BE Computer Engineering (Pokhara University) Numerical Methods (PU, MTH 252) Question Paper 2079

(a) Derive the iterative formula for the Newton-Raphson method for finding a real root of the equation $f(x)=0$, and state its order of convergence. Mention two situations in which the method fails. [6]

(b) Find a real root of the equation $x^3 - 2x - 5 = 0$ correct to four decimal places using the Newton-Raphson method, taking $x_0 = 2$. Compare the number of iterations required with the Bisection method to achieve the same accuracy in the interval $[2, 3]$. [6]

(a) Construct Newton's divided difference table for the following data and hence estimate the value of $f(3)$.

[7]

(b) Explain why Newton's divided difference formula is preferred over Lagrange's interpolation formula when a new data point is added to an existing set. Illustrate your answer briefly. [5]

(a) Derive the composite Simpson's $1/3$ rule for numerical integration and write down the expression for its error term. [6]

(b) Evaluate $\displaystyle\int_{0}^{6}\frac{dx}{1+x^2}$ by taking 6 sub-intervals using (i) the Trapezoidal rule and (ii) Simpson's $1/3$ rule. Compare both results with the exact value and comment on the accuracy. [6]

(a) Describe the fourth-order Runge-Kutta method for solving a first-order ordinary differential equation $\dfrac{dy}{dx}=f(x,y)$ with the initial condition $y(x_0)=y_0$, clearly writing all four slope expressions. [6]

(b) Using the fourth-order Runge-Kutta method, compute $y(0.2)$ for the initial value problem $\dfrac{dy}{dx}=x+y,\; y(0)=1$, taking a step size $h=0.1$. [6]

Solve the following system of linear equations using Gauss elimination with partial pivoting, and explain why partial pivoting is necessary:

State the condition for convergence of the Gauss-Seidel iterative method. Apply the method to solve the following system, performing three iterations starting from $(0,0,0)$:

Fit a straight line of the form $y = a + bx$ to the following data using the method of least squares, and estimate $y$ when $x = 6$.

The following table gives the value of a function $f(x)$. Using Newton's forward difference formula, find the first and second derivatives $f'(x)$ and $f''(x)$ at $x = 1.0$.

(a) Distinguish between round-off error and truncation error with a suitable example of each. [3]

(b) The number $\pi = 3.14159265$ is rounded to $3.1416$. Compute the absolute error, the relative error, and the percentage error. [3]

(a) Write down the iterative formula of the Secant method and explain how it differs from the Newton-Raphson method. [3]

(b) Perform two iterations of the Secant method to find a root of $f(x) = \cos x - x e^{x} = 0$ taking initial approximations $x_0 = 0$ and $x_1 = 1$. [3]

Using Newton's forward difference interpolation formula, estimate the population in the year 1925 from the following census data, and state the assumption on which the formula is based.

Using the Modified Euler (Heun's) method, compute $y(0.1)$ and $y(0.2)$ for the initial value problem $\dfrac{dy}{dx} = x^2 + y,\; y(0) = 1$ with step size $h = 0.1$, iterating the corrector until two successive values agree to three decimal places.