## Third Semester | Second Year | Tribhuvan University

Old Question Collection | Question Bank

Numerical Method, Year: 2068

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 types of errors in numerical calculations. Derive the formula for secant method and illustrate the method by figure.
**(4+4)** - Define the linear least squares approximations. Give the data set (x
_{i}, y_{i}) as (20.5, 765), (32.7, 826), (51.0, 873), (73.2, 942), (95.7, 1032) find the linear least square to fit given data.**(2+6)** - Evaluate I = dx using trapezoidal rule with n = 10. Also evaluate the same integral using Grossion 3 point formula and compare the result.
**(4+4)** - Solve the following system of linear equations using Gauss-elimination method (use partial pivoting if necessary);

2x_{2}+ x_{4}= 0

2x_{1}+ 2x_{2}+ 3x_{3}+ 2x_{4}= -2

4x_{1}– 3x_{2}+ x_{4}= -7

6x_{1}+ x_{2}– 6x_{3}-5x_{4}= 6**(8)**

**OR**

What do you mean by eigen-value eigen-vector problems? Find the largest eigen value correct to two significant digits and corresponding eigen vectors of the following matrix using power method.

A =**(2+6)** - Write an algorithm and program to solve system of linear equations using Gauss-Jordan method.
**(4+8)** - Apply Runge Kutta method of second order and fourth order to find an approximate value of y when x = 0.2 given that

= x + y and y(0)=1.**(8)** - How can you solve Laplace’s equation? Explain. The steady-state two dimensional heat flow in a metal plate is defined by =0.

A steel plate of size 30 x 30cm is given. Two adjacent sides are placed at 100ºC and other side at held at 0ºC. Find the temperature at interior points, assuming the grid size of 10 x 10cm.**(3+5)**

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