Giới thiệu về các thuật toán - lec24

Giới thiệu về các thuật toán -

6.006 Introduction to Algorithms

Spring 2008

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Lecture 24: Numerics II

Lecture Overview

e Review:

— high precision arithmetic

— multiplication

e Division

— Algorithm

— Error Analysis

e Termination

Review:

Want millionth digit of 2:

[v2 - 102 đ = 109

Compute |./a]| via Newton’s Method

xo = 1 (initial guess) Xi+1 = ae Tự — division!

Converges quadratically; ÿ correct digits doubles each step

Multiplication:

1 Naive Divide & Conquer method: 6(d?) time

2 Karatsuba: 0(d198: 3) — 60(d1584 )

3 Toom-Cook generalizes Karatsuba (break into k > 2 parts )

T(d) = 5T(d/3) + 0(d) = 0 (a?) = 9 (a4)

4 Schonhage-Strassen - almost linear! 6(d lgdlglgd) using FFT All of these are in gmpy package

5 Furer(2007): @ (n log n20 los" ”) where log” ?› is iterated logarithm ‡ times log needs

to be applied to get a number that is less than or equal to 1

High Precision Division

a

We want high precision rep of 3

1

e Compute high-precision rep of ọ first

e High-precision rep of ; means lz! where R is large value s.t it is easy to divide by

R

Ex: R = 2" for binary representations

Division

Rk Newton’s Method for computing ?

f(z) = = RB (zero atx = 2)

—Ì

¿-9

Xi+1 — Xi — Fx) = Xi — —

1 6b by;? > multiply

2 2

Example

R 2'© 65536

t — = — = — =13107.2

216

Try initial guess —— = 21

Xo = 2'* = 16384

Xxi = 2- (16384) — 5(16384)?/65536 = 12288 x2 = 2- (12288) — 5(12288)?/65536 = 13056 x3 = 2- (13056) — 5(13056)?/65536 = 13107

Error Analysis

Yur = XA R Assume x; = — (1+ €;) b

= 2 (1+4) 2— 7) —ms | x} n1) (1+ 4) Gta

= (2+ 24) = (Lt 26+ i ))

= x=ø7)=T (+) where €j41 = —e;"

Quadratic convergence; { digits doubles at each step

Therefore complexity of division = complexity of multiplication

Termination

Iteration: yj;41 = it Lebel)

Do floors hurt? Does program terminate?

Iteration is

Nit -@

Nit

= n7 wherey = 5 + Sand0<7<1

Since 5 > Vab, 5 > Va, so subtracting + always leaves us > |vwœ| This won't stay stuck above ife; <1 (good initial guess)