This course contains the concepts of numerical method techniques for solving linear and nonlinear equations, interpolation and regression, differentiation and integration, and partial differential equations. The main objective of the course is to provide the knowledge of numerical method techniques for mathematical modeling.
Tribhuvan University
Institute of Science and Technology
2075
Bachelor Level / third-semester / Science
Computer Science and Information Technology( CSC212 )
Numerical Method
Full Marks: 60 + 20 + 20
Pass Marks: 24 + 8 + 8
Time: 3 Hours
Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.
Group A
Attempt any Two questions: (10 x 2 = 20)
What is non-linear equation? Derive the required expression to calculate the root of non-linear equation using secant method. Using this expression find a root of following equation.
\(x^{2} + cos(x) – e^{-x} – 2 = 0\)
What is matrix factorization? Factorize the given matrix A into LU using Doolittle algorithm and also solve Ax = b for given b using L and U matrices.
A = \(\begin{bmatrix}2 & 4 & -4 & 0\\ 1 & 5 & -5 & -3\\ 2 & 3 & 1 & 3\\ 1 & 4 & -2 & 2\end{bmatrix}\) and b = \(\begin{bmatrix}12\\ 18\\ 8\\ 8\end{bmatrix}\)
What is initial value problem and boundary value problem? Write an algorithm and program to solve the boundary value problem using shooting method.
Group B
Attempt any Eight questions:(5 x 8 = 40)
Calculate a real negative root of following equation using Newton’s method for polynomial.
\(x^4 + 2x^3 + 3x^2 + 4x = 5\)
What is least squares approximation of fitting a function? How does it differ with polynomial interpolation? Explain with suitable example.
Find the lowest degree polynomial, which passes through the following points:
| X | -2 | -1 | 1 | 2 | 3 | 4 |
| F(x) | -19 | 0 | 2 | -3 | -4 | 5 |
Using this polynomial estimate f(x) at x = 0
The fit function of type y = a + bx for the following points using the least square method.
| X | -1 | 1.2 | 2 | 2.7 | 3.6 | 4 |
| F(x) | 1 | 20 | 27 | 33 | 41 | 45 |
Calculate the integral value of the function given below from x = 1.8 to x = 3.4 using Simpson’s 1/3 rule.
| X | 1.8 | 2.0 | 2.2 | 2.4 | 2.6 | 2.8 | 3.0 | 3.4 |
| F(x) | 0.003 | 0.778 | 1.632 | 2.566 | 3.579 | 4.672 | 7.097 | 8.429 |
Evaluate the following integration using Romberg integration.
\(\int_{0}^{1} \frac{sinx}{x} dx\)
Solve the following set of equations using Gauss Seidel method.
x + 2y + 3z = 4
6x – 4y + 5z = 10
5x + 2y + 2z = 25
From the following differential equation estimate y(1) using RK 4th order method.
\(\frac{dy}{dx} + 2x^2y = 4 \enspace with \enspace y(0) = 1\)
[Take h = 0.5]
Solve the Poison’s equation over the square domain 0 ≤ x ≤ 1.5, 0 ≤ y ≤ 1.5 with f = 0 on the boundary and h = 0.5.
