Assignment01SupplementSolutionFall2010

# Assignment01SupplementSolutionFall2010 - 525.412 Computer...

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Assignment 1 Extra Problem Solutions Babbage’s Diﬀerence Engine Show how to compute y = x 3 + 4 x 2 - 1 using the approach Bab- bage used in programming his diﬀerence engine. Solution For every positive integer i we have y j = x 3 j + 4 x 2 j - 1 . (1) Therefore y j +1 = x 3 j +1 + 4 x 2 j +1 - 1 = ( x j + 1) 3 + 4( x j + 1) 2 - 1 = x 3 j + 3 x 2 j + 3 x j + 1 + 4 x 2 j + 8 x j + 4 - 1 y j +1 = x 3 j + 7 x 2 j + 11 x j + 4 , (2) y j +2 = x 3 j +2 + 4 x 2 j +2 - 1 = ( x j + 2) 3 + 4( x j + 2) 2 - 1 = x 3 j + 6 x 2 j + 12 x j + 8 + 4 x 2 j + 16 x j + 16 - 1 y j +2 = x 3 j + 10 x 2 j + 28 x j + 23 . (3) and y j +3 = x 3 j +3 + 4 x 2 j +3 - 1 = ( x j + 3) 3 + 4( x j + 3) 2 - 1 = x 3 j + 9 x 2 j + 27 x j + 27 + 4 x 2 j + 24 x j + 36 - 1 y j +3 = x 3 j + 13 x 2 j + 51 x j + 62 . (4) We can compute the ﬁrst diﬀerences as Δ 1 y j = y j +1 - y j (5) = ( x 3 j + 7 x 2 j + 11 x j + 4 ) - ( x 3 j + 4 x 2 j - 1 ) Δ 1 y j = 3 x 2 j + 11 x j + 5 (6) Δ 1 y j +1 = y j +2 - y j +1

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Assignment01SupplementSolutionFall2010 - 525.412 Computer...

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