Transcription of 4.3 Least Squares Approximations
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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.
4.3. Least Squares Approximations 221 Figure 4.7: The projection p DAbx is closest to b,sobxminimizes E Dkb Axk2. In this section the situation is just the opposite. There are no solutions to Ax Db. Instead of splitting up x we are splitting up b. Figure 4.3 shows the big picture for least squares. Instead of Ax Db we solve Abx Dp.
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