 x_{3}=3 \\
2 x_{1}+x_{2}+4 x_{3}=7 \\
3 x_{1}+5 x_{2}-2 x_{3}=6 .
\end{gathered}
\]

Store the multipliers and also write the pivoting vectors.

4. a) Solve the system of equations

\[
\begin{aligned}
8 x_{1}-x_{2}+2 x_{3} & =4 \\
-3 x_{1}+11 x_{2}-x_{3}+3 x_{4} & =23
\end{aligned}
\]

\[
\begin{array}{r}
-x_{2}+10 x_{3}-x_{4}=-13 \\
-2 x_{1}+x_{2}-x_{3}+8 x_{4}=13
\end{array}
\]

with \(\mathrm{x}^{(0)}=\left[\begin{array}{llll}0 & 0 & 0 & 0\end{array}\right]^{\mathrm{T}}\), by using the Gauss Jacboi and Gauss Seidel method. The exact

the same accuracy is obtained by both the methods. What conclusions can you draw from the results obtained?

b) Starting with \(\mathrm{x}^{(0)}=\left[\begin{array}{lll}1 & 1 & 1\end{array}\right]^{\mathrm{T}}\), find the dominant eigenvalue and corresponding eigenvector for the matrix \(A=\left[\begin{array}{ccc}4 & -1 & 1 \\ 4 & -8 & 1 \\ -2 & 1 & 5\end{array}\right]\) using the power method.

5. a) The solution of the system of equations \(\left(\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}4 \\ -2\end{array}\right)\) is attempted by the Gauss Jacobi and Gauss Seidel iteration schemes. Set up the two schemes in matrix form. Will the iteration schemes converge? Justify your answer.

b) Obtain an approximate value of \(\int_{0}^{1} \frac{\mathrm{dx}}{1+\mathrm{x}^{2}}\) using composite Simpson’s rule with \(\mathrm{h}=0.25\) and \(\mathrm{h}=0.125\). Find also the improved value using Romberg integration.

c) Find the minimum number of intervals required to evaluate \(\int_{0}^{1} \mathrm{e}^{-\mathrm{x}^{2}} \mathrm{dx}\) with an accuracy of \(\frac{1}{2} \times 10^{-4}\), by using the Trapezoidal rule.

6. a) From the following table, find the number of students who obtained less than 45 marks.

b) Calculate the third-degree Taylor polynomial about \(x_{0}=0\) for \(f(x)=(1+x)^{1 / 2}\).

c) Use the polynomial in part (a) to approximate \(\sqrt{1.1}\) and find a bound for the error involved.

d) Use the polynomial in part (a) to approximate \(\int_{0}^{0.1}(1+x)^{1 / 2} d x\).

7. a) Using \(\sin (0.1)=0.09983\) and \(\sin (0.2)=0.19867\), find an approximate value of \(\sin (0.15)\) by using Lagrange interpolation. Obtain a bound on the truncation error. b) Consider the following data

Use Stirling’s formula to approximate \(\mathrm{f}(1.5)\) with \(x_{0}=1.6\).

c) Solve the I.V.P., \(\mathrm{y}^{\prime}=-\mathrm{y}+\mathrm{t}+1,0 \leq \mathrm{t} \leq 1, \mathrm{y}(0)=1\) using R-K method of \(0\left(h^{4}\right)\) with \(\mathrm{h}=0.1\) and obtain the value of \(y(0.2)\). Also find the error at \(t=0.2\), if the exact solution is \(y(t)=t+e^{-t}\).

8. a) The position \(f(x)\) of a particle moving in a line at various times \(x_{k}\) is given in the following table. Estimate the velocity and acceleration of the particle at \(\mathrm{x}=1.2\)

b) A solid of revolution is formed by rotating about the \(x\)-axis the area bounded between \(x=0, x=1\) and the curve given by the table

Find the volume of the solid so formed using

i) Trapezodial rule ii) Simpson’s rule

c) Take 10 figure logarithm to base 10 from \(\mathrm{x}=300\) to \(\mathrm{x}=310\) by unit increment. Calculate the first derivative of \(\log _{10} \mathrm{x}\) when \(\mathrm{x}=310\).

9. a) For the table of values of \(f(x)=x e^{x}\) given by

Find \(f^{\prime \prime}(2.0)\) using the central difference formula of \(0\left(h^{2}\right)\) for \(h=0.1\) and \(h=0.2\). Calculate T.E. and actual error.

b) Suppose \(f_{n}\) denotes the value of \(f(t)\) at \(t=t_{n}\). If \(f(t)=t^{3}\) then find the value of \(\frac{\left(f_{n+1}-2 f_{n}+f_{n-1}\right)}{h^{2}}\).

c) Use Runge-Kutta method of order four to solve \(y^{\prime}=x+y\). Start with \(x=1, y=0\) and carry to \(\mathrm{x}=1.5\) with \(\mathrm{h}=0.1\).

d) Find the solution of the difference equation \(y_{k+2}-4 y_{k+1}+4 y_{k}=0, k=0,1, \ldots\). Also find the particular solution when \(\mathrm{y}_{0}=1\) and \(\mathrm{y}_{1}=6\).

10. a) The iteration method

\[
x_{n+1}=\frac{1}{8}\left[6 x_{n}+\frac{3 N}{x_{n}}-\frac{x_{n}^{3}}{N}\right], n=0,1,2
\]

where \(\mathrm{N}\) is positive constant, converges to some quantity. Determine this quantity. Also find the rate of convergence of this method. b) Determine the spacing \(\mathrm{h}\) in a table of equally spaced values for the function \(f(x)=(2+x)^{4}, 1 \leq x \leq 2\), so that the quadratic interpolation in this table satisfies | error \(\mid \leq 10^{-6}\).

c) Determine a unique polynomial \(\mathrm{f}(\mathrm{x})\) of degree \(\leq 3\) such that \(f\left(x_{0}\right)=1, f^{\prime}\left(x_{0}\right)=2, f\left(x_{1}\right)=2, f^{\prime}\left(x_{1}\right)=3\) where \(x_{1}-x_{0}=h\).