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Unformatted text preview: CS205 Homework #6 Problem 1
1. Let A be a symmetric and positive deﬁnite n × n matrix. If x, y ∈ Rn prove that the operation x, y A = xT Ay = x · Ay is an inner product on Rn . That is, show that the following properties are satisﬁed (a) u + v, z (b) αu, v (c) u, v (d) u, u
A A A A = u, z A A + v, z A = α u, v
A = v, u ≥ 0 and equality holds if and only if u = 0 2. Which of those properties, if any, fail to hold when A is not positive deﬁnite? Which fail to hold if it is not symmetric? Problem 2
1. Let x1 , x2 , . . . , xk be an A-orthogonal set of vectors, that is xT Axj = 0 for i = j. i Show that if A is symmetric and positive deﬁnite, then x1 , x2 , . . . , xk are linearly independent. Does this hold when A is symmetric but not positive deﬁnite? 2. Let x1 , x2 , . . . , xn be n linearly independent vectors of Rn and A a n × n symmetric positive deﬁnite matrix. Show that we can use the Gram-Schmidt algorithm to create a full A-orthogonal set of n vectors. That is, subtracting from xi its A-overlap with x1 , x2 , . . . , xi−1 will never create a zero vector. Problem 3
Let A be a n × n symmetric positive deﬁnite matrix. Consider the steepest descent method for the minimization of the function 1 f (x) = xT Ax − bT x + c 2 1. Let xmin be the value that minimizes f (x). Show that 1 f (xmin ) = c − bT A−1 b 2 2. If xk is the k-th iterate, show that 1 f (xk ) − f (xmin ) = rT A−1 rk 2 k 1 3. Show that rk+1 = 4. Show that I− Ark rT k rT Ark k rk [f (xk+1 ) − f (xmin )] = [f (xk ) − f (xmin )] 1 − 5. Show that (rT rk )2 k (rT Ark )(rT A−1 rk ) k k σmin σmax [f (xk+1 ) − f (xmin )] ≤ [f (xk ) − f (xmin )] 1 − where σmin , σmax are the minimum and maximum singular values of A, respectively. 6. What does the result of (5) imply for the convergence speed of steepest descent? [Note: Even if you fail to prove one of (1)-(6) you may still use it to answer a subsequent question] 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