Q$1)$a$)$Explain what LU method is by giving a particular example

and design a program of the LU method for any $3$x$3$ linear equation

system.

b$)$Run the program for the same example and add your output.

c$)$Note that the system has unique solution. Also, state that what

is the condition for the system to have unique solution.

Q$2)$a$)$For a particular $4$x$4$ linear equations system explain Jacobi

and Gauss Siedel Methods and design a program of the Gauss Siedel

method for any $4$x$4$ equations system.

b$)$Run the program for the same example and add your output.

c$)$Write the necessary theorem and proof for convergence of

successive methods.