## Third Semester | Second Year | Tribhuvan University

Old Question Collection | Question Bank

Numerical Method, Year: 2073

Computer Science and Information Technology (CSc 204)

Full Marks: 60 | Pass Marks: 24 | Time: 3 hours

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Candidates are required to give their answer in their own words as far as practicable.

**The figures in the margin indicate full marks.**

**Attempt all Questions:**

- Explain the idea of the secant method to estimate the root of any equation. Using the secant method, estimate the root of the equation

x^{2}– 4x – 10 = 0

with the initial estimates of x_{1}=4 and x_{2}=2. Do these points bracket a root? (3+4+1) - Given the data

x 1.2 1.3 1.4 1.5 f(x) 1.063 1.091 1.119 1.145 Calculate f(1.35) using Newton’s interpolating polynomial of order 1 through 3. Choose base points to attain good accuracy. Comment on the accuracy of results on the order of polynomial. (5+3)

- How do you find the derivative if the function values are given in a tabulated form? The distance travelled by a vehicle at the intervals of 2 minutes are given as follows. Evaluate the velocity and the acceleration of the

Time(sec) 0 2 4 6 8 10 12 14 16 Distance(km) 0 0.25 1 2.2 4 6.5 8.5 11 13 Vehicle at time T = 5, 10, 13 (3+5)

- What do you mean by ill-conditioned systems? Solve the following system using Dolittle L.U decomposition method.

3x_{1}+ 2x_{2}+ x_{3}= 10

2x_{1}+ 3x_{2}+ 2x_{3}= 14

x_{1}+ 2x_{2}+ 3x_{3}= 14 (2+6) - Obtain y(1.5) to the following differential equation using Runge-Kutta 4
^{th}order method.

+ 2x^{2}y = 1, with y(1) = 0 taking h = 0.25 (8) - Write the finite difference formula for solving Poisson’s equation. Hence solve the Poisson equation.

∇^{2}f = 3x^{2}y

Over the domain 0≤x≤1.5 and 0≤y≤3 with f=0 on the boundary and h=0.5 (1+7) - Write an algorithm and a C program for the secant method to find the roots of non-linear equation. (4+8)

**OR**

Write an algorithm and a C program for the Simpson’s 1/3 rule to integrate a given function. (4+8)

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