BCA Semester 4 – Numerical Methods – Unit 4: Solution of Linear Algebraic Equations (10Hrs.)

What is meant by ill-conditioned system? Find the Cholesky decomposition of the following matrix:

$\quad \quad \quad \begin{bmatrix} 4 & 1 & 1 \\ 1 & 5 & 2 \\ 1 & 2 & 3 \end{bmatrix}$

a) Solve the following system of equations using Gauss-Seidel method:

$\quad\quad\quad 2x - 7y - 10z = -175$

$\quad\quad\quad x + y + 3z = 14$

$\quad\quad\quad x + 10y + 9z = 7$

b) Write an expansion of Taylor’s Theorem. Solve the equation

$\quad\quad\quad y′(x)=x2+y2, \quad y(0)=1$

for $x=0.5$ using Taylor method.

Find the dominant eigenvalue of

$A = \begin{bmatrix} 2 & 3 \\ 5 & 4 \end{bmatrix}$

by the power method. Apply the Gauss--Seidel method to solve the system of equations:

$\begin{aligned} 6x_1 - 2x_2 + x_3 &= 11, \\ -2x_1 + 7x_2 + 2x_3 &= 5, \\ x_1 + 2x_2 - 5x_3 &= -1, \end{aligned}$

with the initial vector of $(0, 0, 0).$

What is the form of resultant matrix using Gauss-Jordan method? Solve the following system of equations using Gauss-Jordan Method.  

$\quad \quad  x + 2y - 3z = 4$

$\quad \quad2x + 4y - 6z = 8$

$\quad \quad  x - 2y - 5z = 4$

Solve the following equations by using Gauss-Jordan method.

$\quad \quad2x+3y+4z=5$

$\quad \quad3x+4y+5z=6$

$\quad \quad4x+5y+6z=7$

Solve the given set of linear equations using Dolittle LU decomposition method:  

$\quad \quad3x 1 + 2x 2 + x 3 = 10$

$\quad \quad2x 1 + 3x 2 + 2x 3 = 14$

$\quad \quad3x 1 + 2x 2 + 3x 3 = 14$