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
2079
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 secant method can approximate the root of a non-linear equation? Explain with necessary derivation. Estimate a real root of following equation using secant method. Assume error precisionn of 0.01.
x3 + 2x – cos(x) = 4
How spline interpolation differs with the Langrage’s interpolation? Estimate the value of f(0) and f(4) using cubic spline interpolation from the following data.
| x | -1 | 1 | 2 | 3 |
| f(x) | -10 | -2 | 14 | 86 |
What is pivoting? Why is it necessary? Write an algorithm and program to solve the set of n linear equations using Gaussian elimination method.
Section B
Attempt any eight questions.
Calculate a real root of the following function using bisection method correct upto 3 significant figures.
x2 – e-x = 3
What is fixed point iteration method? How can it converge to the root of a non-linear equation? Also explain the diverging cases with suitable examples.
Write down the program for solving ordinary differential equation using Heun’s method.
Fit the quadratic function for the data given below using least square method.
| x | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 |
| f(x) | 2.7 | 4 | 5.8 | 8.3 | 11.2 | 15 | 19 |
Estimate the integral value of following function from x =1.2 to 2.4 using Simpson’s 1/3 rule.
| x | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 | 2.4 | 2.6 |
| f(x) | 1.53 | 2.25 | 3.18 | 4.32 | 5.67 | 7.23 | 8.98 | 10.94 | 13.08 |
What is Gaussian integration formula? Evaluate the following integration using Gaussian integration three ordinate formula
0∫1 Sinx/x dx
Solve the following set of equations using Gauss Siedal method.
x + 2y + 3z = 4
6x + 4y + 5z = 16
5x + 2y + 3z = 12
Solve the following differential equation for 0 ≤ x ≤ 1 taking h=0.5 using Runge Kutta 4th order method.
y'(x) + y = 3x with y(0)=2
Solve the Poisson’s equation ∇2f=3x2y over the square domain 0 ≤ x ≤ 3, 0 ≤ y ≤ 3 with f=0 on the boundary and h=1.