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# lec22 - 6.006 Introduction to Algorithms Lecture 22 Piotr...

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6.006- Introduction to Algorithms Lecture 22 Piotr Indyk

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Outline “Numerics” - algorithms for operations on large numbers – high precision – cryptography, simulations, etc We will see: – irrationals – large number operations: • multiplication • division • matrix multiplication 3.14159265358979323846264338327950288419 7169399375105820974944592307816406286208 9986280348253421170679821480865132823066 4709384460955058223172535940812848111745 0284102701938521105559644622948954930381 9644288109756659334461284756482337867831 6527120190914564856692346034861045432664 8213393607260249141273724587006… 2.4142135623730950488016887242096980785696718753769 480731766797379907324784621070388503875343276415727 350138462309122970249248360558507372126441214970999 358314132226659275055927557999505011527820605714701 095599716059702745345968620147285174186408891986095 523292304843087143214508397626036279952514079896872 533965463318088296406206152583523950547457502877599 61729835575220337531857011354374603 ...
Computing to lots of digits ... why? h 1 1 2 1. 414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 376 948 073 176 679 ...

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Computing to lots of digits ... why? High precision may be needed in some applications Consider Dijkstra for paths between points on plane: – lengths have form – where Is > ? h i i h a i b i h i h i 2 = a i 2 + b i 2 1 + 40 + 60 12 + 17 + 56 1 + 40 + 60 = 15.07052201275 12 + 17 + 56 = 15.07052201430
c. 10 AD Computing ; Babylonian method Iterative approach Also called the Heron’s method y 0 =h; x 0 =1 y 1 =(x 0 +y 0 )/2; x 1 =h/(y 1 ) In general y i+1 =(x i +y i )/2 x i+1 =h/(y i+1 ) h c. 1700 BC Since x 0 +y 0 - 2 x 0 1/2 y 0 1/2 =( x 0 1/2 - y 0 1/2 ) 2 0 , we have (x 0 +y 0 )/2 x 0 1/2 y 0 1/2 = h 1/2

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Computing ; Babylonian method y 0 =h; x 0 =1 y
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lec22 - 6.006 Introduction to Algorithms Lecture 22 Piotr...

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