## Third Semester | Second Year | Tribhuvan University

Old Question Collection | Question Bank

Numerical Method, Year: 2066

Computer Science and Information Technology (CSc 204)

Full Marks: 60 | Pass Marks: 24 | Time: 3 hours

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Candidates are required to give their answer in their own words as far as practicable.

**The figures in the margin indicate full marks.**

**Attempt all Questions:**

- Define the fixed-point iteration method. Given the function ?(x) = x
^{2}-2x-3 = 0, rearrange the function in such a way that the iteration method converses to its roots.**(2+3+3)** - What do you mean by interpolation problem? Define divided difference table and construct the table from the following data set.
**(2+2+4)**

Xi 3.2 2.7 1.0 4.8 5.6 Fi 22.0 17.8 14.2 38.3 51.7 **OR**

Find the least squares line that fits the following data.x 1 2 3 4 5 6 y 5.04 8.12 10.64 13.18 16.20 20.o4 What do mean by linear least squares approximation?

- Derive the composite formula for the trapezoidal rule with its geometrical figure. Evaluate using this rule with n = 5, upto 6 decimal places.
**(4+4)** - Solve the following system of algebraic linear equations using Jacobi or Gauss-Seidel iterative method. (8)

6x_{1}-2x_{2}+x_{a}=11

-2x_{1}+7x_{2}+2x_{a}=5

X_{1}+2x_{2}-5x_{3}=-1 - Write an algorithm and computer program to fit a curve y=ax
^{2}+bx+c for given sets of (x_{i}, y_{i}, g. o = 1, …., x) values by least square method.**(4+8)** - Derive a difference equation to represent an Poisson’s equation. Solve the Poison’s equation ∇
^{2}f=2x^{2}y^{2}over the square to main 0≤?≤3, 0≤?≤3 with f = 0 on the boundary and h = 1.**(3+5)** - Define ordinary differential equation of the first order. What do you mean by initial value problem? Find by Taylor’s series method, the values of y at x = 0.1 and x = 0.2 to fine places of decimal form .

=x^{2}y-1, y (0)=1**(2+6)**

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