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notes6-5 - solve to produce a least-squares solution of A x...

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SECTION 6.5 LEAST-SQUARES PROBLEMS, OR BEST APPROXIMATIONS THE BASIC PROBLEM. We want to solve A x = b but the system is inconsistent. So we want to find x so that A x is as close as possible to b . Where is A x ? No matter what x is, A x is in Col A . So, we already know what A x must be – it must be the projection ˆ b of b onto Col A . Suppose ˆ x satisfies A ˆ x = ˆ b . Then by our projection stuff we know that b - ˆ b = b - A ˆ x is orthogonal to each column of A . Express this by making the columns of A into rows. We get A T A ˆ x = A T b (called . the normal equations ), so this is the system we need to
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Unformatted text preview: solve to produce a least-squares solution of A x = b . Can we simplify this if we have the QR factorization of A ? EXAMPLE. Find a least-squares solution of A x = b when A = 1-2-1 2 3 2 5 and b = 3 1-4 2 . Now find the least-squares error associated with this least-squares solution. This is the distance from b to A ˆ x . HOMEWORK: SECTION 6.5...
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