(a) (i) Find the LU decomposition of the matrix 1 1 2 A -2 -3 -4...

 

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(a) (i) Find the LU decomposition of the matrix 1 1 2 A -2 -3 -4 1 4 -1 (ii) Use this decomposition to find the solution of the equation Ax = b where b [3, -8, 1]". (b) Consider the following system of equations: 2x1 - 4x, + x3 = 1 X + 2r, - 5x,3 1 4x, - x2 - 2r, 1 (i) Reorder the equations to obtain a system of equations for which the Gauss-Seidel method is guaranteed to converge. Explain why convergence is guaranteed. (ii) Perform two iterations of the Gauss-Seidel method with the starting value of x) = %3D [0, 0, 0]', to find an approximation to the solution of the equations. (a) (i) Find the LU decomposition of the matrix 1 1 2 A -2 -3 -4 1 4 -1 (ii) Use this decomposition to find the solution of the equation Ax = b where b [3, -8, 1]". (b) Consider the following system of equations: 2x1 - 4x, + x3 = 1 X + 2r, - 5x,3 1 4x, - x2 - 2r, 1 (i) Reorder the equations to obtain a system of equations for which the Gauss-Seidel method is guaranteed to converge. Explain why convergence is guaranteed. (ii) Perform two iterations of the Gauss-Seidel method with the starting value of x) = %3D [0, 0, 0]', to find an approximation to the solution of the equations.