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)**

1

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.

\(x^{2} + cos(x) – e^{-x} – 2 = 0\)

2

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 = \(\begin{bmatrix}12\\ 18\\ 8\\ 8\end{bmatrix}\)

3

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)**

4

Calculate a real negative root of following equation using Newton’s method for polynomial.

\(x^4 + 2x^3 + 3x^2 + 4x = 5\)

5

What is least squares approximation of fitting a function? How does it differ with polynomial interpolation? Explain with suitable example.

6

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

7

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 |

8

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 |

9

Evaluate the following integration using Romberg integration.

\(\int_{0}^{1} \frac{sinx}{x} dx\)

10

Solve the following set of equations using Gauss Seidel method.

x + 2y + 3z = 4

6x – 4y + 5z = 10

5x + 2y + 2z = 25

11

From the following differential equation estimate y(1) using RK 4^{th} order method.

\(\frac{dy}{dx} + 2x^2y = 4 \enspace with \enspace y(0) = 1\)

[Take h = 0.5]

12

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.

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