2 The Truncated Fourier Transform and Applications The TFT permits to speed up the multiplication of univariate polynomials with a constant factor bet

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1 The Truncated Fourier Transform and Applications Joris van der Hoeven D pt. de Math matiques (B t. 425) Universit Paris-Sud Orsay Cedex France joris@texmacs.org January 9, 2004 In this paper, we present a truncated version of the classical Fast Fourier Transform. When applied to polynomial multiplication, this algorithm has the nice property of eliminating the jumps in the complexity at powers of two. When applied to the multiplication of multivariate polynomials or truncated multivariate power series, we gain a logarithmic factor with respect to the best previously known algorithms. Keywords: Fast Fourier Transform, jump phenomenon, truncated multiplication, FFT-multiplication, multivariate polynomials, multivariate power series. A.M.S. subject classification: 42-04, 68W25, 42B99, 30B10, 68W Introduction Let R 3 1/2 be an eective ring of constants (i.e. the usual arithmetic operations, and can be carried out by algorithm). If R has a primitive n-th root of unity with n = 2 p, then the product of two polynomials P Q 2 R[X] with deg P Q < n can be computed in time O(n log n) using the Fast Fourier Transform or FFT [CT65]. If R does not admit a primitive n-th root of unity, then one needs an additional overhead of O(log log n) in order to carry out the multiplication, by articially adding new root of unity [SS71, CK91]. Besides the fact that the asymptotic complexity of the FFT involves a large constant factor, another classical drawback is that the complexity function admits important jumps at each power of two. These jumps can be reduced by using (k 2 p )-th roots of unity for small k. They can also be smoothened by decomposing (n ) (n )-multiplications as n n-, n - and (n ) -multiplications. However, these tricks are not very elegant, cumbersome to implement, and they do not allow to completely eliminate the jump problem. In section 3, we present a new kind of Truncated Fourier Transform or TFT, which allows for the fast evaluation of a polynomial P 2 R[X] in any number n of well-chosen roots of unity. This algorithm coincides with the usual FFT if n is a power of two, but it behaves smoothly for intermediate values. In section 4, we also show that the inverse operation of interpolation can be carried out with the same complexity. 1

2 2 The Truncated Fourier Transform and Applications The TFT permits to speed up the multiplication of univariate polynomials with a constant factor between 1 and 2. In the case of multivariate polynomials, the repeated gain of such a constant factor leads to the gain of a non-trivial asymptotic factor. More precisely, assuming that R admits suciently 2 p -th roots of unity, we will show in section 5 that the product of two multivariate polynomials P Q 2 R[z 1 z d ] can be computed in time O(s log s), where s = and r = deg P Q 1. The best previously r d 1 d known algorithm [CKL89], based on sparse polynomial multiplication, has time complexity O(s log 2 s). In section 6 we nally give an algorithm for the multiplication of truncated multivariate power series. This algorithm, which has time complexity O(s log 2 s), again improves the best previously known algorithm [LS01] by a factor of O(log s). Moreover, both in the cases of multivariate polynomials and power series, we expect the corresponding constant factor to be much better. 2. The Fast Fourier Transform Let R be an eective ring of constants, n = 2 p with p 2N and! 2 R a primitive n-th root of unity (i.e.! n/2 = 1). The discrete Fast Fourier Transform (FFT) of an n-tuple (a 0 a n 1 ) 2 R n (with respect to!) is the n-tuple (a^0 a^n 1 ) = FFT! (a) 2 R n with n 1 a^i =X a j! ij : j=0 In other words, a^i = A(! i ), where A 2 R[X] denotes the polynomial A = a 0 a 1 X a n 1 X n 1. The F.F.T can be computed eciently using binary splitting: writing (a 0 a n 1 ) = (b 0 c 0 b n/2 1 c n/2 1 ) we recursively compute the Fourier transforms of (b 0 Then we have FFT! (a 0 FFT! 2(b 0 FFT! 2(c 0 b n/2 1 ) = (b^0 b^n/2 1) c n/2 1 ) = (c^0 c^n/2 1 ): b n/2 1 ) and (c 0 c n/2 1 ) a n 1 ) = (b^0 c^0 b^n/2 1 c^n/2 1! n/2 1 b^0 c^0 b^n/2 1 c^n/2 1! n/2 1 ): This algorithm requires n p = n log 2 n multiplications with powers of! and 2 n p additions (or subtractions). In practice, it is most ecient to implement an in-place variant of the above algorithm. We will denote by [i] p the bitwise mirror of i at length p (for instance, [11] 5 = 26). At step 0, we start with the vector x 0 = (x 00 x 0n 1 ) = (a 0 a n 1 ):

3 Joris van der Hoeven 3 At step s 2 f1 pg, we set x simj! = 1![i] sm x s(i1)mj 1! [i] sm! x s 1imj x s 1(i1)mj!: (1) for all i 2 f0 2 n/m 2g and j 2 f0 m 1g, where m = 2 p s. Using induction over s, it can easily be seen that for all i 2 f0 x simj = (FFT! m(a j a mj a n mj )) [i] s n/m 1g and j 2 f0 m 1g. In particular, x pi = a^[i] p a^i = x p[i] p for all i 2 f0 n 1g. This algorithm of repeated crossings is illustrated in gure 1. Figure 1. Schematic representation of a Fast Fourier Transform for n = 16. The black dots correspond to the xsi, the upper row being (x00 x015) = (a0 a15) and the lower row (x40 x415) = (a^0 a^8 a^4 a^12 a^15 ). A classical application of the FFT is the multiplication of polynomials A = a 0 a n 1 X n 1 and B = b 0 b n 1 X n 1. Assuming that deg A B < n, we rst evaluate A and B in 1!! n 1 using the FFT: (A(1) (B(1) A(! n 1 )) = FFT! (a 0 a n 1 ) B(! n 1 )) = FFT! (b 0 b n 1 ) We next compute the evaluations (A(1) B(1) A(! n 1 ) B(! n 1 ))) of A B at 1! n 1. We nally have to recover A B from these values using the inverse FFT. But the inverse FFT with respect to! is nothing else as 1/n times the direct FFT with respect to! 1. Indeed, for all (a 0 a n 1 ) 2 R n and all i 2 f0 n 1g, we have n 1 n 1 FFT! 1(FFT! (a)) i =X X a i! (i k)j = n a i (2) k=0 j =0 since n 1 X j=0! (i k)j = 0

4 4 The Truncated Fourier Transform and Applications whenever i k. This yields a multiplication algorithm of time complexity O(n log n) in R[X], when assuming that R admits enough primitive 2 p -th roots of unity. In the case that R does not, then new roots of unity can be added articially [SS71, CK91, vdh02] so as to yield an algorithm of time complexity O(n log n log log n). 3. The Truncated Fourier Transform The algorithm from the previous section has the disadvantage that n needs to be a power of two. If we want to multiply two polynomials A B 2 R[X] such that deg A B 1 = n, then we need to carry out the FFT at precision 2 n, thereby losing a factor of 2. This factor can be reduced using several tricks. For instance, one may decompose the (n ) (n )- product into an n n product, an n -product and an (n ) -product. This is ecient for small, but not very good if n/2. In the latter case, one may also use an FFT at precision 3 n/2, by using 3 3-matrices at one step of the FFT computation. However, all these tricks of the trade require a large amount of hacking and one always continues to lose a non-trivial factor between 1 and 2. The idea behind the Truncated Fourier Transform is to provide an ecient algorithm for the evaluation of polynomials in any number of distinct points. Moreover, the inverse operation of interpolation can be carried out with the same complexity. This technique will eliminate the jumps in the complexity of FFT multiplication. So let n = 2 p, l < n (usually, l>n/2) and let! be a primitive n-th root of unity. Given an l-tuple (a 0 a l 1 ), we will evaluate the corresponding polynomial A = a 0 a l 1 X l 1 in! [0]p! [1]p! [l 1] p. We call (A(! [0]p ) A(! [l 1]p )) the Truncated Fourier Transform (TFT) of (a 0 a l 1 ). Now consider the completion of the l-tuple (a 0 a l 1 ) into an n-tuple (a 0 a l 1 0 0). When using the in-place algorithm from the previous section in order to compute (A(! [0]p ) A(! [l 1]p )), we claim that many of the computations of the x si can actually be skipped (see gure 2). Indeed, at stage s, it suces to compute the vector (x s0 x sbl/mc(m1) 1 ). Besides x s0 x sl 1, we therefore compute at most m = 2 p s additional values. In total, we therefore compute at most p l 2 p 1 2 p 2 1 < p l n values x si. This proves the following result: Theorem 1. Let n = 2 p, l < n and let! 2 R be a primitive n-th root of unity in R. Then the Truncated Fourier Transform of an l-tuple (a 0 a l 1 ) w.r.t.! can be computed using at most l p n additions (or subtractions) and d(l p n)/2e multiplications with powers of!. Figure 2. Schematic representation of a TFT for n = 16 and l = 11.

5 Joris van der Hoeven 5 = ) Remark 2. Assume that R admits a privileged primitive n-th root of unity! n for every n 2 2 N 2, such that! 2n =!n for all n. Then the TFT (a^0 a^l 1 ) of an l-tuple (a 0 a l 1 ) w.r.t.! n with n > l does not depend on the choice of n. We call (a^0 a^l 1 ) the TFT of (a 0 a l 1 ) w.r.t. the privileged sequence (! 1! 2! 4 of roots of unity. Remark 3. Since the only operations we need for computing the TFT are additions, subtractions and multiplications by powers of!, the algorithm naturally combines with Sch nhage-strassen's algorithm when! is a symbolically added root of unity. Remark 4. If f 0 = f l 0 1, then the TFT of (f 0 f l 1 ) can be computed using O((l l 0 ) p 2 n) operations using a similar algorithm as above. More generally, this allows the rapid transformation of unions of segments. 4. Inverting the Truncated Fourier Transform Unfortunately, the inverse TFT cannot be computed using a similar formula as (2). Indeed, starting with the x li, we need to compute an increasing number of x si when s decreases. Therefore we will rather invert the algorithm which computes the TFT, but with this dierence that we will sometimes need x s 0 i 0 with s0 < s in order to compute x si. We will use the fact that whenever one value among x simj, x s 1imj and one value among x s(i1)mj, x s 1(i1)mj are known in the cross relation (1), then we can deduce the others from them using one multiplication by a power of! and two shifted additions or subtractions (i.e. the results may have to be divided by 2). More precisely, let us denote k s = bl/mc m and l s = k s m at each stage s. We use a recursive algorithm which takes the values x pks x pl 1 and x sl x sls on input, and which computes x sks x sl 1. If s = p, then we have nothing to do. Otherwise, we distinguish two cases: If l s = l s1, then we rst compute x sks x sks1 1 from x pks x pks1 1 using repeated crossings. We next deduce x si and x s1im/2 from x s1i and x sim/2 for all i 2 fl m/2 k s1 1g. Invoking our algorithm recursively, we now obtain x s1ks1 x s1l 1. We nally compute x si and x sim/2 from x s1i and x s1im/2 for i 2 fk s l m/2 1g. If l s > l s1, then we rst compute x s1i from x si and x sim/2 for i 2 fl l s1 1g. Invoking our algorithm recursively, we next compute x s1ks1 x s1l 1. We nally deduce x si from x s1i and x sim/2 for all i 2 fk s l 1g. The two cases are illustrated in gures 3 resp. 4. Since x 0l = = x 0n 1 = 0, the application of our algorithm for s = 0 computes the inverse TFT. We notice that the values x si with i < l are computed in decreasing order (for s) and the values x si with i > l in increasing order. In other words, the algorithm may be designed in such a way to remain in place. We have proved: Theorem 5. Let n = 2 p, l < n and let! 2 R be a primitive n-th root of unity in R. Then the l-tuple (a 0 a l 1 ) can be recovered from its Truncated Fourier Transform w.r.t.! using at most l p n shifted additions (or subtractions) and d(l p n)/2e multiplications with powers of!.

6 6 The Truncated Fourier Transform and Applications Figure 3. Schematic representation of the recursive computation of the inverse TFT for n = 16, l = 11. The dierent images show the progression of the known values xij (the black dots) during the dierent computations at stage s = 0. Since l0 = l1 = 16, we fall into the rst case of our algorithm and the recursive invocation of the algorithm is done between the third and the fourth image. Figure 4. Schematic representation of the recursive computation of the inverse TFT for n = 16, l = 11 at stage s = 1. Since l 1 = 16 and l 2 = 12, we now fall into the second case of our algorithm and the recursive invocation of the algorithm is done between the third and the fourth image. 5. Multiplying multivariate polynomials Let R be a ring with a privileged sequence (! 1! 2! 4 ) of roots of unity (see remark 2). Given a non-zero multivariate polynomial f = X f i1 i i 1 i d i d z 1 1 z in d > 1 variables, we dene the total degree of f by deg f = max fi 1 d i d 2 R[z 1 z d ] i d : f i1 i d 0g 2N We let deg 0 = 1. Now let f g 2 R[z 1 z d ] be such that deg f g < r. In this section we present an algorithm to compute f g, which has a good complexity in terms of the number s = r d 1 d

7 Joris van der Hoeven 7 TFT of expected coecients of f g. When computing f g using the classical FFT with respect to each of the variables z 1 z d, we need a time O(d (2 r) d log r)) which is much bigger than s, in general. When using multiplication of sparse polynomials [CKL89], we need a time O(s log 2 s) with a very large constant factor. Our algorithm is based on the TFT w.r.t. all variables and we will show that it has a complexity O(s log s). Given f 2 R[z 1 z d ] with deg f < r, the TFT of f with respect to one variable z v at order r is dened by [i TFT vr (f ) = f i1 i v 1 i v1 i d (z v =! v v ]n i 1 i d <r where n = 2 p > r. We recall that the result does not depend on the choice of n. The TFT with respect to all variables z 1 z d at order r is dened by [i TFT r (f ) = f (! 1 ]n [i n! d ]n n ) where n = 2 p > r (see gure 5). We have Given f g 2 C[z 1 TFT r (f ) = TFT dr ( i 1 i d <r 1r (f ) ): z d ] with deg f g < r, we will use the formula in order to compute the product f g. f g = TFT r 1 (TFTr (f ) TFT r (g)) f 05 f (1! 3 ) f 04 f 14 f (1! 5 ) f(! 4! 5 ) f 03 f 13 f 23 f (1! 6 ) f(! 4! 6 ) f(! 2! 6 ) f 02 f 12 f 22 f 32 f (1! 2 ) f(! 4! 2 ) f(! 2! 2 ) f(! 6! 2 ) f 01 f 11 f 21 f 31 f 41 f (1! 4 ) f(! 4! 4 ) f(! 2! 4 ) f(! 6! 4 ) f(! 5! 4 ) f 00 f 10 f 20 f 30 f 40 f 50 f(1 1) f(! 4 1) f (! 2 1) f (! 6 1) f (! 5 1) f(! 3 1) Figure 5. Illustration of the TFT in two variables (! =! 8 ). In order to compute TFT vr (f ), say for v = 1, we compute the TFT of (f 0i2 i d f l 1i2 i d ) with l = r i 2 i r for all i 2 i d with i 2 i n < r 1 (if i 2 i n = r 1, then the TFT of (f 0i2 i d ) is given by itself, so we have nothing to do). One such computation takes a time 6C l log l for some universal constant C, by using the TFT w.r.t.! n with minimal n = 2 p > l (so n may vary as a function of i 2 i d, but not C). The computation of TFT r (f ) therefore takes a time T dr with r T dr 6 C dx r l d 2 l log l: Dividing by s, we obtain T dr s r 6 C d 2 (d 1)X Xr 6 C d 3 l=2 l=2 l=2 d 2 (r d l 2)! (r d 1)! (r d l)! (r 1)! (r d 1)! (r l)! (r 1)! (r l)! l log l l log l (r l d) (r l d 1) (3)

8 8 The Truncated Fourier Transform and Applications TFT TFT If r 6 d, then the summand rapidly deceases when l > 2, so that T dr s = r O(d3 ) = O(r) = O(log s): (r d) 3 Consequently, T dr = O(s log s) and even T dr = O(s) for xed r. If r > d, then for d = " r and l = r, Stirling's formula yields log (r d l)! (r 1)! (r d 1)! (r l)! = " r : It follows that only the rst O(r/d) = O(1/") terms in (3) contribute to the asymptotic behaviour of T dr, so that s T dr = O(d 3 1 " log (1/") " r 2 ) = O(d log (r/d)) = O(log s): Again, we nd that T dr = O(s log s). We have proved: Theorem 6. Let R be a ring with a privileged sequence (! 1! 2! 4 Let f g 2 R[z 1 z d ] be polynomials with deg f deg g < r and let s = product f g can be computed using O(s log s) operations in R. ) of roots of unity.. Then the r d 1 r 6. Multiplying multivariate power series Since power series have innitely many terms, implementing an operation on power series really corresponds to implementing the operation for polynomial approximations at all degrees. As P usual, multiplication is a particularly important operation. Given f g 2 C[z 1 z d ] with deg f < r and deg g < r, we will show how to compute the truncated product h = i 1 i d <d (f g) i i 1 i d z 1 i 1 z d d 2 R[z 1 z d ] of f and g. The rst idea [LS01] is to use homogeneous coordinates instead of the usual ones: f ~ (z 1 z 2 g~(z 1 z 2 z d ) = f (z 1 z 1 z 2 z 1 z d ) z d ) = g(z 1 z 1 z 2 z 1 z d ): This transformation takes no time since it corresponds to some re-indexing. We next compute the TFTs f^ and g^ in z 2 z d at order r: f^ = TFT dr ( g^ = TFT dr ( 2r (f ~ ) ) 2r (g~) ): We next compute the s 0 r d 2 = d 1 truncated products h^i2 i d (z 1 ) of the obtained polynomials f^i 2 i d (z 1 ) and g^i2 i d (z 1 ). After transforming the results of these multiplication back using h ~ 1 = TFT 2r( we obtain the truncated product h of f and g by h(z 1 z 2 1 TFT dr(h^) ) z d ) = h ~ (z 1 z 2 /z 1 z d /z 1 ): The total computation time is bounded by O(r s 0 log s 0 r s 0 log r). Using the fact that r s 0 = O(s log s), we have proved the following theorem: Theorem 7. Let R be a ring with a privileged sequence (! 1! 2! 4 ) of roots of unity. Let f g 2 R[z 1 z d ] be polynomials of degrees <r and let s =. Then the truncated r d 1 r product of f and g at degree <r can be computed using O(s log 2 s) operations in R.

9 Joris van der Hoeven 9 Remark 8. In practice, if the coecients f i1 i d have dierent growths in i 1 i d, then it may be useful to consider truncations along more general degrees of the form deg f = max f 1 i 1 d i d : f i1 i d 0g: The slicing technique from section in [vdh02] may then be used in order to obtain complexity bounds of the same type. Remark 9. Using remark 4, the polynomial and truncated multiplication algorithms can be used in combination with the strategy of relaxed evaluation [vdh97, vdh02, vdh03b] for solving partial dierential equations in multivariate power series with an additional overhead of O(log r). A recent technique [vdh03a] allows to reduce this overhead even further and it would be interesting to study more precisely what happens in the multivariate case. Bibliography [CK91] D.G. Cantor and E. Kaltofen. On fast multiplication of polynomials over arbitrary algebras. Acta Informatica, 28:693701, [CKL89] J. Canny, E. Kaltofen, and Y. Lakshman. Solving systems of non-linear polynomial equations faster. In Proc. ISSAC '89, pages , Portland, Oregon, A.C.M., New York, ACM Press. [CT65] J.W. Cooley and J.W. Tukey. An algorithm for the machine calculation of complex Fourier series. Math. Computat., 19:297301, [HQZ00] Guillaume Hanrot, Michel Quercia, and Paul Zimmermann. Speeding up the division and square root of power series. Research Report 3973, INRIA, July Available from [HQZ02] Guillaume Hanrot, Michel Quercia, and Paul Zimmermann. The middle product algorithm I. speeding up the division and square root of power series. Submitted, [HZ02] Guillaume Hanrot and Paul Zimmermann. A long note on Mulders' short product. Research Report 4654, INRIA, December Available from [LS01] G. Lecerf and. Schost. Fast multivariate power series multiplication in characteristic zero. Technical Report , GAGE, cole polytechnique, Palaiseau, France, [SS71] A. Sch nhage and V. Strassen. Schnelle Multiplikation grosser Zahlen. Computing 7, 7: , [vdh97] J. van der Hoeven. Lazy multiplication of formal power series. In W. W. K chlin, editor, Proc. ISSAC '97, pages 1720, Maui, Hawaii, July [vdh02] Joris van der Hoeven. Relax, but don't be too lazy. JSC, 34:479542, [vdh03a] Joris van der Hoeven. New algorithms for relaxed multiplication. Technical Report , Univ. d'orsay, [vdh03b] Joris van der Hoeven. Relaxed multiplication using the middle product. In Manual Bronstein, editor, Proc. ISSAC '03, pages , Philadelphia, USA, August 2003.