Question

Solve each item supporting your answer with clear explanation.

1. Consider the following augmented matrix of a system of linear equations.

( 7 −3 4 6

−3 2 6 2

2 5 3 −5 )

a. Solve the system with the Jacobi method. First rearrange to make it diagonally dominant if possible. Use [0,0,0] as the starting vector. Find the condition number of the matrix of coefficients κ_∞(A), and compute how many iterations are required to get the solution accurate to five significant digits?

b. Repeat part a) using the Gauss-Seidel method. Are fewer iterations required?

c. Is convergence faster in parts a) and b) if the starting vector is [-0.26602, -0.26602,-0.26602]?

Answer 1

a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is larger than or equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row.

To make it Diagonally dominant we get.

Condition number is calculted by the formula

for our rearranged Matrix .

The Gauss-Jacobi Iterative formula for above Equations is

This method requires 32 iterations to get accuracy level of 10^-5

The Iterative Algorithm for Gauss Seidel method is

This method requires 14 iterations to get accuracy level of 10^-5

The Convergence of Gauss Sedel method is faster than Jacobi Method.

c)

With initial values as [-0.26602, -0.26602,-0.26602]

Gauss Jacobi Method takes same 32 itarations to reach the rquired accuracy

Gauss Seidel Method takes same 14 itarations to reach the rquired accuracy

No change in speed of convergence or number of iterations required.