Question

Consider the following system of linear equations: 3x1+x2+5x3=−1 x1+x2+2x3=2 5x1+ax2+4x3=1 Here, a is a real number. Use x(0)=0 as the initial guess vector and apply the Jacobi method twice to get an approximate solution x(2) to this system. Let r be the residual vector for x(2) with respect to this system. Determine all the possible numerical values of a if ∥r∥2=10.

Answer 1

Hello,

To determine the possible numerical values of a if ∥r∥2 = 10, we need to apply the Jacobi method twice to obtain an approximate solution x(2) and then calculate the residual vector r. By varying the value of a, we can check which values satisfy ∥r∥2 = 10.

Let's start by applying the Jacobi method twice using the given initial guess vector x(0) = 0 and the system of linear equations:

3x1 + x2 + 5x3 = -1
x1 + x2 + 2x3 = 2
5x1 + ax2 + 4x3 = 1

After applying the Jacobi method twice, we obtain the approximate solution x(2). Then, by calculating the residual vector r, we can determine the possible numerical values of a that satisfy ∥r∥2 = 10.