Polynomial Arithmetic

Transcription

1 Polynomal Arthmetc Stefan Jerg Zentrum Mathematk Technsche Unverstät München (TUM) 1

2 Overvew Polynomal Arthmetc Polynomal Arthmetc Generaltes Polynomal Addton Polynomal Multplcaton Fast Polynomal Algorthms Polynomal Exponentaton Polynomal Substtuton Polynomal Greatest Common Dvsors

3 Generaltes Assume R s a rng, a,..., 0 ad R d d 1 P X = adx + a X a unvarate polynomal n X d 1 0 a : coeffcents of P( X ) ax : monomals / terms of P( X ) R: coeffcent doman of P( X ) R[ X ]: set of all polynomals n X wth coeffcents n R deg P( X ): degree of P( X ) ( 0 deg a P X = n) d 3

4 Generaltes Assume R s a rng, a,..., 0 ad R d d 1 P X = adx + a X a unvarate polynomal n X d 1 0 lc( P( X )): leadng coeffcent - lc P( X ) lt P( X ) : leadng term - lt P( X ) = = ad deg P lc( P) X f lc( P( X )) = 1, we call P( X ) a monc polynomal 4

5 Generaltes e (,..., ) t e t e nt e e e = n P X X a X X X a X X X 1 n t 1 n 0 1 n multvarate polynomal n X,..., 1 X n e1 e en ax 1 X Xn P X e en: total degree of a monomal deg P( X1,..., X n ): maxmum of the monomal s total degrees [ ] 1,..., n : monomals / terms of R X X : set of all multvarate polynomals n X,..., 1 X n 5

6 Generaltes dfferent representatons of polynomals expanded / recursve representaton 3 3 P = X Y + X Z + YZ + YZ + Z X, Y, Z [ ] [ ] (( )[ ])[ ] P = Y + Z X + Z + Z Y + Z Z Y X varable sparse / varable dense representaton P = X Y Z + X Y Z + X YZ + X YZ + X Y Z 3 4 ( ) P = Z Y + Z Y X + Z + Z Y + Z Y X degree sparse / degree dense representaton P = Z Y + 0Y + 0Y + Z + 0Z Y X + 0X + 5 ( ) ( ) + ( + 0 ) Z Z Z Z Y Z Z Y X

7 Polynomal Addton algorthm for polynomal addton usng a recursve, varable sparse, degree sparse representaton usng lnear lsts 7 5 some short functons we need: lt(p), lc(p), le(p) empty(lst) scoef(p) var(p), terms(p) eg..: F X = X + 3X 13X + 3 X, 7,1, 5,3,, 13, 0,3 = F :leadng term/coeffcent/exponent of P :decde whether the lst s empty :decde whether P s a coeffcent :man varable of P / lst of terms of P :the lst wthout the frst element :concat / create lsts PolyCreate (var, terms) := { //creates a new polynomal f empty(terms) then 0 elf ( le(terms) = 0 ) then lc(terms) else terms) } 7

8 Polynomal Addton TermsPlus (FTerms, GTerms) := { //addton of two term-lsts f empty(fterms) then GTerms; elf empty(gterms) then FTerms; elf le(fterms) > le(gterms) then TermsPlus(rest(FTerms), GTerms)); elf le(fterms) < le(gterms) then TermsPlus(rest(GTerms), FTerms)); else { tempc := PolyPlus(lc(FTerms), lc(gterms)); f (tempc = 0) then TermsPlus(rest(FTerms), rest(gterms)); else ((le(fterms), TermsPlus(rest(FTerms), rest(gterms)); } } 8

9 Polynomal Addton when havng dfferent man varables: see one as constant term e. g.: X Y = X YX 0 PolyPlus (F, G) := { //addton of two polynomals f scoef(f) then f scoef(g) then F+G else PolyCreate(var(G),TermsPlus(((0,F)),terms(G))); elf scoef(g) then PolyCreate(var(F),TermsPlus(((0,G)),terms(F))); elf ( var(f) > var(g) ) then PolyCreate(var(F),TermsPlus(((0,G)),terms(F))); elf ( var(f) < var(g) ) then PolyCreate(var(G),TermsPlus(((0,F)),terms(G))); else PolyCreate(var(F),TermsPlus(terms(F),terms(G))); } 9

10 Polynomal Multplcaton m 0... n 0... F X = a X + + a m G X = b X + + b n = F( X) G( X) H X (, m + n l ) coef H X = ab + a b + + ab + ab e.g.: l l l l X 3X + X 5 X + X + = X X + X 9X X 10

11 Polynomal Multplcaton polynomal type PolyTmes (F(X), G(X)) := { H := (); foreach a X e n F(X) foreach b k X f k n G(X) coef(h, X e +f k ) := coef(h, X e +f k ) + a *b k ; return(h); } number of terms n H after the th pass through: total exponent comparsons dense t+ 1 sparse max. t ( 1) t t t t+ 1 = t t+ = Ot t ( t+ 1) t 3 t = Ot = 11

12 Fast Polynomal Algorthms use of effcent data structures e.g. balanced trees or Hash tables TermsPlus (FTerms, GTerms) := { HTerms := FTerms; foreach (e,c) n GTerms { oldc := lookup(hterms, e); f ( oldc = () ) then nsert((e,c), HTerms); else { delete(e, HTerms); nsert((e,polyplus(c, oldc)), HTerms); } } return HTerms; } TermsPlus usng data abstracton operators 1

13 Fast Polynomal Algorthms use of effcent data structures max. Comparsons Comparsons Arth. Ops. Lnear Lst k O(n) O(n) Balanced tree log (k) O(n log(n) ) O(n) Hash table O(1) O(n) O(n) Operaton counts for addton of polynomals max. Comparsons Comparsons Arth. Ops. Lnear Lst k O(n 3 ) O(n ) Balanced tree log (k) O(n log(n) ) O(n ) Hash table O(1) O(n ) O(n ) Operaton counts for multplcaton of polynomals 13

14 Fast Polynomal Algorthms Dvde and Conquer Algorthm let FG, are polynomals wth degree k 1 = + F X f X X f X 0 1 k 1 = + G X g X X g X 0 1 n = k deg f,deg g k 1 k k ( ) 1 1 k ( ) F X G X = f g X + f g + f g X + f g = = fgx + f+ f g+ g fg fg X + fg k 1 14

15 Fast Polynomal Algorthms Dvde and Conquer Algorthm M ( n ): number of coeffcent multplcatons n n x n M( n) = 3M = 3 M =... = 3 M x 4 n wth = 1 and M () 1 = 1 x n 3 M n = = n n log log O n multplcatons 15

16 Polynomal Exponentaton repeated multplcaton vers. repeated squarng algorthm n (...) F = F F F F (( ) n ( F) = ( F )... ) PolyExptRm (P, s) := { n ( Q := ) 1; (...) F = F F F F loop for 1 s do { Q := P * Q; } return Q; } repeated multplcaton algorthm PolyExptSq (P, s) := { Q := 1; M := P loop whle s > 0 do { f odd(s) then Q := Q * M M := M * M; s := floor(s/); } return Q; } repeated squarng algorthm 16

17 Polynomal Exponentaton repeated multplcaton vers. repeated squarng algorthm wth P = ( 1 + t + t t ) and costs of multplyng P r 1 L P = P s : n n+ n L P =number of terms n P (proof wth nducton) n+ r n+ s CSq ( r, s) = L( Pr ) L( Ps ) = n n r+ s 1 r+ s 1 n + j n+ r+ s n+ r CMul ( r, s) = L( P1 ) L( Pj ) = ( n+ 1) = ( r+ s) r j= r j= r n n n 17

18 Polynomal Exponentaton repeated multplcaton vers. repeated squarng algorthm ( + 1) r n+ r n r r CSq ( r, r) =... 1 O n n = = + + ( r! ) n ( + ) r n+ r r r r 1 CMul ( 1,r 1) = r 1 =... = 1+ + n n O n r ( r)! n r r CSq r, r 1 r r = > 1,large values of 1, 1 On r n CMul r r r n 18

19 Polynomal Exponentaton algorthm based on the bnomal theorem bnomal formula: set a: ( 1) n n a+ b = a + na b+ a b b n n n 1 n n d d 1 let F( X) = fdx + ( fd 1X f0) d = fd X and : ( d b f 1 d 1X... f0) = + + to receve a fast algorthm d d ( 1 n n d n d n 1 d f ) ( 1 dx fd 1X f0 fd X n fdx fd 1X f0) =

20 Polynomal Substtuton let F( X) R[ X], G R-module (could also be R[ X] d d 1 F X = f X + f X f d ( d d d ) d 1 0 (... ) ) F G = f G+ f G+ f G+ f G+ f TermsHorner (FTerms, G) := { H := 0; f := le(fterms); foreach (e,c) n FTerms { H := G f-e * H + c; f := e; } return H * G e ; } (Horner s rule) 0

21 Overvew Polynomal GCD s Polynomal Arthmetc Polynomal Greatest Common Dvsors Generaltes GCD of Several Quanttes Polynomal Contents Coeffcent Growth Pseudo-Quotents Subresultant Polynomal Remander Sequence 1

22 Generaltes ntegral doman: commutatve rng wth 0 1 and no zero dvsors wth f, g R ntegral doman f ah, Rwth f = a h a, h dvsors f abh,, Rwth f = a h, g = b h h common dvsor of f, g gcd ( f, g ) : greatest common dvsor f g lcm ( f, g) : = least common multple gcd, ( f g) (( ) ) gcd f, f,..., f = gcd f,gcd f,..., f = gcd f,..., f f 1 n 1 n 1 n 1 n

23 Generaltes assume FG RX [ X ] H j, 1,... n, R ntegral doman : = common dvsor F, G wth maxmal degree n X ( H H ) lcm, s common dvsor wth maxmal degree n X and X j j common dvsors( F, G) wth maxmal degree n all X these polynomals we call gcd ( F, G) up to elements of R 3

24 GCD of several Quanttes [,... ],, F {,..., }, gcd (,... ) gcd ( F1,gcd ( F,..., Fl )) l 1 GCD calculatons F R X X R ntegral doman = F F H = F F 1 n 1 l 1 l more effcent algorthm: 1) F = F1 + k3f3 + k5f5 +..., G = F + kf + k4f4 +..., k chosen randomly ) H': = gcd ( F, G) 3) remove all elements F F wth H' F, add H ' to F 4) repeat choosng new k untl F = 1 4

25 GCD of several Quanttes ( ) res x F X, G X : resultant of F and G res x F X, G X = 0 F, G have a common factor wth G( k) res ( F, F kf ) = + we get: x 1 3 ( F F F ) ( F F + kf ) k s zeroes of G( k) gcd,, gcd, expected GCD calculatons tends to 1 5

26 Polynomal Contents n n 1 F X = f X + f X f, f n 0 cont F: = gcd f,..., f " content of F" 0 cont F = 1 " prmtve polynomal"/" trval content" F X prm F : = " prmtve part of F " cont F n cont F: = gcd coeffcents of monomals of F " scalar content" R 6

27 Polynomal Contents Proposton (Gauss): If F and G are prmtve polynomals over an entre rng R then FG s also prmtve. Dealng wth polynomals over an ntegral doman the prmtve part of a reducble polynomal s stll reducble Proof: Let m m 1 F X = f0x + f1x fm n n 1 G X = g0x + g1x gn and p Rprme, p/ cont FG 7

28 Polynomal Contents Proposton (Gauss): If F and G are prmtve polynomals over an entre rng R then FG s also prmtve. the mages of F and G n R [ ] and G are prmtve: / p ˆ r ˆ r 1 ˆ m r m r 1 m Fˆ X = f X + f X f ˆ s s 1 = ˆ ˆ ˆ n s + n s n G X g X g X g X are both non zero, snce F wth f ˆ, gˆ 0 m r n s but f ˆ gˆ = 0 snce p/contfg m r n s contradcton R ( s ntegral doman! ) 8

29 Polynomal Contents Assume F G R[ X X ], H : = gcd ( FG, ) f, 1,..., n X only occurs n G, not n F then H not nvolves 0 ( ) ( ) ( ) X H= gcd FG, = gcd F,coef GX,,coef GX,,coef GX,,... we can remove all varables that do not occur n every polynomal cont H = gcd cont F,cont G = gcd f,..., f, g,..., g F G gcd, cont H cont H 0 m 0 s prmtve (although f F, G are not prmtve) m 9

30 Coeffcent Growth Let, F X F X be polynomals over a feld 1 = + deg < deg F X q X F X F X F F quotent remander = + F X q X F X F X 3 4 = + = + F X q X F X F X n 3 n 3 n n 1 F X q X F X F X n n n 1 n n : F = 0 F = gcd F, F n+ 1 n 1 30

31 Coeffcent Growth example 1 = F1 X X X X X X 6 4 = F X X X X F3 = X + X F4 = X 9X F5 = X F6 =

32 Coeffcent Growth example 4 3 F1 = X + X W 3 F = X + X + 3WX F = 3W + X + 4W 1 X W + 1 F F 3 3 7W W 11W + 3 9W W + 4W = X + 9W 1W + 4 9W 1W W 738W 474W + 75W + 81W 16W + 68W 8 5 = W 108W 590W + 06W + 109W 66W + 9 3

33 Pseudo-Quotents Eucldean GCD algorthm = lc, Let F1( X), F( X ) be polynomals over a rng R, f F( X) = deg( F ) deg( F ) δ + 1 = + deg < deg δ 1 + f 1 F X Q X F X R X R F 1 R( X ) always has ntegral coeffcents 33

34 Pseudo-Quotents example F1 X X X X X X 6 4 F = 3X + 5X 4X 9X F3 = 15X + 3X 9 F4 = 15795X X = F5 = X F6 = Eucldean PRS F1 X X X X X X 6 4 F = 3X + 5X 4X 9X F3 = 5X + X 3 F4 = 13X + 5X 49 6 = F5 = 4663X 6150 F = 1 Prmtve PRS (n each step dvson by cont F ) 34

35 Pseudo-Quotents polynomal remander sequences A sequence of polynomals F F F R[ X],,... n 1 that satsfes = deg < deg β F X α F X F X F F 1 1 where, R q X s a polynomal over R, s called a polynomal remander sequence generated from F 1 and F. α β, lc wth f = F( X), δ deg( F) deg( F+ 1 ) that F = prem ( F, F ). ( 1 cont prem ( F, F ) β 1 = we declare f δ α = + 1 so ( 1 ) ) β 1 35

36 Subresultant Poly. Remander Seq. accordng to Collns and Brown We choose the β usng followng equaltes f = lc( F( X) ), δ = deg( F) deg( F+ 1 ), α 1 f δ 1 = + δ1 δ 1 1 δ 1 h = f, h = f h 1 3,..., = k δ1+ 1 δ + 1 δ ( 1 ), β ( 1) f h ( 4,..., k 1) β = = = + 3 to get the best GCD algorthm that uses a full polynomal remander seqence 36

37 Subresultant Poly. Remander Seq. example F1 X X X X X X 6 4 F = 3X + 5X 4X 9X F3 = 15X 3X 9 F4 = 65X + 15X 45 F F 5 6 = = 936X 1300 =

38 Polynomal Arthmetc Stefan Jerg Thanks for lstenng 38