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Unformatted text preview: 18.06 Problem Set 6 Solutions Total: 100 points Section 4.3. Problem 4: Write down E = Ax − b 2 as a sum of four squares– the last one is ( C + 4 D − 20) 2 . Find the derivative equations ∂E/∂C = and ∂E/∂D = 0. Divide by 2 to obtain the normal equations A T Ax = A T b . Solution (4 points) Observe ⎞ ⎛ ⎞ ⎛ 1 A = ⎜ ⎜ ⎝ 1 1 1 3 ⎟ ⎟ ⎠ , b = ⎜ ⎜ ⎝ 8 8 ⎟ ⎟ ⎠ , and define x = C D . 1 4 20 Then ⎞ ⎛ C C + D − 8 ⎜ ⎜ ⎝ ⎟ ⎟ ⎠ Ax − b = , C + 3 D − 8 C + 4 D − 20 and Ax − b 2 = C 2 + ( C + D − 8) 2 + ( C + 3 D − 8) 2 + ( C + 4 D − 20) 2 . The partial derivatives are ∂E/∂C = 2 C + 2( C + D − 8) + 2( C + 3 D − 8) + 2( C + 4 D − 20) = 8 C + 16 D − 72 , ∂E/∂D = 2( C + D − 8) + 6( C + 3 D − 8) + 8( C + 4 D − 20) = 16 C + 52 D − 224 . On the other hand, A T A = 4 8 , A T b = 36 . 8 26 112 Thus, A T Ax = A T b yields the equations 4 C +8 D = 36 , 8 C +26 D = 112. Multiply ing by 2 and looking back, we see that these are precisely the equations ∂E/∂C = 0 and ∂E/∂D = 0. Section 4.3. Problem 7: Find the closest line b = Dt , through the origin , to the same four points. An exact fit would solve D = 0, D 1 = 8, D 3 = 8, D 4 = 20. · · · · 1 Find the 4 by 1 matrix A and solve A T Ax = A T b . Redraw figure 4.9a showing the best line b = Dt and the e ’s. Solution (4 points) Observe ⎞ ⎛ ⎞ ⎛ A = ⎜ ⎜ ⎝ 1 3 ⎟ ⎟ ⎠ , b = ⎜ ⎜ ⎝ 8 8 ⎟ ⎟ ⎠ , A T A = (26) , A T b = (112) . 4 20 Thus, solving A T Ax = A T b , we arrive at D = 56 / 13 . Here is the diagram analogous to figure 4.9a. Section 4.3. Problem 9: Form the closest parabola b = C + Dt + Et 2 to the same four points, and write down the unsolvable equations Ax = b in three unknowns 2 x = ( C,D,E ). Set up the three normal equations A T Ax = A T b (solution not required). In figure 4.9a you are now fitting a parabola to 4 points–what is happening in Figure 4.9b? Solution (4 points) Note ⎛ ⎞ ⎛ ⎞ 1 ⎛ ⎞ ⎜ 1 1 1 ⎟ ⎜ 8 ⎟ C ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ A = ⎠ , b = ⎠ , x = D . ⎝ 1 3 9 ⎝ 8 E 1 4 16 20 Then multiplying out Ax = b yields the equations C = 0 , C + D + E = 8 , C + 3 D + 9 E = 8 , C + 4 D + 16 E = 20 ....
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 Fall '10
 Strang
 Linear Algebra, Algebra, Equations, Derivative

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