BE Computer Engineering (Pokhara University) Numerical Methods (PU, MTH 252) Question Paper 2078

(a) Derive the iterative formula of the Newton-Raphson method for finding a real root of the equation $f(x)=0$, and state the geometric interpretation of the method. Discuss the conditions under which the method may fail to converge. (6)

(b) A real root of the equation $x^3 - 2x - 5 = 0$ lies between 2 and 3. Using the Newton-Raphson method, perform three iterations starting from $x_0 = 2$ and obtain the root correct to four decimal places. Compare the rate of convergence with that of the bisection method. (6)

(a) Derive Newton's forward difference interpolation formula and clearly state the situation in which it should be preferred over Newton's backward difference formula. (6)

(b) The following table gives the population (in thousands) of a town in different census years:

Using an appropriate interpolation formula, estimate the population of the town in the year 1976. (8)

(a) Derive the fourth-order Runge-Kutta method for solving the first-order ordinary differential equation $\frac{dy}{dx}=f(x,y)$ with the initial condition $y(x_0)=y_0$. (6)

(b) Using the fourth-order Runge-Kutta method, find $y(0.2)$ for the initial value problem $\frac{dy}{dx}=x+y^2,\; y(0)=1$, taking step size $h=0.1$. Carry out two steps and retain four decimal places. (8)

(a) Explain the Gauss-Seidel iterative method for solving a system of linear equations. State the condition of diagonal dominance and explain why it guarantees convergence. (5)

(b) Solve the following system of equations using the Gauss-Seidel iterative method, performing iterations until the values are correct to three decimal places:

(7)

Using the secant method, find a real root of the equation $\cos x - x e^{x} = 0$ correct to three decimal places. Take initial approximations $x_0 = 0$ and $x_1 = 1$, and tabulate your iterations.

Fit a curve of the form $y = a e^{bx}$ to the following data by the method of least squares and hence estimate $y$ at $x = 6$:

Evaluate the integral $\int_{0}^{1}\frac{dx}{1+x^{2}}$ by dividing the interval into six equal subintervals, using (a) the Trapezoidal rule and (b) Simpson's $1/3$ rule. Compare both results with the exact value and comment on the accuracy.

From the following table of values, find the first and second derivatives of the function $y=f(x)$ at $x = 1.1$ using suitable numerical differentiation formulae:

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

(b) If $u = \frac{5xy^{2}}{z^{3}}$ and errors in $x, y, z$ are each $0.001$ at $x = y = z = 1$, compute the maximum absolute error and the maximum relative error in $u$. (3)

Using the Modified Euler method, solve the differential equation $\frac{dy}{dx} = x + y$ with $y(0) = 1$ to find $y(0.1)$ and $y(0.2)$, taking $h = 0.1$. Perform the corrector iteration until two successive values agree to four decimal places.

Using the LU (Doolittle) decomposition method, factorize the coefficient matrix and hence solve the system of equations: