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Unformatted text preview: 6.079/6.975, Fall 2009-10 S. Boyd & P. Parrilo Homework 9 1. Ecient numerical method for a regularized least-squares problem. We consider a reg- ularized least squares problem with smoothing, k n 1 n minimize ( a T i x b i ) 2 + ( x i x i +1 ) 2 + x 2 i , i =1 i =1 i =1 where x R n is the variable, and , > are parameters. (a) Express the optimality conditions for this problem as a set of linear equations involving x . (These are called the normal equations.) (b) Now assume that k n . Describe an ecient method to solve the normal equations found in (1a). Give an approximate op count for a general method that does not exploit structure, and also for your ecient method. (c) A numerical instance. In this part you will try out your ecient method. Well choose k = 100 and n = 2000, and = = 1. First, randomly generate A and b with these dimensions. Form the normal equations as in (1a), and solve them using a generic method. Next, write (short) code implementing your ecient method, and run it on your problem instance. Verify that the solutions found by the two methods are nearly the same, and also that your ecient method is much faster than the generic one. Note: Youll need to know some things about Matlab to be sure you get the speedup from the ecient method. Your method should involve solving linear equations with tridiagonal coecient matrix. In this case, both the factorization and the back sub- stitution can be carried out very eciently. The Matlab documentation says that banded matrices are recognized and exploited, when solving equations, but we found this wasnt always the case. To be sure Matlab...
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- Fall '06