BE Computer Engineering (IOE, TU) Numerical Methods (IOE, SH 553) Question Paper 2078

(a) Derive the iterative formula for the Newton-Raphson method for finding a real root of a nonlinear equation $f(x)=0$, and give a clear geometrical interpretation of the method. State two situations in which the method fails to converge. [7]

(b) A real root of the equation $x^3 - 5x + 1 = 0$ lies between 0 and 1. Using the Newton-Raphson method, compute the root correct to four decimal places. [5]

(a) Explain the construction of a natural cubic spline that interpolates a set of $(n+1)$ data points. Clearly state the continuity conditions imposed at the interior knots and the boundary conditions for the natural spline. [6]

(b) Using Newton's divided difference interpolation, estimate the value of $f(3)$ from the following data:

(a) Describe the Gauss elimination method with partial pivoting for solving a system of linear equations. Explain why partial pivoting is necessary and how it improves numerical stability. [6]

(b) Solve the following system of equations using the Gauss-Seidel iterative method, performing three iterations and starting with the initial guess $(0,0,0)$:

[6]

(a) Derive the standard five-point finite difference formula for the Laplace equation $\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} = 0$ over a rectangular region. [5]

(b) A square plate is divided into a mesh as shown, with the boundary values of $u$ specified. Set up the system of linear equations for the interior nodal values $u_1, u_2, u_3, u_4$ using the five-point formula, and solve for the interior values by the Gauss-Seidel (Liebmann) iteration, taking the boundary on all four edges as $u = 100$ on the top edge and $u = 0$ on the remaining three edges. [7]

Find a real root of the equation $x e^{x} = 2$ correct to three decimal places using the Bisection method, taking the initial interval as $[0, 1]$. Tabulate your iterations clearly.

Evaluate the integral $\displaystyle\int_{0}^{1} \frac{dx}{1+x^2}$ using Simpson's 1/3 rule with $h = 0.25$, and compare your result with the exact value. Briefly explain how the accuracy could be improved using Romberg integration.

Fit a straight line $y = a + bx$ to the following data using the method of least squares:

Hence estimate the value of $y$ when $x = 6$.

Derive the central difference formulae for the first and second derivatives of a function from Taylor's series expansion, and state their orders of accuracy. Using the data below, compute $f'(2.0)$ and $f''(2.0)$:

Using the fourth-order Runge-Kutta method, solve the initial value problem $\dfrac{dy}{dx} = x + y$, with $y(0) = 1$, and find $y(0.2)$ taking a step size $h = 0.1$.

Using Lagrange's interpolation formula, find the value of $f(2)$ from the following data:

Explain the shooting method for solving a two-point boundary value problem of a second-order ordinary differential equation. How does it convert a boundary value problem into an initial value problem?

Explain the LU (Doolittle) decomposition method for solving a system of linear equations $[A]\{x\} = \{b\}$. State its main advantage over Gauss elimination when the system must be solved for several different right-hand-side vectors $\{b\}$.