Numerical Method TU Board 2078 question paper

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.

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 = $\left[\begin{array}{ccc}1& 2& 3\\ 2& 8& 11\\ 3& 22& 36\end{array}\right]$ and b = $\left[\begin{array}{c}4\\ 12\\ 28\end{array}\right]$

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

$\frac{d^{2}y}{d{x}^2}+3\frac{dy}{dx}+5y=0$, width y(1) = 1 and y^'(1) = 2

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

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

2x^3 – 2x = 5

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

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

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

Evaluate the following integration using Romberg integration.
${\int }_0^1\frac{si{n}^{2}x}{x}dx$

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

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

y^'(x) + x^2y = 3x, with y(1) = 1

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