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(b) Now assume that k ≪ n. Describe an eﬃcient method to solve the normal equations found in
part (a). Give an approximate ﬂop count for a general method that does not exploit structure,
and also for your eﬃcient method.
(c) A numerical instance. In this part you will try out your eﬃcient method. We’ll choose k = 100
and n = 4000, and δ = η = 1. First, randomly generate A and b with these dimensions. Form
the normal equations as in part (a), and solve them using a generic method. Next, write
(short) code implementing your eﬃcient 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 eﬃcient
method is much faster than the generic one.
75 Note: You’ll need to know some things about Matlab to be sure you get the speedup from the
eﬃcient method. Your method should involve solving linear equations with tridiagonal coeﬃcient
matrix. In this case, both the factorization and the back substitution can be carried out very
eﬃciently. The Matlab documentation says that banded matrices are recognized and exploited,
when solving equations, but we found this wasn’t always the case....
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.
 Fall '13
 F.Borrelli
 The Aeneid

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