ANSWER ALL THE PROBLEMS

Consider a birth and death process X(t), t 2 0, such as the branching process, that has state space {0, 1, 2, ...} and birth and death rates of the form Ax = x1 and Hx = XH, x 2 0, where 1 and u are nonnegative constants. Set my(t) = E.(X(t)) = > yP x,(t). )=0 (a) Write the forward equation for the process. (b) Use the forward equation to show that my(t) = (2 - u)m.(t). (c) Conclude that my(t) = xe(2-4)tConsider the linear system of equations 5x1 -$1 + - T2 + 5x3 (a) Write down the components of the Jacobi iteration equation. Do not do any iterations, just write down the iteration mechanism. (b) With an initial guess x) = [1, 0, 0] apply, by hand, the Jacobi iteration until the approximate relative error falls below 0.7%.Q13 (1 mark). For liner system , the following expression is WRONG. (a) If is strictly diagonally dominant, then Jacobi method converged. (b) If is strictly diagonally dominant, then Gauss- Seidel method converged. (c) If Jacobi method converged, then Gauss-Seidel method converged. (d) If Jacobi method converged, then Gauss-Seidel method may not converged.Rewrite the system to be diagonally dominant: Jacobi iterations after rewriting: n 0 0 1 3 4Exercise Write down the iterative schemes for the Jacobi, Gauss-Seidel and SOR methods. Explain how SOR is obtained from the Gauss-Seidel method. Explore convergence property of the Jacobi and SOR method for the system Anr = b 0 . .. An = 0 b = [1.. . 1] n=30 2 Use z() = [000. . . 0]", Wopt - 1+ sin Iterate until |x - x) | 5 0.00005 The exact solution r can be found as a = A\\b Implement the SOR method (.m file should be submitted). Jacobi and Gauss-Seidel can be found on the webpage code Create a table k Error Jacobi Error SOR Ratio Jacobi (c) Ratio SOR (c) E ESOR O ... N 10 K where Ejacobi - error estimate for Jacobi, ESOR - error estimate for SOR, K number of iterations which SOR method needed to reach the prescribed accuracy