linear algebraic euations

12 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 = x}. and p, = xu, x 2 0, where A and ,u are nonnegative constants. Set (I) mix) = Ex(X(t)) = 2 ny,(t). r=0 (a) Write the forward equation for the process. (b) Use the forward equation to show that m;(t) = (A ,u)m,,(t). (c) Conclude that m1\") = atria'0'. 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 system of linear algebraic equations below. Start with, Po = (0,0), to implement Jacobi and Gauss-Seidel iteration to find Pr for k = 1, 2, 3. Will Jacobi or Gauss-Seidel iteration converge to the solution? Explain. Linear system: 2x + 3y = 1, 7x - 2y =1.Exercise 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 2 -1 2 An = b = 1. . .1 n =30 0 0 2 Use x() = [000 . . . 0]' , Wopt = 1+ sin - Iterate until |r - x*)||