Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Tribhuwan University

Institute of Science and Technology

Model

Bachelor Level / Second Year / Third Semester / Science

Bachelors in Information Technology (BIT203)

(Numerical Methods)

Full Marks: 60

Pass Marks: 24

Time: 3 Hours

Candidates are required to give their answers in their own words as for as practicable.

The figures in the margin indicate full marks.

Section A

Long Answers Questions

Attempt any TWO questions.
[2*10=20]
1.
How Secant methods differs from Newton Raphson method? Derive the formula for Secant Method. Solve the equation cosx+2sinx−x2=0cosx+2sinx-x^2=0 using Secant method. Assume error precision is 0.01. [10]
2.
How interpolation differs from regression? Write down algorithm and program for Lagrange interpolation. [10]
3.
Explain the working of Jacobi Iteration method? Solve the following system of equations using the method. Assume error precision is 0.01. Compare Jacobi Iteration method with Gauss-Seidel method. 5x−2y+3z=−15x-2y+3z=-1, −3x+9y+z=2-3x+9y+z=2, 2x−y−7z=−32x-y-7z=-3 [10]
Section B

Short Answers Questions

Attempt any Eight questions.
[8*5=40]
4.
Define the terms true error and relative error? Write down algorithm for Horner' method to evaluate polynomial and use the method to evaluate the polynomial 2x3−3x2+5x−22x^3-3x^2+5x-2 at x=3. [5]
5.
Construct Newton's backward difference table for the given data points and approximate the value of f(x) at x=45.
X1020304050f(x)0.1730.3420.50.6430.766\begin{array}{|c|c|c|c|c|c|}\hline X & 10 & 20 & 30 & 40 & 50 \\ \hline f(x) & 0.173 & 0.342 & 0.5 & 0.643 & 0.766 \\ \hline \end{array}
[5]
6.
Fit the quadratic curve through the following data points and estimate the value of f(x) at x=2.
x13456y27875\begin{array}{|c|c|c|c|c|c|}\hline x & 1 & 3 & 4 & 5 & 6 \\ \hline y & 2 & 7 & 8 & 7 & 5 \\ \hline \end{array}
[5]
7.
Derive formula for the Doolittle LU decomposition matrix factorization method. [5]
8.
How can we calculate derivatives of continuous functions? Write down algorithm and program for differentiating continuous function using two point forward difference formula. [5]
9.
Find following integral using composite trapezoidal rule using 2 segments (k=2) and 4 segments (k=4). ∫28(x+3)2dx\int_{2}^{8} (x+3)^2 dx [5]
10.
Approximate the solution of y'=2x+y, y(0)=1 using Euler's method with step size of 0.1. Approximate the value of y(0.4). [5]
11.
Solve the Poisson's equation ∇2f=xy\nabla^2 f = xy with f=2f=2 on boundary by assuming square domain 0≤x≤30 \leq x \leq 3, 0≤y≤30 \leq y \leq 3 and h=1. [5]
12.
How boundary value problems differs from initial value problems? Discuss shooting method for solving boundary value problem. [5]