Numerical Method
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
2078
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.
Section A
Attempt any TWO questions:
How can Horner’s rule be used to evaluate the f(x) and f(x) of a polynomial at a given point? Explain. Write an algorithm and program to calculate a real root of a polynomial using Horner’s rule.
Write matrix factorization? How can be used to solve a system of linear equations? Factorize the given matrix A and solve the system of equations Ax = b for given b using L and U matrices.
A = \(\begin{bmatrix}1 & 2 & 3\\ 2 & 8 & 11\\ 3 & 22 & 36\end{bmatrix}\) and b = \(\begin{bmatrix}4\\ 12 \\28\end{bmatrix}\)
What is a higher-order differential equation? How can you solve the higher-order differential equation? Explain. Solve the following differential equation for 1 ≤ x ≥ 2, taking h = 0.25
\(\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 5y = 0\), width y(1) = 1 and y‘(1) = 2
Section B
Attempt any EIGHT questions:
How the half-interval method can be estimate a root of a non-linear equation? Find a real root of the following equation using the half-interval method to correct up to two decimal places.
x2 – e-x – x = 1
Calculate the real root of the given equation using fixed point iteration correct up to 3 significant figures.
2x3 – 2x = 5
What is Newton’s interpolation? Obtain the divided difference table from the following data set and estimate the f(x) at x = 2 and x = 5.
| x | 3.2 | 2.7 | 1.0 | 4.8 | 5.6 |
| f(x) | 22.0 | 17.8 | 14.2 | 38.3 | 51.7 |
What is linear regression? Fit the linear function to the following data
| x | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 | 2.4 |
| f(x) | 2.0 | 2.6 | 3.9 | 6.0 | 9.3 | 15 | 20.6 | 30.4 |
What are the problems with polynomial interpolation for a large number of data set? How such problems are addressed? Explain with an example.
Evaluate the following integration using Romberg integration.
\(\int_{0}^{1} \frac{sin^2 x}{x} dx\)
Solve the following set of linear equations using the Gauss-Jordan method.
x2 + 2x3 + 3x4 = 9
7x1 + 6x2 + 5x3 + 4x4 = 33
8x1 + 9x2 + x4 = 27
2x1 + 5x2 + 4x3 + 3x4 = 23
Solve the following differential equation for 1 ≤ x ≤ 2, taking h = 0.25 using Heun’s method.
y‘(x) + x2y = 3x, with y(1) = 1
Consider a metallic plate of size 90cm by 90cm. The two adjacent sides of the plate are maintained at a temperature of 1000C and the remaining two adjacent sides are held at 2000C. Calculate the steady-state temperature at interior points assuming a grid size of 30 cm by 30 cm.