matrix numerical analysis

(a) For R E Rnxn upper triangular and b E R" the backward substitution algorithm for solving Rx = b is C;= bi for i = n, n - 1, . .., 2, 1. ri j=i+1 State the algorithm or equations that define forward substitution to solve Lx = b when L E Rnxn is lower triangular. Determine how many arithmetic operations are required to perform one of these algorithms. (The answer is the same for both algorithms, so it does not matter which you do.) (b) State the Cholesky algorithm for finding an upper triangular matrix R E Rnx such that A = RTR when A E Rnx is a positive definite matrix. How many operations does the algorithm require? (c) Given A E Rmxn briefly state the steps required to solve A" Ax = Ab using Cholesky factor- ization and forward and backward substitution and the operations required for each. Which step is the most computationally expensive? (d) Let 16 4 8 4 32 4 10 8 4 26 AT A = Ab = 8 8 12 10 38 4 4 10 12 30 Find an upper triangular matrix R such that A"A = RTR. Hence find x and y such that R*y = A"b and AT Ax = ATb