Relaxed Multiplication Using the Middle Product

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1 Relaxed Multiplication Usin the Middle Product Joris van der Hoeven Département de Mathématiques (bât. 425) Université Paris-Sud Orsay Cedex France ABSTRACT In previous work, we have introduced the technique o relaxed power series computations. With this technique, it is possible to solve implicit equations almost as quickly as doin the operations which occur in the implicit equation. In this paper, we present a new relaxed multiplication alorithm or the resolution o linear equations. The alorithm has the same asymptotic time complexity as our previous alorithms, but we improve the space overhead in the divide and conquer model and the constant actor in the F.F.T. model. Cateories and Subject Descriptors G. Mathematics o Computin [G.1. Numerical Analysis]: G.1.0. General Multiple Precision Arithmetic General Terms Alorithms Keywords power series, relaxed multiplication, middle product 1. INTRODUCTION Let R be an eective rin and consider two power series = 0 + 1z + and = 0 + 1z + in R[[z]]. In this paper we will be concerned with the eicient computation o the irst n coeicients o the product h = = h 0+h 1z+. I the irst n coeicients o and are known beorehand, then we may use any ast multiplication or polynomials in order to achieve this oal, such as divide and conquer multiplication [6, 7], which has a time complexity K(n) = O(n lo 3/ lo 2 ), or F.F.T. multiplication [2, 9, 1, 11], which has a time complexity M(n) =O(n lo n lo lo n). For simplicity, time complexity stands or the required number o operations in R. Similarly, space complexity will stand or the number o elements o R which need to This paper is in the public domain. Permission is ranted to reproduce it on all media, either in its entirety or in any derived orm. ISSAC 03, Auust 3 6, 2003, Philadelphia, Pennsylvania, USA.. be stored. The required number o multiplications K (n) in the divide and conquer alorithm satisies the ollowin recurrence relations: K (1) = 1 K (n) = 2K ( n/2 )+K ( n/2 ) When perormin computin only the product truncated at order n, then the number o multiplications K needed by the divide and conquer alorithm becomes K (1) = 1 K (n) = K ( n/2 )+2K ( n/2 ) For certain computations, and most importantly the resolution o implicit equations, it is interestin to have so called relaxed alorithms which output the irst i coeicients o h as soon as the irst i coeicients o and are known or each i 6 n. This allows or instance the computation o the exponential =exp o a series with 0 = 0 usin the ormula =. (1) In [10, 11], we proved the ollowin two theorems: Theorem 1. There exists a relaxed multiplication alorithm o time complexity K(n) and space complexity O(lo n), andwhichusesk (n) multiplications. Theorem 2. There exists a relaxed multiplication alorithm o time complexity O(M(n)lon) and space complexity O(n). Althouh these theorems are satisactory rom a theoretical point o view, they can be improved in two directions: by removin the loarithmic space overhead in the divide and conquer model and by improvin the constant actor in the F.F.T. model. In this paper, we will present such an improved alorithm in the case o relaxed multiplication with a ixed series. More precisely, let and be power series, such that is known up to order n. Then our alorithm will compute the product h = up to order n and output () i as soon as 0,..., i are known, or all i<n. We will prove the ollowin: Theorem 3. There exists a relaxed multiplication alorithm with ixed series o time complexity O(K(n)), ospace complexity O(n), andwhichusesk (n) multiplications.

2 Fiure 1: Illustration o the middle product. Fiure 2: The subdivision used or the new relaxed multiplication alorithm. γ = 0( 1 + 2) h 0 = α β h 1 = γ + β This trick may be applied recursively in order to yield an alorithm which needs exactly the same number o multiplications K (n) as the divide and conquer alorithm or the computation o the product o two polynomials o de We also obtain a better constant actor in the asymptotic complexity in the F.F.T. model, but this result is harder to stateinaprecisetheorem. The alorithm is useul or the relaxed resolution o linear dierential or dierence equations. For instance, the exponential o a series can be computed usin 6 K (n) multiplications in R. Moreover, the new alorithm is very simple to implement, so it is likely to require less overhead than the alorithm rom theorem 1. Our alorithm is based on the recent middle product alorithm [3, 4], which is recalled in section 2. In section 3 we present our new alorithm and in section 5 we ive some applications. In our alorithms we will use the ollowin notations: the data type TPS(n) stands or truncated power series o order n, like = n 1z n 1. Given TPS(n) and 0 6 i<j6 n, we will denote i...j = i + + j 1z j i 1 TPS(j i). Given TPS(m) and TPS(n), we also denote 1 = + z m TPS(m + n). We will denote by Re(TPS(n)) the type, whose elements are reerences to elements o type TPS(n). I TPS(n) and06 i<j6 n, then we assume that i...j Re(TPS(n)). 2. THE MIDDLE PRODUCT Let = n 1z n 1 and = n 2z 2n 2 be two truncated power series at orders n resp. 2n 1. The middle product and is deined to be the truncated power series P h = = h h n 1z n 1 o order n, such that n 1 h i = j=0 jn 1+i j or all i {0,...,n 1}. In iure 1, h corresponds to the colored reion. The middle product o = 0+ 1z and = 0+ 1z+ 2z 2 can be computed usin only three multiplications, usin the ollowin trick: α = 1( 0 + 1) β = ( 1 0) 1 ree n 1. More precisely, the ollowin recursive alorithm comes rom [3, 4]. Alorithm Input: TPS(n) and TPS(2n 1) Output: their middle product TPS(n) i n =1then return 0 0 k := n/2,l:= n/2 α := k...n ( 0...2l 1 + l...3l 1 ) i n is even then β := ( l...n 0...k ) l...3l 1 else β := [ k 1 ( l...n 0...k )] l...3l 1 γ := 0...k ( l...l+2k 1 + 2l...2n 1 ) return (α β) 1 (γ + β 0...k ) In [4] it is also shown that, in the F.F.T. model, the middle product can still be computed in essentially the same time as the product o two polynomials. 3. RELAXED MULTIPLICATION WITH A FIXED SERIES Let and be power series, such that is known up to order n. In this section, we present an alorithm which computes the product h = up to order n. Foreachi<n, the alorithm outputs () i as soon as 0,..., i are known. The idea o our alorithm is similar to the idea behind ast relaxed multiplication in [10, 11] and based on a subdivision o the trianular area which corresponds to the computation o the truncated power series product. This subdivision is shown in iures 2 and 3, where each paralleloram corresponds to the computation o a middle product. More precisely, let l = n/2 and assume that 0,..., l 1 are known. Then the contribution o 0...l n+1 2l...n to may be computed usin the middle product alorithm rom the previous section. The relaxed truncated products 0...k 0...k and l...n 0...k may be computed recursively. In order to implement this idea, we will use an in-place alorithm, which adds the result o h = to a reerence ϕ to an element o TPS(n). Denote by ϕ init the initial value o ϕ. Then the in-place alorithm should be called successively or i =1,...,n. Ater the last call, we have ϕ = ϕ init + h. Takin ϕ init = 0, the alorithm computes h.

3 Alorithm relaxed-muladd(,,ϕ,i) Input:, TPS(n), ϕ Re(TPS(n)), i 6 n. Action: we have ϕ 0...i = ϕ init i 0...i on exit. i i = n =1then ϕ 0 += 0 0 and return k := n/2,l:= n/2 i i 6 k then relaxed-muladd( 0...k, 0...k,ϕ 0...k,i) i i = k +1then ϕ k...n += 0...l n+1 2l...n i i>lthen relaxed-muladd( l...n, 0...k,ϕ l...n,i l) The number o multiplications R (n) used by relaxedmuladd is determined by the relations R (1) = 1 ; R (n) = K ( n/2 )+2R ( n/2 ). By induction, it ollows that R (n) =K (n). The overall time complexity satisies R(n) 6 K( n/2 ) +2R( n/2 ) +O(n), so R(n) = O(K(n)). The alorithm bein in-place, its space complexity is clearly O(n). This proves theorem 3. In it is also interestin to use the above alorithm in the F.F.T. model. We then have the estimation R(n) 6 M( n/2 ) +2R( n/2 ) +O(n) or the asymptotic complexity R(n). I M(n) cn lo n lo lo n, this yields R(n) 1 2 M(n)lo 2 n. This should be compared with the complexity R(n) M(n)lo 2 n o the previously best alorithm and with the complexity L(n) 2M(n)lo 2 n o the standard ast relaxed multiplication alorithm. Notice that we rarely obtain the complexity M(n) cn lo n lo lo n in practice. In the rane where M(n) cn α,weobtain R(n) 1 2 α 2 M(n). 4. A WORKED EXAMPLE Let us consider the computation o =e z/(1 z) up till order 7 = n + 1 usin our alorithm and the ormula =, with 0 =1and =1+2z +3z 2 +. We start with ϕ =0 in relaxed-muladd and perorm the ollowin computations at successive calls or i =1,...,6: 1. We set ϕ 0 += 0 0 =1,sothat and 1 =1. ϕ := 1 2. We recursively apply relaxed-muladd to 0...3, 0...3, ϕ and i = 2. This requires the computation o Fiure 3: Illustration o an order 6 relaxed multiplication =(1+z) (1 + 2z +3z 2 )=3+5z. We thus increase ϕ = 3 + 5z, sothat and 2 = 3 2. ϕ := 1 + 3z +5z 2 3. The two nested recursive calls to relaxed-muladd now lead to the increase o ϕ 2 by 2 0 = 3 2,sothat and 3 = ϕ := 1 + 3z z2 4. We now both have i = k +1=4 andi>l=3. Sowe irst compute = z +17z2 and set ϕ = z +17z2. We next recursively apply relaxed-muladd to 3...6, 0...3, ϕ and i =1,which leads to an increase o ϕ 3 by 3 0 = Alltoether, we obtain ϕ := 1 + 3z z z z4 +17z 5 and 4 = We recursively apply relaxed-muladd to 3...6, 0...3, ϕ and i = 2. This leads to the increase ϕ = = z,sothat 8 12 ϕ := 1 + 3z z z z z5 and 5 = The two nested recursive calls lead to the increase ϕ 5 += 5 0 = 167,sothat 40 ϕ := 1 + 3z z z z z5 and 6 = The entire computation is represented schematically in iure 3.

4 5. APPLICATIONS First o all, let us consider the problem o relaxed division by a ixed power series. In other words, we are iven two power series and, where is known up to order n and 0 = 1. We want an alorithm or the computation o h = / up to order n, such that h i is computed as soon as 0,..., i are known or each i<n.nowwehave h = z(ϕh), where ϕ =( 1)/z R[[z]]. We may thus compute h in a relaxed way usin the alorithm rom the previous section. Computin h up till n terms will then necessitate 6 K (n) multiplications in R. Let us next consider a linear dierential equation L r (r) + + L 0 =0, (2) with L 0,...,L r R[[z]] and L r(0) = 1. Given initial conditions or 0,..., r 1, there exists a unique solution to this equation. We may compute this solution usin the relaxed alorithm rom the previous section, the above alorithm or relaxed division, and the ormula = L 1 r... r (L r (r) + + L 0). In order to compute n coeicients, we need to perorm (r + 1)K (n) multiplications in R and O(n) multiplications and divisions by inteers. I L r = 1, then we only need rk (n) multiplications. For instance, the exponential o a series with 0 =0 satisies the equation =0, so can be computed usin K (n) multiplications, usin the ormula (1). More enerally, consider the solution to (2) with the prescribed initial conditions, and let be another series with 0 = 0. Then the composition h = aain satisies a linear dierential equation. Indeed, we have the relations = h = h = h h 2 3. Postcomposin (2) with and usin these relations, we obtain a linear dierential equation or h. In act, our alorithm may be used to solve ar more eneral linear equations, such as linear partial dierential equations, or linear dierential-dierence equations. In the case o dierence equations, we notice that the relaxed multiplications in the alorithms rom [11] or relaxed riht composition with a ixed series all have one ixed arument. So we may indeed apply the alorithm rom section 3. We inally notice that our alorithm can even be used in a non-linear context. Indeed, ater computin n/2 coeicients o a truncated relaxed product, the computation o the remainin products reduces to the computation o two truncated relaxed products with one ixed arument. Actually, this corresponds to an implicit application o Newton s method. Fiure 4: Usin Mulders trick in combination with the middle product. 6. CONCLUSION AND OPEN QUESTIONS We have presented a new alorithm or relaxed multiplication. Althouh the new alorithm does not yield a siniicant improvement rom the asymptotic complexity point o view, we do expect it to be very useul or practical applications, such as the exponentiation o power series. First o all, the alorithm is easy to implement. Secondly, it only needs a linear amount o memory in the rane where divide and conquer multiplication is appropriate. In combination with F.F.T. multiplication, the alorithm yields a better constant actor in the asymptotic complexity. When implementin a library or power series computations, it is interestin to incorporate a mechanism to automatically detect relaxed and ixed multiplicands in a complex computation. This is possible by examinin the dependency raph. With such a mechanism, one may use the new alorithm whenever possible. Some interestin questions remain open in the divide and conquer model: can we apply Mulders trick [8, 5] or the computation o short products in our settin while maintainin the linear space complexity (see iure 4)? In that case, we miht improve the number o multiplications in theorem 3 to K(n). In a similar vein, does there exist a relaxed multiplication alorithm o time complexity 6 K(n) and linear space complexity? This would be so, i the middle product alorithm could be made relaxed in an in-place way (the alorithm is already essentially relaxed in the sense o [10, 11] in the divide and conquer model). As it stands now, with the above questions still unanswered, the oriinal relaxed multiplication alorithm rom theorem 1 remains best rom the time complexity point o view in the divide and conquer model. Moreover, Mulders trick can be applied in this settin, so as to yield a short relaxed multiplication alorithm o complexity K(n), or even better [5]. This has surprisin consequences or the complexities o several operations like short division and square roots: we obtain alorithms o time complexities K(n) and 1 K(n) when usin O(n lo n) space, while the best known 2 alorithms which use linear space have time complexities K(n) and 3 K(n). In order to obtain the complexity 4 o 1 K(n) in the case o square roots, one should use 2

5 a relaxed version o the ast squarin alorithm rom [4], which is based on middle products. We inally remark that this relaxed version o squarin usin middle products is also interestin in the F.F.T. model. In this case, the relaxed middle product corresponds to a ull relaxed product with one ixed arument. Such products can be computed in time 2R(n), so that we obtain a relaxed squarin alorithm o time complexity 2R(n). This is twice as ood as eneral relaxed multiplication. In the non-relaxed settin, squares can be computed in a time between 1 M(n) and 2 M(n), dependin on whether most 2 3 time is spent on inner multiplications or ast Fourier transorms respectively. 7. REFERENCES [1] Cantor, D., and Kaltoen, E. On ast multiplication o polynomials over arbitrary alebras. Acta Inormatica 28 (1991), [2] Cooley, J., and Tukey, J. An alorithm or the machine calculation o complex Fourier series. Math. Computat. 19 (1965), [3] Hanrot, G., Quercia, M., and immermann, P. Speedin up the division and square root o power series. Research Report 3973, INRIA, July Available rom [4] Hanrot, G., Quercia, M., and immermann, P. The middle product alorithm I. speedin up the division and square root o power series. Submitted, [5] Hanrot, G., and immermann, P. A lon note on mulders short product. Research Report 4654, INRIA, Dec Available rom [6] Karatsuba, A., and Oman, J. Multiplication o multidiit numbers on automata. Soviet Physics Doklady 7 (1963), [7] Knuth, D. The Art o Computer Prorammin, 3-rd ed., vol. 2: Seminumerical Alorithms. Addison-Wesley, [8] Mulders, T. On short multiplication and division. AAECC 11, 1 (2000), [9] Schönhae, A., and Strassen, V. Schnelle Multiplikation rosser ahlen. Computin 7 7 (1971), [10] van der Hoeven, J. Lazy multiplication o ormal power series. In Proc. ISSAC 97 (Maui, Hawaii, July 1997), W. W. Küchlin, Ed., pp [11] van der Hoeven, J. Relax, but don t be too lazy. JSC 34 (2002),