MIT18_06S10_pset6_s10_soln

# MIT18_06S10_pset6_s10_soln - 18.06 Problem Set 6 Solutions...

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 11

MIT18_06S10_pset6_s10_soln - 18.06 Problem Set 6 Solutions...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online