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:**

1

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.

2

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 = \(\begin{bmatrix}4\\ 12 \\28\end{bmatrix}\)

3

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:**

4

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.

x^{2} – e^{-x} – x = 1

5

Calculate the real root of the given equation using fixed point iteration correct up to 3 significant figures.

2x^{3} – 2x = 5

6

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 |

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 |

8

What are the problems with polynomial interpolation for a large number of data set? How such problems are addressed? Explain with an example.

9

Evaluate the following integration using Romberg integration.

\(\int_{0}^{1} \frac{sin^2 x}{x} dx\)

10

Solve the following set of linear equations using the Gauss-Jordan method.

x_{2} + 2x_{3} + 3x_{4} = 9

7x_{1} + 6x_{2} + 5x_{3} + 4x_{4} = 33

8x_{1} + 9x_{2} + x_{4} = 27

2x_{1} + 5x_{2} + 4x_{3} + 3x_{4} = 23

11

Solve the following differential equation for 1 ≤ x ≤ 2, taking h = 0.25 using Heun’s method.

y^{‘}(x) + x^{2}y = 3x, with y(1) = 1

12

Consider a metallic plate of size 90cm by 90cm. The two adjacent sides of the plate are maintained at a temperature of 100^{0}C and the remaining two adjacent sides are held at 200^{0}C. Calculate the steady-state temperature at interior points assuming a grid size of 30 cm by 30 cm.

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