371 Math Numerical Methods
Homework 4. Due: Mar 05 Computational problems are marked with . Please include your code and output but limit the print-out within four pages. 1. Let L be an n-by-n lower triangular matrix. Its inverse L1 is also lower triangular. Assume both matrices are dense with O(n2 ) nonzero entries. Find the operation counts for the following two algorithms that are theoretically equivalent (a) Use forward substitution to nd y in Ly = b; (b) Use matrix multiplication to nd y = L1 b. 2. Generate an (2m + 1)-by-(2m + 1), lower triangular matrix L using Matlab code L=diag(1:(2*m+1))+6*diag(ones(2*m,1),-1). Its inverse can be found using code inv(L). (a) Let m = 5 and run the above codes. Descibe the main dierence you see in the pattern of nonzero entries in L and L1 . (b) What are the operation counts for two (theoretically) equivalent approaches of nding y: Ly = b and y = L1 b? Please use notation O(mk ) with k specied and indicate which approach is computationally faster for general ms. (c) Does your conclusion contradict problem 1? Explain. 3. Consider a 10-by-10 matrix A. (Note the diagonal entries are all 5s and there are only 4 nonzero o-diagonal entries.) 5 0 0 0 0 0 0 0 3 0 0 5 0 0 0 0 0 0 0 -1 0 0 5 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 0 5 0 0 0 0 -2 0 0 0 0 0 5 0 0 0 2 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 5
(a) Do a LU factorization by hand on A and then use Matlab code [L U]=lu(A) to verify your result; (b) Solve Ax = b by hand using the above LU decomposition for b = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)T . Do NOT use the Gaussian elimination. 4. In class, we discussed a 1D boundary value problem that models the equilibrium temperature of a rod heated by some external source. Now, consider a similar physical problem with thermal advection y + 2y + 3y = 2x 1, y(0) = 0, y(1) = 0;
Discretize this problem with mesh size h = 1/n to arrive at a linear system Ax = b. Please discribe A and b with enough details so itll help you coding in later homework. No programming is required in this problem. 5. Write matlab code to sol...