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.**

1

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.

x^{3} + 2x – cos(x) = 4

2

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 |

3

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.**

4

Calculate a real root of the following function using bisection method correct upto 3 significant figures.

x^{2} – e^{-x} = 3

5

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.

6

Write down the program for solving ordinary differential equation using Heun’s method.

7

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 |

8

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 |

9

What is Gaussian integration formula? Evaluate the following integration using Gaussian integration three ordinate formula

_{0}∫^{1} Sinx/x dx

10

Solve the following set of equations using Gauss Siedal method.

x + 2y + 3z = 4

6x + 4y + 5z = 16

5x + 2y + 3z = 12

11

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

12

Solve the Poisson’s equation ∇^{2}f=3x^{2}y over the square domain 0 ≤ x ≤ 3, 0 ≤ y ≤ 3 with f=0 on the boundary and h=1.

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