lec23 - Introduction to Algorithms 6.006 Lecture 23 Jelani...

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Introduction to Algorithms 6.006 Lecture 23 Jelani Nelson
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Introduction to Algorithms 5/5/11 2 Menu “Numerics” - algorithms for operations on large numbers – Cryptography, simulations, etc Operations – Addition – Multiplication – Division – Matrix multiplication 3.14159265358979323846264338327950288419 7169399375105820974944592307816406286208 9986280348253421170679821480865132823066 4709384460955058223172535940812848111745 0284102701938521105559644622948954930381 9644288109756659334461284756482337867831 6527120190914564856692346034861045432664 8213393607260249141273724587006…
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Introduction to Algorithms 5/5/11 3 Division a/b Inversion: – Given positive n -digit number b (radix r ) – Compute 1/b I.e., for some R=r k , compute Σ R/b Φ Iterative algorithm: – Start from some x 0 – Keep computing x i+1 from x i – Show this converges to the answer
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Introduction to Algorithms 5/5/11 4 Newton’s method Iterative approach to solving f(x)=0 – In our case f(x)=1/x-b/R Iterative step: – Find a line y= f(x i )+f’(x i )(x- x i ) tangent to f(x) at x i – Set x i+1 to the solution of f(x i )+f’(x i )(x- x i ) =0 I.e., x i+1 =x i -f(x i )/f’(x i ) -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 x i x i+1 f
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lec23 - Introduction to Algorithms 6.006 Lecture 23 Jelani...

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