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Unformatted text preview: CS205 Homework #7 Solutions Problem 1 We have seen the application of the conjugate gradient algorithm on the solution of sym- metric, positive definite systems. Now assume that in the system Ax = b , the n × n matrix A is symmetric positive semi-definite with a nullspace of dimension p < n . This problem illustrates that one can use a modified version of conjugate gradients to solve this system as well. 1. Prove that we can write A as A = M ˜ AM T where M is an n × ( n- p ) matrix with orthonormal columns that form a basis for the column space of A , while ˜ A is an ( n- p ) × ( n- p ) symmetric positive definite matrix (no nullspace) [Hint: Use the diagonal form of A = QΛQ T ] 2. Let the n × n matrix P be defined as P = MM T . Explain (no formal proof required) why this is a projection matrix and onto what space it projects. How can we compute P without knowledge of the eigenvalues-eigenvectors of A ? 3. Show that, in order to have a solution to Ax = b , we must be able to write b = M ˜ b for an appropriate vector ˜ b ∈ R n- p 4. Let ˜x be the solution to the system ˜ A˜x = ˜ b and explain why ˜x is unique. Show that any solution to the original system Ax = b can be written as x = M˜x + x where x is in the nullspace of A . 5. Consider the conjugate gradients algorithm for solving ˜ A˜x = ˜ b ˜x = initial guess ˜ s = ˜ r = ˜ b- ˜ A˜x for k = 0 , 1 , . . . , 2 ˜ α k = ˜ r T k ˜ r k ˜ s T k ˜ A˜ s k ˜x k +1 = ˜x k + ˜ α k ˜ s k ˜ r k +1 = ˜ r k- ˜ α k ˜ A˜ s k ˜ s k +1 = ˜ r k +1 + ˜ r T k +1 ˜ r k +1 ˜ r T k ˜ r k ˜ s k end 1 Show that we can compute a solution to the original system Ax = b by using the following modification of the algorithm x = initial guess s = r = P ( b- Ax ) for k = 0 , 1 , . . . , 2 α k = r T k r k s T k As k x k +1 = x k + α k s k r k +1 = r k- α k PAs k s k +1 = r k +1 + r T k +1 r k +1 r T k r k s k end [Hint: Show that x k = M˜x k , r k = M˜ r k , s k = M˜ s k , ˜ α k = α k ] Solution 1. Since A is symmetric and positive definite it can be written as A = QΛQ T = q 1 q 2 ··· q n λ 1 λ 2 ....
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This homework help was uploaded on 01/29/2008 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.
- Fall '07