Multiplication of polynomials

Transcription

1 Multiplication of polynomials M. Gastineau IMCCE - Observatoire de Paris - CNRS UMR828 77, avenue Denfert Rochereau 7514 PARIS FRANCE gastineau@imcce.fr Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

2 Different products Full All terms are computed Truncated in the partial or total degree of the variables Special truncation to select terms satisfying a rule Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

3 Available methods Naive method efficient for low degree or for sparse polynomials Karatsuba s algorithm efficient for intermediate degree and dense polynomials reduce the number of multiplications FFT method efficient only for large degree Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

4 Naive multiplication A(x) = da max i=da min a i x i and B(x) = db max i=db min b i x i Perform the multiplication of all terms C(x) = da max +db max k=da min +db min c k x k with c k = if A, B and C, have r, s and t terms i+j=k a i b j rs multiplications and rs-t additions complexity : O(rs) Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

5 Multiplication for univariate polynomials stored as vector Algorithm 1: Compute the full product of univariate polynomials A and B represented with a dense vector Input: A: polynomial {da min, da max, array of coefficients a } Input: B: polynomial {db min, db max, array of coefficients b } Output: C: polynomial {dc min, dc max, array of coefficients c } dc min da min + db min dc max da max + db max C create a polynomial with minimal degree dc min and maximal degree dc max for k dc min to dc max do c[k] a[da min ] b[k da min ] for j da min + 1 to da max do c[k] c[k] + a[j] b[k j] return C Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

6 Multiplication for recursive dense vector 1/2 Function mulfull(a,b) Compute the full product of multivariate polynomials A and B represented with a recursive dense vector Input: A: multivariate polynomial {da min, da max, array of coefficients a} Input: B: multivariate polynomial {db min, db max, array of coefficients b} Output: C: multivariate polynomial {dc min, dc max, array of coefficients c} dc min da min + db min dc max da max + db max C create a polynomial with minimal dc min and maximal dc max degree for k dc min to dc max do c[k] mulfull (a[da min ], b[k da min ]) for j da min + 1 to da max do fmafull (a[j], b[k j], c[k]) return C Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

7 Multiplication for recursive dense vector 2/2 Procedure fmafull(a,b,c ) Compute the full fused multiplicationaddition C = C + A B with A, B and C multivariate polynomials represented as recursive dense vector Input: A: multivariate polynomial {da min, da max, array of coefficients a} Input: B: multivariate polynomial {db min, db max, array of coefficients b} Input: C: multivariate polynomial {dc min, dc max, array of coefficients c } Output: C: multivariate polynomial newdc min da min + db min newdc max da max + db max if newdc min < dc min or dc max < newdc max then dc min min(newdc min, dc min ) dc max max(newdc max, dc max ) resize C for k dc min to dc max do c[k] mulfull (a[da min ], b[k da min ]) for j da min + 1 to da max do fmafull (a[j], b[k j], c[k]) if c contains at its beginning or at its then adjust dc min adjust dc max resize C Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

8 Multiplication for univariate polynomials stored as list Function mulfull(a,b) Compute the full product of univariate polynomials A and B represented with a list Input: A: polynomial { list of ( coefficients a, degree δ a ) } Input: B: polynomial { list of ( coefficients b, degree δ b ) } Output: C: polynomial { list of ( coefficients c, degree δ c )} C create a empty polynomial foreach element in A do D create a empty polynomial foreach element in B do add to the tail of D an element (a b, δ a + δ b ) C C + D return C Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

9 Multiplication for recursive list Procedure fmafull(a,b,c ) Compute the full fused multiplicationaddition C = C + A B with A, B and C multivariate polynomials represented as recursive list Input: A: polynomial { list of ( coefficients a, degree δ a ) } Input: B: polynomial { list of ( coefficients b, degree δ b ) } Input: C: polynomial { list of ( coefficients c, degree δ c ) } Output: C: polynomial { list of ( coefficients c, degree δ c ) } iter head of C foreach element in A do // avoid to scan to C when the loop on B is finished iterb iter foreach element in B do // find after iterb in C if the degree δ a + δ b is present while current degree δ c referenced by iterb < δ a + δ b do iterb next element after iterb if δ c = δ a + δ b then fmafull (a, b, c) if c = then remove the element referenced by iterb else insert an element (mulfull (a, b), δ a + δ b ) just before iterb if current element is the first element of B then iter iterb Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

10 Multiplication for flat vector 1/2 variables exponents coefficients X X 1 2 m 1 2 m 1 n 1 st term 2 nd term 65 th term 66 th term 64*(m-1)+1 term 64*(m-1)+2 term 64 th term 128 th term 64*m term How to sort terms? search and shift operations too slow need an intermediate and adjustable storage : BURST TRIE Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

11 Multiplication for flat vector 2/2 Burst tries trie node = dense container leaf node = sparse container 3 + 5z + 7z y + 9zy + 13zyx + 8z 2 x 2 + 9x 4 Burst trie Coe!cients Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

12 Homogeneous block A = da max δ=da min BH δ (a), B = db max δ=db min BH δ (b) a i X d 1 1 Xd Xd n n b j X d 1 1 Xd Xd n n = a i b j X d 1+d 1 1 X d 2+d X d n+d n n with C = A B = dc max δ=dc min BH δ (c) dc min = da min + db min dc max = da max + db max BH δ (c) = BH i (a) BH j (b) i+j=δ Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

13 Homogeneous block Function fmafull(bh δ (a), BH δ (b), BH δ+δ (c)) Compute the full fused multiplication-addition BH δ+δ (c) = BH δ+δ (c) + BH δ (a) BH δ (b) Input: BH δ (a) : homogeneous blocks { degree δ, a : array of r coeff. } Input: BH δ (b) : homogeneous blocks { degree δ, b : array of s coeff. } Input: BH δ+δ (c) : homog. blocks { degree δ + δ, c : array of t coeff. } Output: BH δ+δ (c) : homog. blocks { degree δ + δ, c : array of t coeff.} for i 1 to r do for j 1 to s do l get location of the term in BH δ+δ (c) BH δ+δ (c)[l] BH δ+δ (c)[l] + BH δ (a)[i] BH δ (b)[j] Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

14 Homogeneous block using functions Function fmafull(bh δ (a), BH δ (b), BH δ+δ (c)) Compute the full fused multiplication-addition BH δ+δ (c) = BH δ+δ (c) + BH δ (a) BH δ (b) using functions to compute location Input: BH δ (a) : homogeneous blocks { degree δ, a : array of r coeff. } Input: BH δ (b) : homogeneous blocks { degree δ, b : array of s coeff.s } Input: BH δ+δ (c) : homog. blocks { degree δ + δ, c : array of t coeff. } Output: BH δ+δ (c) : homog. blocks { degree δ + δ, c : array of t coeff.} for i 1 to r do expoa get array of exponents from the location i in BH δ for j 1 to s do expob get array of exponents from the location j in BH δ expoc expoa + expob l get location of the term with exponents expoc in BH δ+δ BH δ+δ (c)[l] BH δ+δ (c)[l] + BH δ (a)[i] BH δ (b)[j] Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

15 Homogeneous block using addressing tables Construction of the addressing table for the product of blocks in 3 variables with exponent tables of degree 1 and 2. T addr 2,1 = t T addr 1,2 T exp 1 + T exp 2 = T exp T addr 1, Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

16 Execution time to build the tables of exponents build the tables of exponents for homogeneous blocks in 1 variables up to the degree time (s) number of terms (in Millions) Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

17 Execution time to build the addressing tables product of two homogeneous blocks in 5 variables up to the total degree 4 1 initialization after initialization time (s) muliplication operations (in Millions) Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

18 Overhead to load the addressing tables from disk 3 25 execution time overhead (%) total degree of the addressing table Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

19 Homogeneous blocks BH(X,Y,Z) BH BH 1 BH 2 BH 3 BH P (x, y, z) = 3 + 5z + 7z y + 9yz + 13xyz + 8x 2 z 2 + 9x 4 Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

20 Compacted homogeneous blocks BHC BHC(X,Y,Z) BHC 1 BHC 2 BHC 3 BHC Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

21 Homogeneous block using addressing tables Function fmafull(bh δ (a), BH δ (b), T addr δ,δ, BH δ+δ (c)) Compute the full fused multiplication-addition BH δ+δ (c) = BH δ+δ (c) + BH δ (a) BH δ (b) using the addressing table Input: BH δ (a) : homogeneous blocks { degree δ, a : array of r coeff. } Input: BH δ (b) : homogeneous blocks { degree δ, b : array of s coeff. } Input: BH δ+δ (c) : homog. blocks { degree δ + δ, c : array of t coeff. } Input: T addr δ,δ : addressing table of degree δ, δ Output: BH δ+δ (c) : homog. blocks { degree δ + δ, c : array of t coefficients } for i 1 to r do for j 1 to s do l T addr δ,δ [i, j] BH δ+δ (c)[l] BH δ+δ (c)[l] + BH δ (a)[i] BH δ (b)[j] Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

22 Homogeneous block using addressing tables Function fmafull(bh δ (a), BH δ (b), T addr δ,δ, BH δ+δ (c)) Compute the full fused multiplication-addition BH δ+δ (c) = BH δ+δ (c) + BH δ (a) BH δ (b) using the addressing table Input: BH δ (a) : homogeneous blocks { degree δ, a : array of r coeff. } Input: BH δ (b) : homogeneous blocks { degree δ, b : array of s coeff. } Input: BH δ+δ (c) : homog. blocks { degree δ + δ, c : array of t coeff. } Input: T addr δ,δ : addressing table of degree δ, δ Output: BH δ+δ (c) : homog. blocks { degree δ + δ, c : array of t coefficients } for i 1 to r do for j 1 to s do ll T addr δ,δ δ,δ [BHC [i, j] δ (a).index[i], BHC δ (b).index[j]] BH δ+δ (c)[l] BH δ+δ (c)[l] + BH δ (a)[i] BH δ (b)[j] Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

23 Homogeneous block using addressing tables Function fmafull(bh δ (a), BH δ (b), T addr δ,δ, BH δ+δ (c)) Compute the full fused multiplication-addition BH δ+δ (c) = BH δ+δ (c) + BH δ (a) BH δ (b) using the addressing table Input: BH δ (a) : homogeneous blocks { degree δ, a : array of r coeff. } Input: BH δ (b) : homogeneous blocks { degree δ, b : array of s coeff. } Input: BH δ+δ (c) : homog. blocks { degree δ + δ, c : array of t coeff. } Input: T addr δ,δ : addressing table of degree δ, δ Output: BH δ+δ (c) : homog. blocks { degree δ + δ, c : array of t coefficients } for i 1 to r do for j 1 to s do l T addr δ,δ [i, j] BH δ+δ (c)[l] BH δ+δ (c)[l] + BH δ (a)[i] BH δ (b)[j] Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

24 Cache blocking technique BH! (a) BH! (a) BH! (b) BH! (b) normal flow flow with cache blocking Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

25 Cache blocking technique Input: BH δ (a) : homogeneous blocks { degree δ, a : array of r coeff. } Input: BH δ (b) : homogeneous blocks { degree δ, b : array of s coeff. } Input: T addr δ,δ : addressing table of degree δ, δ Input: BH δ+δ (c) : homog. blocks { degree δ + δ, c : array of t coeff. } Input: chunksize : chunk size { arrays of two integers} Output: BH δ+δ (c) : homog. blocks { degree δ + δ, c : array of t coeff.} iteri r/chunksize[1] /* number of chunks for the loop i */ iterj s/chunksize[2] /* number of chunks for the loop j */ for ci to iteri 1 do for cj to iterj 1 do for bi 1 to chunksize[1] do for bj 1 to chunksize[2] do i ci iteri + bi j cj iterj + bj l T addr δ,δ [i, j] BH δ+δ (c)[l] BH δ+δ (c)[l] + BH δ (a)[i] BH δ (b)[j] /* if s not divisible by chunksize[2] */ for bi 1 to chunksize[1] do for j iterj chunksize[2] to s do i ci iteri + bi l T addr δ,δ [i, j] BH δ+δ (c)[l] BH δ+δ (c)[l] + BH δ (a)[i] BH δ (b)[j] Function fmafull(bh δ (a), BH δ (b), T addr δ,δ, chunksize, BH δ+δ (c)) Compute the full fused multiplication-addition using the addressing table and cache blocking technique /* if r not divisible by chunksize[1] */ for i iteri chunksize[1] to r do for j 1 to s do l T addr δ,δ [i, j]; BH δ+δ (c)[l] BH δ+δ (c)[l] + BH δ (a)[i] BH δ (b)[j] Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

26 Benchmark of the cache blocking technique 1 Factor of the reduction of the execution time for the product of two homogeneous blocks in 8 variables of degree 7 G5 processor Core2 duo processor Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

27 Benchmark of the cache blocking technique 1 Factor of the reduction of the execution time of the product of two homogeneous blocks in 8 variables of degree 9 double precision quadruple precision Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

28 Benchmarks s (s + 1) with s = (1 + x + y + z + t + u) 14 CAS representation time (s) Ginac tree 4137 Maple 1 DAG Singular 3..2 list Maxima recursive list Mathematica 5.2 tree TRIP.99 recursive vector 13.5 TRIP.99 recursive list TRIP.99 flat vector 28.1 TRIP.99 homogeneous blocks (with initialization) 5.44 (after initialization).57 Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

29 Effect of the sparsity of the polynomials s s with s containing some terms of (1 + x 1 + x 2 + x 3 + x 4 + x 5 ) recursive list recursive vector flat vector homogeneous block compacted homogeneous block.6.5 time (s) number of terms Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

30 full product VxV with different representations The serie V has 352 terms and the result has terms V (λ, λ, X, X, Y, Y, X, X, Y, Y ) = X d 1 X d 2 Y d 3 Y d 4 X d 5 X d 6 Y d 7 Y d 8 e ı(k 1λ+k 2 λ ) d 1 + d 2 + d 3 + d 4 + d 5 + d 6 + d 7 + d 8 = 7 CAS representation time (s) Maple 1 DAG Mathematica 5.2 tree TRIP.99 recursive vector 2.91 TRIP.99 recursive list 2.32 TRIP.99 flat vector 1.97 TRIP.99 homogeneous blocks (with initialization) (after initialization) TRIP.99 compacted homogeneous blocks (with initialization) 17.7 (after initialization) TRIP.99 d alembert blocks (with initialization) (after initialization) 8.55 TRIP.99 compacted d alembert blocks (with initialization) 11.1 (after initialization) 1.25 Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

31 full product VxV for different degrees 3 25 recursive vector recursive list flat vector BH BHC BHDAL BHDALC 2 time (s) degree Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

32 Karatsuba's algorithm Idea : one multiplication could be avoided for polynomial of degree 1 Let A and B polynomials A(X) = a + a 1 X and B(X) = b + b 1 X The naive multiplication C = AB is C(X) = a b + (a b 1 + a 1 b )X 1 + a 1 b 1 X 2 But the coefficient of X 1 could be written as a b 1 + a 1 b = (a + a 1 )(b + b 1 ) a b a 1 b 1 we need to perform 3 multiplications and 4 additions instead of 4 multiplications and 1 addition. could be applied recursively to polynomials of degree 2 k -1 complexity O(n 1.59 ) Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

33 Karatsuba's algorithm Algorithm 11: Compute the full product of two polynomials A and B using the Karatsuba s multiplication algorithm Input: A : polynomial of degree at most n 1 with n = 2 k for k N Input: B : polynomial of degree at most n 1 Output: C : polynomial if n = 1 then return C AB C 1 A () B () by a recursive call C 2 A (1) B (1) by a recursive call C 3 A () + A (1) C 4 B () + B (1) C 5 C 3 C 4 by a recursive call C 6 C 5 C 1 C 2 C C 1 + C 6 X n/2 + C 2 X n return C Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

34 Univariate polynomials Truncated product (a + a 1 X a n X n + O(X n )) (b + b 1 X b n X n + O(X n )) = (c + c 1 X c n X n + O(X n )) Multivariate polynomials keep the term a i X d 1 1 Xd Xd n n bj X d 1 1 Xd Xd n n if d 1 + d 1 + d 2 + d d n + d n T if truncation is performed only on some variables, truncated variables must be ordered Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

35 Truncated product on polynomials stored as recursive list Function fmatruncated(a,b,c, T ) Compute the truncated fused multiplication-addition C = C + A B with A, B and C multivariate polynomials represented as recursive list Input: A: polynomial { list of ( coefficients a, degree δ a ) } Input: B: polynomial { list of ( coefficients b, degree δ b ) } Input: C: polynomial { list of ( coefficients c, degree δ c )} Input: T : degree of the truncation Output: C: polynomial { list of ( coefficients c, degree δ c )} if variable of A is truncated then iter head of C foreach element in A such that δ a T do /* avoid to scan to C when the loop on B is finished */ iterb iter foreach element in B such that δ a + δ b T do /* find after iterb in C if the degree δ a + δ b is present */ while current degree δ c referenced by iterb < δ a + δ b do iterb next element after iterb if δ c = δ a + δ b then fmatruncated (a, b, c, T δ a + δ b ) if c = then remove the element referenced by iterb else insert an element (multruncated (a, b, T δ a + δ b ), δ a + δ b ) just before iterb if current element is the first element of B then iter iterb else fmafull (A,B,C) Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

36 Truncated product on homogeneous blocks Let A = BH δ (a), B = BH δ (b) Truncated product on the total degree C = A B = T BH δ (c) with BH δ (c) = δ= δ BH n (a) BH δ n (b) n= use the full product of 2 homogeneous blocks use less addressing tables than for the full product Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

37 Truncated product - benchmarks 6 5 recursive vector recursive list flat vector BH BHC BHDAL BHDALC degree Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

38 Special truncated product in degree Perform the product of two Poisson series but we only want to keep terms which have specific values for k1 and k2 e.g., S 1 S 2 = S, we want only terms such that and S = a i X d 1 1 Xd Xd n n exp ı(k 1λ+k 2 λ ) k 1 = k 2 = very easy for the recursive representation if the series are correctly ordered. Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

39 Special truncated product on magnitude is kept if a i X d 1 1 Xd Xd n n b j X d 1 1 Xd Xd n n a i b j ɛ brut-force method Algorithm 1: Compute the truncated product of the series A and B in the amplitude of their coefficients Input: A: serie a i x i ordered by decreasing amplitude Input: B: serie b i x i ordered by decreasing amplitude Input: ɛ : thresold > Output: C: series C = AB with all coefficients greater than ɛ C create an empty polynomial foreach coefficient a i such that a i b ɛ do foreach coefficient b j such that b j ɛ / a i do C C + a i b j x i+j return C order the series on the magnitude of the coefficient? Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

40 Special truncated product on magnitude Let a variable ɛ, and a small parameter ɛ such that ɛ p = ɛ with p N. Each coefficient a i of the serie A(x) could be written as a i = a ix i ɛ k with k = log a i log ɛ and a i = a i ɛ k A (ɛ, x) = k ( j a jx j )ɛ k A(x) = A (ɛ, x) The truncated product A(x) B(x) on the amplitude of the coefficient is transformed to a truncated product A (ɛ, x) B (ɛ, x) on the variable ɛ. The degree of truncation is p. Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS

41 Special truncated product in amplitude source code s1 =(1+.5 x ) ˆ 3 ; s2 =(1+.4 x ) ˆ 4 ; / i n t r o d u c e the v a r i a b l e eps in s1 and s2 / s1e=s e r e p s ( s1, eps,. 1 ) ; s2e=s e r e p s ( s2, eps,. 1 ) ; / d e f i n e the t r uncature on amplitude to (. 1 ) ˆ 2 / t r =({eps, 2 } ) ; usetronc ( t r ) ; / perform the product / s3e=s1e s2e ; / remove the v a r i a b l e eps from s3e / s3=i n v s e r e p s ( s3e, eps,. 1 ) ; Execution of the previous source code by trip s1(x) = *x +.75*x** *x**3 s2(x) = *x +.96*x** *x** E-6*x**4 s1e(x,eps) = *x +.75*x**2*eps** *x**3*eps**3 s2e(x,eps) = *x +.96*x**2*eps** *x**3*eps** *x**4*eps**5 tr = ( { eps, 2 } ) s3e(x,eps) = *x +.24*x** *x**2*eps** *x**3*eps**2 s3(x) = *x +.411*x** *x**3 Table 4.6: Source code and execution by TRIP of the truncated product of two series Advanced School on Specific Algebraic Manipulators, 27, M. Gastineau, ASD/IMCCE/CNRS