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4.3 Least Squares Approximations

218 Chapter 4. Least Squares ApproximationsIt often happens thatAxDbhas no solution. The usual reason is:too many matrix has more rows than columns. There are more equations than unknowns(mis greater thann). Thencolumns span a small part ofm-dimensional space. Unless allmeasurements are perfect,bis outside that column space. Elimination reaches animpossible equation and stops. But we can t stop just because measurements include repeat: We cannot always get the erroreDb Axdown to zero. Wheneis zero,xis an exact solution the length ofeis as small as possible,bxis aleast Squares goal in this section is to computebxand use it. These are realproblems and they need an previous section emphasizedp(the projection). This section emphasizesbx(theleast Squares solution). They are connected bypDAbx. The fundamental equation is stillATAbxDATb. Here is a short unofficial way to reach this equation:WhenAxDbhas no solution, multiply byATand solveATAbxDATb:Example 1A crucial application of Least Squares is fitting a straight line with three points:Find the closest line to the ; 6/.

This 3 by 2 system has no solution: b D.6;0;0/is not a combination of the columns.1;1;1/and .0;1;2/. Read off A;x; and b from those equations: A D 2 4 10 11 12 3 5 x D C D b D 2 4 6 0 0 3 5 Ax Db is not solvable. The same numbers were in Example 3 in the last section. We computed bx D.5;3/. Those numbers are the best C and D,so5 3t will be the ...

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