Advanced Algorithmic Problem Solving Le 3 Arithmetic. Fredrik Heintz Dept of Computer and Information Science Linköping University

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1 Advced Algorthmc Prolem Solvg Le Arthmetc Fredrk Hetz Dept of Computer d Iformto Scece Lköpg Uversty

2 Overvew Arthmetc Iteger multplcto Krtsu s lgorthm Multplcto of polyomls Fst Fourer Trsform Systems of ler equtos Nïve Guss Elmto

3 Arthmetc Rge of defult teger dt types (C++) usged t = usged log: (9 dgts) usged log log: (9 dgts) How to represet 777! Opertos o Bg Iteger Bsc: dd, sutrct, multply, dvde, etc Use hgh school method

4 Arthmetc Gretest Commo Dvsor (Euclde Algorthm) GCD(, ) = GCD(, ) = GCD(, mod ) t gcd(t, t ) { retur ( ==? : gcd(, % )); } Lest Commo Multpler LCM(, ) = * / GCD(, ) t lcm(t, t ) { retur ( / gcd(, ) * ); } // Q: why we wrte the lcm code ths wy? GCD/LCM of more th umers: GCD(,, c) = GCD(, GCD(, c)) Fd d,, y such tht d = + y d d = GCD(,) (Eteded Euclde Algorthm) EGCD(,) = (,,) EGCD(,) (d,,y ) = EGCD(, mod ) (d,,y) = (d,y, /*y )

5 Arthmetc 5 Represetg rtol umers. Prs of tegers, where GCD(,)=. Represetg rtol umers modulo m. The oly dffcult operto s verse, = (mod m), where verse ests f d oly f d m re co-prme (gcd(,m)=). C e foud usg the Eteded Euclde Algorthm = (mod m) => = qm => qm = (d,, y) = EGCD(,m) => s the soluto ff d =.

6 Krtsu s lgorthm Usg the clsscl pe d pper lgorthm two dgt tegers c e multpled O( ) opertos. Krtsu cme up wth fster lgorthm. Let A d B e two tegers wth A = A k + A, A < k B = B k + B, B < k C = A*B = (A k + A )(B k + B ) = A B k + (A B + A B ) k + A B Isted ths c e computed wth multplctos T = A B T = (A + A )(B + B ) T = A B C = T k + (T - T - T ) k + T

7 Complety of Krtsu s Algorthm 7 Let T() e the tme to compute the product of two -dgt umers usg Krtsu s lgorthm. Assume = k. T() = ( lg() ), lg().58 T() T(/) + c (T(/) + c(/)) + c = T(/ ) + c(/ + ) (T(/ ) + c(/)) + c(/ + ) = T(/ ) + c( / + / + ) T(/ ) + c( - / / + )... c k + c[((/) k - )/(/ -)] --- Assumg T() c c k + c( k - k ) c lg() = c lg()

8 Fst Fourer Trsform 8 See seprte presetto

9 9 Systems of Ler Equtos Mtr form Stdrd form dfferet forms c epreseted ler equtos A system of

10 Solutos of Ler Equtos s 5 soluto to the followg equtos :

11 Solutos of Ler Equtos A set of equtos s cosstet f there ests o soluto to the system of equtos: 5 These equtos re cosstet

12 Solutos of Ler Equtos Some systems of equtos my hve fte umer of solutos hve fte.5( umer of ) s solutos soluto for ll

13 Grphcl Soluto of Systems of Ler Equtos 5 Soluto =, =

14 Crmer s Rule s Not Prctcl Crmer' s Rule c e used to solve 5, 5 the system Crmer' s Rule s ot prctcl for lrge systems. To solve N y N system requres (N )(N -)N! multplctos. To solve y system,.8 multplctos re eeded. It c eused f the determts re computed effcet wy 5

15 5 Nve Guss Elmto The method cossts of two steps: Forwrd Elmto: the system s reduced to upper trgulr form. A sequece of elemetry opertos s used. Bckwrd Susttuto: Solve the system strtg from the lst vrle. ' ' ' ' '

16 Elemetry Row Opertos Addg multple of oe row to other Multply y row y o-zero costt

17 7 Emple: Forwrd Elmto 8 7 8,, from equtos Elmte Step: Forwrd Elmto : Prt

18 8 Emple: Forwrd Elmto 9 5 from equto Elmte Step : 9 5, from equtos Elmte Step :

19 9 Emple: Forwrd Elmto Elmto : the Forwrd Summry of

20 Emple: Bckwrd Susttuto () ) ( (), () ) ( 5 9, for,...solve for the solve, for Solve 9 5

21 Forwrd Elmto j j j j j j j j ) ( To elmte ) ( To elmte

22 Forwrd Elmto s elmted. Cotue utl ) ( To elmte k kk k kj kk k j j k k j k

23 Bckwrd Susttuto j j j,,,,,,,,

24 Determt The elemetry opertos do ot ffect the determt Emple : Elemetry opertos A A' det (A) det (A')

25 How My Solutos Does System of Equtos AX=B Hve? 5 Uque det(a) reduced mtr hs o zero rows No soluto det(a) reduced mtr hs oe or more zero rows correspodg B elemets Ifte det(a) reduced mtr hs oe or more zero rows correspodg B elemets

26 Emples.5!.5 # : solutos # of fte No soluto Uque X mpossle X solutos Ifte No soluto soluto X X X X X X

27 Pseudo-Code: Forwrd Elmto 7 do k = to - do = k+ to fctor =,k / k,k do j = k+ to ed do,j =,j fctor * k,j = fctor * k ed do ed do

28 Pseudo-Code: Bck Susttuto 8 = /, do = - dowto sum = do j = + to sum = sum,j * j ed do = sum /, ed do

29 9 Prolems wth Nve Guss Elmto o The Nve Guss Elmto my fl for very smple cses. (The pvotg elemet s zero). o Very smll pvotg elemet my result serous computto errors

30 How Do We Kow If Soluto s Good or Not Gve AX=B X s soluto f AX-B= Compute the resdul vector R= AX-B Due to roudg error, R my ot e zero The soluto s cceptle f m r

31 How Good s the Soluto?....5 Resdues : R soluto

32 Summry Arthmetc Iteger multplcto Krtsu s lgorthm Multplcto of polyomls Fst Fourer Trsform Systems of ler equtos Nïve Guss Elmto