A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS

Transcription

1 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS GREGORIO MALAJOVICH AND JORGE P ZUBELLI Abstract A new algorithm for splitting polynomials is presented This algorithm requires O(d log ɛ ) +δ floating point operations, with O(log ɛ ) +δ bits of precision As far as complexity is concerned, this is the fastest algorithm known by the authors for that problem An important application of the method is factorizing polynomials or polynomial root-finding Introduction The main motivation of this paper is to develop fast algorithms for solving univariate polynomials Although this is a rather old subject, important progress has taken place in the last years (See the review by Pan []) The problem of solving univariate polynomials may be reduced to factorization Once a factorization f = gh is known, factors g and h may be factorized recursively, until one obtains degree or 2 factors Fast algorithms for factorizing a degree d polynomial may proceed by performing a change of coordinates and a small number (say O(log log d)) iterations of Graeffe s transformation Graeffe s transformation (also Date: January 22, 996 Key words and phrases Polynomial equations, Factorization, Splitting Sponsored by CNPq (Brasil) Typeset using AMS-L A TEX

2 2 GREGORIO MALAJOVICH AND JORGE P ZUBELLI studied by Dandelin and by Lobatchevski) was developed and extensively used before the advent of digital computers See Ostrowski [2] and Uspensky [3, p38-33] For more recent applications, see Schönhage [4], Kirrinis [5] and Pan [6, 7] Graeffe s transformation replaces a polynomial f(x) by the polynomial Gf(x) = ( ) d f( x)f( x) The roots of the new polynomial are the square of the roots of the old polynomial The effect of Graeffe s iteration is to pack the roots into clusters near zero and infinity, leaving a wide enough root-free annulus The next (and crucial) step is to factorize this new polynomial into factors having all the roots close to zero (resp infinity) This is called splitting Finally, it is necessary to return from Graeffe s iteration Explicitly, if Gf(x) = Gg(x)Gh(x), one may set g(x) = gcd(gg(x 2 ), f(x)) and h(x) = f(x)/g(x) Known splitting algorithms require O(d +δ ) arithmetic operations, with O((d(d log ɛ)) +δ ) bits of precision They produce approximate factors g and h such that f gh 2 < ɛ We will prove instead: Main Theorem Let R > 64(a+b) 3, a, b N Let f be a polynomial of degree d = a + b, such that a roots are contained inside the disk ζ < R and b roots are contained outside the disk ζ > R Then, for ɛ = o(/d), an ɛ-approximation of the factors g and h of f may be computed within O(d log ɛ log log ɛ ) floating-point arithmetic operations, performed with: O(log ɛ ) bits of precision

3 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 3 The algorithm we propose in this paper requires a more moderate precision than classical ones Therefore, its bit complexity is significantly lower Moreover, we define the ɛ-approximation in order to guarantee the stronger result (2 a i g i g i ) 2 ɛ (2i h i h i ) 2 ɛ where f = g h is the exact factorization The price to pay is a larger splitting radius (meaning O(log log d) extra Graeffe s iterations) It is assumed that f is given as a vector of floating point numbers The output is also given as a vector of floating point numbers, and we assume as a model of computation correctly rounded floating-point arithmetic 2 Sketch of the proof 2 General Idea We will show that factoring a polynomial f is equivalent to solving a certain system of polynomial equations Under the hypotheses of the Main Theorem, this system will be solvable by Newton iteration It turns out that, with a proper choice of the initial guess, one has quadratic convergence since the first iteration 22 Background of α-theory Let ϕ : C n C n be a system of polynomials The Newton operator N ϕ is defined by N ϕ : C n C n x x Dϕ(x) ϕ(x)

4 4 GREGORIO MALAJOVICH AND JORGE P ZUBELLI The following invariants were introduced by Smale in [8]: β(ϕ, x) = Dϕ(x) ϕ(x) 2 γ(ϕ, x) = max k 2 α(ϕ, x) = β(ϕ, x)γ(ϕ, x) Dϕ(x) D k ϕ(x) 2 k! k It was proven in [8] that if α(ϕ, x) < α 0 <, then the sequence 7 (x i ) converges quadratically to a root of ϕ See also [9, 0,, 2, 3] This theorem can be generalized to an approximation of the Newton operator We shall need that generalization in the sequel The space H d is the space of all systems of homogeneous polynomials of degree d = (d,, d n ) In order to work with non-homogeneous polynomials, we may set one coordinate of the variable z = (z 0,, z n ) to be equal to (Say z 0 = ) Then α aff, β aff and γ aff below are precisely α, β and γ defined above Under this notation, we have: Theorem (Malajovich [4], Theorem 2) Let f H d, z (0) C n+, the first coordinate of z (0) be non-zero, and δ 0 be such that: (β aff (f, z (0) )+ δ)γ aff (f, z (0) ) < /6, and γ aff (f, z (0) )δ < /384 Suppose that the sequence (z (i) ), where the first coordinates of z (i) and z (0) are equal, satisfies z (i+) N aff (f, z (i) ) 2 z (i) 2 δ Then, there is a zero ζ of f such that d proj (z (i), ζ) max ( 2 2i, 6δ )

5 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 5 Notice that above we have z (i) 2 2 = z (i) z (i) n 2 = + z (i) z (i) n 2 23 Polynomial systems associated to splitting Let a and b be fixed We want to split a polynomial f of degree d = a + b into factors g and h of degree a and b, respectively This means that we want to factor f = gh, so that the roots of g are inside the disk D(R ) and the roots of h are outside the disk D(R) For convenience, we choose f a = Polynomials with this property shall be called hemimonic We want to solve the system ϕ f (g, h) def = gh f = 0 where g is monic of degree a In vector notation, ϕ f (g, h) = g 0 h 0 f 0 g h 0 + g 0 h f h b + g a h b f d h b f d The system ϕ f (g, h) is a system of d+ = a+b+ non-homogeneous polynomial equations in d + variables In order to simplify the exposition, we shall assume a b The case b < a is similar, mutatis mutandis

6 6 GREGORIO MALAJOVICH AND JORGE P ZUBELLI The derivative of ϕ f is given by Dϕ f (g, h) = h 0 g 0 h h 0 g g 0 h b g0 g h 0 g a h h b g a 0 The second derivative D 2 ϕ f (g, h) is a bilinear operator from C d+ C d+ into C d+ The i-th coordinate of D 2 ϕ f (g, h) can be represented by its Hessian matrix, with ones concentrated along one anti-diagonal For instance, if a = b + and i = b, ( D 2 ϕ f (g, h) ) i = A good metric In this section we shall introduce a few norms that will play an important role in the sequel In some sense, those seem to be the good norms to use for the splitting problem Those will

7 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 7 be the norms referred to in the definition of α, β, γ and in Theorem They correspond to a scaling of the system ϕ f Before doing that we start with some intuitive motivation for such norms First, let s consider the case of a polynomial with roots close to 0, say for example p(x) = x n + t n x n + t 0, with sufficiently small t n and t 0 Then, changes in t 0 will affect much more the roots than changes in t n This lead us to consider a norm that for i > j emphasizes the contribution of the coefficient of x j more than that of x i A similar reasoning holds for polynomials whose roots are close to except that more emphasis has to be put on the higher degree coefficients The problem we have at hand is that of splitting a polynomial f of degree a + b into two factors g and h of degree a and b, respectively, so that the roots of g are close to 0 and the roots of h are close to Therefore, it is natural to consider a norm that captures the relative weight of the different coefficients as far as such coefficients affect the roots Compare with definition (692) in Ostrowski [2], page 209 To make the above heuristics more precise we introduce the following concepts: The polynomial will be called antimonic if h 0 = The polynomial b h(x) = h n x n n=0 f(x) = a+b n=0 f n x n

8 8 GREGORIO MALAJOVICH AND JORGE P ZUBELLI will be called hemimonic (wrt the splitting into factors of degrees a and b) if f a = Definition For g a degree a polynomial we set the monic norm g m = (2 a i g i ) 2 0 i a For h a degree b polynomial, the antimonic norm is defined by h a = (2 i h i ) 2 0 i b For ϕ a polynomial of degree a + b, we set the hemimonic norm with respect to the splitting (a, b) as ϕ h = (2 a i ϕ i ) 2 0 i a+b Notice that if g is a monic polynomial with g x a m sufficiently small, then all its roots are close to 0 Similarly, if h is antimonic with h x b a small, then all its roots are next to As to the concept of (a, b)-hemimonic, the point is that if ϕ belongs to this class, then ϕ x a h small implies that a roots of ϕ are close to 0 and b roots are close to Although the preceding remarks are true for all norms, the definitions above give sharper estimates than the usual 2-norm In order to make the distinction of the norm under consideration g monic more apparent we will try to use the letters h to denote antimonic ϕ hemimonic a polynomials of degree b, respectively a + b

9 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 9 We may estimate the norm of the operator D 2 ϕ f (g, h) : (C a+b+, ma ) (C a+b+, ma ) (C a+b+, h ), () where the norm (g, h) ma def = g m 2 + h a 2, by the following Lemma D 2 ϕ f (g, h) ma h d + 4 The proof of this Lemma is postponed to Section 8 We consider the Newton operator applied to the system ϕ f (g, h) N(f; g, h) = g Dϕ f (g, h) ϕ f (g, h) h Lemma provides the following estimate for Smale s invariants: γ(ϕ f ; g, h) = Dϕ f(g, h) D 2 ϕ f (g, h) ma h 2 Dϕf (g, h) ma h d + 8 β(ϕ f ; g, h) Dϕf (g, h) ma ϕ f (g, h) h h d + α(ϕ f ; g, h) Dϕ f (g, h) 2 ma ϕf (g, h) 8 h h 25 A good starting point In this section, we shall prove that a good starting point for the Newton iteration is g(x) = x a, h(x) = This choice makes the matrix Dϕ f (g, h) equal to the identity This implies

10 0 GREGORIO MALAJOVICH AND JORGE P ZUBELLI Lemma 2 Assume that g = x d and h = Then, Dϕ f (x a, ) ma h = As we are using our non-standard norms, the proof will be postponed to Section 8 We have: α(ϕ f ; g, h) = a + b + ϕ f (g, h) 8 h a + b + f x a 8 h β(ϕ f ; g, h) f x a h a + b + γ(ϕ f ; g, h) 4 Lemma 3 Let s assume that f and ɛ satisfy and ɛ < f x a h < 6 a + b a + b + Then, if the sequences g (k) and h (k) satisfy with (g (0), h (0) ) = (x d, ), (g (i+), h (i+) ) N(ϕ f ; g (i), h (i) ) ma (g (i), h (i) ) ma ɛ 6 there exist g and h such that f = g h, and for we have min λ 0 k > log 2 log 2 ɛ, ma ɛ (g (k), h (k) ) ma (g (k), h (k) ) λ( g, h)

11 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS Proof of Lemma 3: We are under the hypotheses of the Theorem, where we set δ = ɛ/6 Indeed, a + b + (β(ϕ f, (g (0), h (0) )) + δ)γ ( f x a h + δ) 4 4 ( 7 a + b + + ɛ a + b + 6 ) ɛ a + b Furthermore, γ(f; (g (0), h (0) ))δ < a + b a + b + = 384 Therefore, if k > log 2 log 2 (/ ɛ), the final (g (k), h (k) ) are within (projective) distance ɛ from the true factors ( g, h) Moreover, it can be seen that the usual distance between (g (k), h (k) ) and ( g, h) is bounded by 4 ɛ Indeed, (g (k), h (k) ) ma g a Therefore, (g (k), h (k) ) λ( g, h) ɛ ma Since g (k) and g are monic, we should have ɛ λ + ɛ So, (g (k), h (k) ) ( g, h) ɛ( + ( g, h) ) ma ma We will show later the following bound:

12 2 GREGORIO MALAJOVICH AND JORGE P ZUBELLI Lemma 4 Let R > 4(a + b) 2 Let g be monic of degree a, with all roots in D(R ) Let h be of degree b, with all roots outside D(R) Let ϕ = gh be hemimonic Then (g, h) ma 3 Hence, assuming the conditions of Lemmas 3 and 4, we obtain (g (k), h (k) ) ( g, h) 4 ɛ ma 26 End of the proof In order to prove the Main Theorem, we have to show that hemimonic polynomials with a large enough splitting annulus have small hemimonic norm More precisely, Theorem 2 Let ϕ be an (a, b)-hemimonic polynomial, with a roots in the disk D(R ), and b roots outside the disk D(R) If R > 2 max(a, b) then: ϕ x a h, < max(a, b) R In particular, let R > 64(a + b) 3 Then, ϕ x a h, < ( 2 max(a,b) R 3 64(a + b) 2 ) 2 The proof will be given in section 3 Theorem 2 shows that the conditions of the Main Theorem imply those of Lemma 3, and it remains to produce the sequence (g (i), h (i) ) Theorem 3 Assume the hypotheses of the Main Theorem There is an algorithm to compute (g (i+), h (i+) ) of Lemma 3 out of f, g (i) h (i), within O(d log ˆɛ ) floating point operations, performed with precision O(log ˆɛ ), where ˆɛ < o(/d)

13 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 3 Therefore, the ɛ-approximation of g and h, ɛ = 4ˆɛ, may be computed in a total of O(d log ɛ ) floating point operations, with precision O(log ɛ ) 3 Splitting and hemimonicity We will first prove a version of Theorem 2 for monic (resp antimonic) polynomials: Lemma 5 Let g be monic, of degree a, with roots inside the disk D(R ), where R > 2a Then, g x a m, < 2a R Clearly, this lemma implies the analogous result for antimonic polynomials Proof of Lemma 5: Let g be monic, with all roots ζ i inside D(R ) Now, g a i = ( ) i ζ jk j k j < <j i So, 2 i g a i 2 i a i R i

14 4 GREGORIO MALAJOVICH AND JORGE P ZUBELLI Let L i = 2 i a i R i Then L 0 =, and L i+ L i = 2 a i + a i R = 2 a i i + R Therefore L = 2a a i Since in general < a and 2a R i+ R <, one gets 2 i g a i 2a R We may prove now Theorem 2 Assume that g is monic of degree a, with roots inside D(R ) Assume that h is antimonic of degree b, with roots outside D(R) Let ϕ = gh We want to bound 2 a i ϕ i If i a, we may distinguish two cases For instance, assume i > a (The case i < a is analogous) We have Hence, ϕ i = g a h i a + 2 a i ϕ i 2 i a h i a + 2 i a 0 k<a 0<i k b 0 k<a 0<i k b g k h i k g k h i k 2 i a i+a h a, + g x a m, h a, 2 i a a+k i+k 0 k<a 0<i k b < h a, + 3 g xa m, h a, < 4 3 h a,

15 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 5 On the other hand, we may write ϕ a as ϕ a = g a h 0 + g k h a k Therefore, ϕ a 2 2a+2k a b k<a a b k<a g m, h a, g x a m, h a, Hence, ϕ x a 4 ϕ a h, 3 max( g x a m,, h a, ) g x a m, h a, The bound of Lemma 5 may be inserted in the formula above ϕ x a 4 2 max(a, b) ϕ a h, 3 R ( ) 2 max(a,b) 2 R We prove Lemma 4 now Lemma 5 implies Also, g x a m, 2a R h 2b h 0 a, R In order to bound h a we first bound h 0 Since ϕ = gh is hemimonic, Moreover, = g a h 0 + k g a k h k g a k h k 2 2k g x a m h a 4ab h 0 k k 3R 2 It follows that h 0 < 4ab < 2 3R 2

16 6 GREGORIO MALAJOVICH AND JORGE P ZUBELLI Therefore, g, h ma 2 ( + 2a2 R + + 2b2 R ( ) (a2 + b 2 ) R 5 2 = 30 ) 4 Algorithm for solving the linear system We will construct a first version of the algorithm of Theorem 3 The operation count will be O(d 2 ) Error analysis will be dealt with in Section 5 A fast version will be constructed in Section 6 The proof of Theorem 3 finishes in Section 7 Given a, b, g, h and ϕ, we will design an algorithm to solve h 0 g 0 h h 0 g g 0 h b g0 g h 0 g a h h b g a 0 δg δh = ϕ We assume at input that g a = h b = We will write the algorithm recursively, but it can be made recursion-free by usual techniques The

17 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 7 main feature of this algorithm will be to preserve (in some sense) the monic, antimonic and hemimonic structure of the problem This means that when g m, h a and ϕ h are small, δg m and δh a should be small also It will be convenient to shift to the corresponding max norm Those will be denoted by m,, a, and h, Essentially, the algorithm is based on column operations, elimination of one variable, and a special pivoting operation The algorithm below is written in a pseudo-code, and some lines are numbered for convenience The input data are a and b, positive integers; and polynomials g, h and ϕ, of degree respectively a, b and a + b Those polynomials are represented by the vector of their coefficients Index range between i and j is written i : j Therefore, the notation g :a g :a g 0 h :b stands for g i g i g 0 h i, where i a, b The output of the algorithms are polynomials δg and δh, of degree a and b, respectively Algorithm Solve (δg, δh) (a, b, g, h, ϕ) If a b 0 If a = and b = 0, Return (0, ϕ 0 ) ; 20 g :a g :a g 0 h :b ; 30 g g a ; g :a g :a g ; g a ; 40 δg 0 ϕ 0 ; 50 ϕ :b ϕ :b δg 0 h :b ; 60 (δg :a, δh 0:b ) Solve(a, b, g :a, h 0:b, ϕ :a+b ) ; 70 δh 0:b δh 0:b g ;

18 8 GREGORIO MALAJOVICH AND JORGE P ZUBELLI 80 δg 0:a δg 0:a g 0 δh 0:b ; 90 Return (δg, δh) ; Else 00 (δh b:0, δg a:0 ) Solve(b, a, h b:0, g a:0, ϕ a+b:0 ) ; 0 Return(δg, δh); Lines 00 and 0 of the algorithm refer to the following pivoting procedure: One replaces g and h by x b h(x ) and x a g(x ), respectively ϕ is replaced by x a+b ϕ(x ) The algorithm Solve is called again, this time ensuring that a b Then, the results are pivoted back, so that we obtain a solution of the original problem Line 0 deals with the trivial case a =, b = 0 This line is necessary to avoid infinite recursion Line 20 performs a column operation Namely, we add g 0 times column 0 to b to columns a through a + b + (recall b a) Line 30 ensures that g is still monic It is easy to find δg 0 (Line 40), and then to eliminate that variable of the problem (Line 50) Then the algorithm is called recursively (Line 50), and the column operations of lines 20 and 30 are undone at lines 70 and 80 This algorithm exploits the particular structure of the problem (g is almost x a, h is almost ) It has some remarkable stability properties For instance, we can bound the norm of δg and δh in terms of the norm of the input For clarity, we do that first without considering numerical error Let s define N = max { } g x a m,, h a,, ϕ h,

19 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 9 At line 20 we have g 0 g x a m, 2 a N2 a Hence, g 0 h :b a, g 0 (h ) a, N 2 2 a We estimate g 0 h :b m, as follows Assume that was attained at i, i 0: g 0 h :b g 0 h a x a m, = max 0 i<a g 0h i 2 a i g 0 h :b g 0 h a x a m, N 2 2 2i N Therefore, g (20) :a g a (20) x a m, N( + N2 2 ) The norm above is taken as if g :a was a degree a polynomial, with lowest coefficient 0 In the sequel, g :a will be treated as a degree a polynomial This does not increase its norm At line 30, we first perform the operation g g a The value of g a was possibly modified at line 20 (provided a = b) and Hence, Therefore, g (30) = g 20 a g 0 h a N 2 2 2a g i g N + N2 2 N 2 2 2a 2i a g (30) :a x a m, N + N2 2 N 2 2 2a 2i a+a i+ N + N2 2 N 2 2 2a At line 40, we have N + N2 2 (N2 2 ) 2 = N 2 2 N δg 0 = ϕ 0 ϕ h, 2 a N2 a

20 20 GREGORIO MALAJOVICH AND JORGE P ZUBELLI At line 50, h i N2 i So δg 0 h i 2 a i N 2 On the other hand, ϕ i 2 a i N Since we are choosing i b a, then a i = a i Therefore, ϕ i δg 0 h i 2 i a N( + 2 2i N) If ϕ :a+b is considered as a polynomial of degree (a ) + b, one gets ϕ (50) :a+b ϕ(50) a x a h, N( N) Line 60 performs a recursive call to the algorithm solve, where the norm N was replaced by N N δh have norm bounded by some N > At line 70, one obtains Upon return, let s assume that δg and N N δh (70) 0:b δh 0:b a, a, 2 2a N N 2 2 2a N 2 Before execution of line 80 So, δg m, max{n, δg 0 2 a } max{n, N} N δh 0:b m, 2 a δh 0:b a, NN g 0 δh 0:b m, 2 a N 2 Hence, ) δg (80) 0:a N ( N + N 2 2a N 2 N

21 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 2 In order to bound the values of N and N throughout the execution of the recursive algorithm, one may define the recurrence N i+ = N i N i where N i bounds N at recurrence step i a + b and bounds the norm N at step 2a + 2b i, for i > a + b The general term of this recurrence is, for k < N 0, N k = N 0 k If one fixes, for instance, N 0 < 4(a+b), then N 2a+2b < 2(a+b) 5 Error analysis of the algorithm Solve In this section we develop the rigorous error analysis of the Algorithm Solve We assume that this algorithm is executed in finite precision floating point arithmetic The machine arithmetic is is supposed to satisfy the + ɛ property: If is one of the operations +,,, /, and a and b are floating point numbers, the machine computes fl(a b), where fl is a rounding-off operator such that and fl(a b) = (a b)( + ɛ) ɛ = ɛ(a, b, ) < ɛ m, where ɛ m is a small constant called the machine epsilon Let N 0 be the max-norm of the input (g g a x a, h, ϕ) to the Algorithm Solve N 2a+2b is the max norm of δg and δh at output E 2a+2b is the norm of the error at output Norms are taken as m,,

22 22 GREGORIO MALAJOVICH AND JORGE P ZUBELLI a, or h,, according to convenience Also, we will rather look at g g a x a m (resp to h a ) than to g m (resp h a ) Using the above notation, we have the following: Theorem 4 Suppose that N 0 4(a + b) and N 0 0, and that ɛ m < 24(a + b) Let k 2a + 2b Then, N k a + b, and E k 2000ɛ m Theorem 4 implies that the algorithm solve will stay within the error bound of Theorem 3, provided its input satisfies the condition in N 0 To see that, just set ɛ m = 2000ˆɛ Proof of Theorem 4: Consider first the following procedure: w λu + v, where w, λ, u and v are complex numbers Assume we are in the presence of numerical error arising from two sources: the input λ, u and v; and rounding-off So, the actual computation becomes w + δw = ((λ + δλ)(u + δu)( + ɛ ) + (v + δv)) ( + ɛ 2 ), (2)

23 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 23 where ɛ and ɛ 2 are both smaller than ɛ m We subtract the equality w = λu + v from equation (2), and get δw = (λ + δλ)δu( + ɛ )( + ɛ 2 ) + (v + δv)ɛ 2 + δv + δλ u( + ɛ )( + ɛ 2 ) + λu(ɛ + ɛ 2 + ɛ ɛ 2 ) Furthermore, w + δw ( λ + δλ u + δu ( + ɛ m ) + v + δv )( + ɛ m ) So, w + δw ( λ + δλ u + δu + v + δv )( + ɛ m ) 2 δw ( λ + δλ δu + u δλ )( + ɛ m ) 2 + v + δv ɛ m + δv + λ u (2ɛ m + ɛ m 2 ) Hence, 2 w + δw [(2 a λ + δλ )(2 i u + δu )2 2i + 2 a i v + δv )]( + ɛ m ) 2 2 a i δw (2 a λ + δλ 2 i δu 2 2i + 2 i u 2 a δλ 2 2i )( + ɛ m ) 2 +2 a i v + δv ɛ m + 2 a i δv + 2 a λ 2 i u 2 2i (2ɛ m + ɛ 2 m ) Now, assume that v and w (resp u) are endowed with the monic (resp antimonic) norm Now we consider the following line: w 0:a λu 0:a + v 0:a This corresponds to line 80 It also bounds what happens in lines 20 and 50 of the algorithm Summing up, we showed that Lemma 6 Assume that: i u a,, u + δu a, N ii v m,, v + δv m, N iii λ, λ + δλ 2 a N

24 24 GREGORIO MALAJOVICH AND JORGE P ZUBELLI iv δu a, E v δv m, E vi δλ 2 a E Then, w m,, w + δw m, (N + N 2 )( + ɛ m ) 2 and δw m, 2NE( + ɛ m ) 2 + Nɛ m + E + N 2 (2ɛ m + ɛ m 2 ) We consider now the double recurrence N i+ = N i N i 3ɛ m E i+ = E i ( + 4N i+ ) 2ɛ m + 8N i+ ɛ m Lemma 7 The sequences {N i } 2a+2b i=0 and {E i } 2a+2b i=0 bound the norm and the forward error, respectively, of the values of g, h, ϕ, δg and δh (in the appropriate norm) computed by the algorithm Solve Proof Lemma 6 models what happens in lines 20, 50 and 80 We are left with two divisions (lines 30 and 70) that may at most multiply N by g ( + ɛ m ) Let E be the forward error at the beginning of line 20 (resp 50, 80) We denote by E a bound of the error after the operation of line 20 (resp 50, 80) We want a bound E of the error after lines 30 (resp 50, or the combination of lines 70 and 80) E will be replaced by E g E + N g ɛ m Therefore, after lines 20 and 30, we have new values N and E bounded by N (N + N 2 ) ( + ɛ m ) 2 ( + ɛ g m )

25 But, So, A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 25 As g N 2 2 2a we get that Furthermore, E g E + N g ɛ m N 2 4 N 2 4 N 2 4 N N + N N 2 ( + ɛ m) 3 N ( + ɛ m) 3 N N N 3ɛ m ( E( + 2N)( + ɛm ) 2 + 4Nɛ m + N ɛ m ) ( E( + 2N)( + ɛm ) 2 + 4Nɛ m + N ɛ m ) = + N N N 2 2 E E( + 4N )( + ɛ m ) 2 + 8N ɛ m E( + 4N ) + 8N ɛ m 2ɛ m Composition of lines 70 and 80 is similar, since dividing δh by g is certainly less problematic that dividing the final result by g So, for norm-increase and error-bound estimates, we can invert the order of those operators and use the previous bound The general term for N i is given by the following: Set ρ = 3ɛ m

26 26 GREGORIO MALAJOVICH AND JORGE P ZUBELLI Then, N i = We will show the following bounds: ρ i N 0 ( + ρ + ρ ρ i ) Lemma 8 Let k N be fixed, and assume that 0 i k, N 0 min( 2k, /0), and ɛ m < 2k Then, N i is well defined 2 N i < 2 k 3 i<k N i < 2 4 i k( + 4N i ) < 5 4 = 625 Lemma 8 may be used to bound E i, as follows: E i = E 0 0 p i ( ) + 4Np + 8 2ɛ m ɛ m N j 0 j i j+ p i ( ) + 4Np 2ɛ m Since E 0 = 0 we bound E i < 8ɛ m N j 0 j i ( ) i ( + 4N p ) 0 p i 2ɛ m Using the results of Lemma 8 one obtains Moreover, we have ( 2ɛ m E i < 0000ɛ m ( ) i This follows from the fact that 2ɛ m ) i 2iɛ m x y (x+y) and small, and from induction on i So, E i < 2000ɛ m, x and y positive Since there are a+b recursion calls, we set k = 2a+2b, and Theorem 4 is proven

27 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 27 Proof of Lemma 8 Formula can be used to estimate ρ i ρ ρ i < 3iɛ m (3) 3iɛ m 3ɛ m 3ɛ m 3iɛ m 3iɛ m 3ɛ m 3ɛ m i 3iɛ m i 3iɛ m Therefore, using the hypothesis ɛ m < 2k, we have 3iɛ m 3kɛ m < 4 and that ρ i ρ i = 4 3 i (4) 4 Using equation (4), we may bound the general term N i by This implies item N i ρ i N 0 4i 3 Formula 3, together with hypothesis ɛ m < 2k implies: ρi < 4 3 Thus, Using N 0 > 2k, we obtain N i 4 3 N 0 4i 3 N i 2 k This proves item 2 Item 3 is now trivial To prove item 4, notice + 4N i N 0 4i = 3 N 0 4i N 0 4i 3

28 28 GREGORIO MALAJOVICH AND JORGE P ZUBELLI This can be rewritten + 4N i (i 4) N 0 4i 3 N Therefore, for k 4, the product of ( + 4N i ) may be bounded by: ( + 4N i ) < i k < < < < ( + ) ( 6 N0 3 4 (k 4)) N0 3 ( ) ( N0 4 N0 3 (k)) ( + ) ( 6 + ) ( 2 + ) ( N0 3 N0 3 N0 3 N0 3) ( 4(k ( N0 3 3)) N 0 4(k ( 3 2)) N 0 4(k ( 3 )) N (k)) ( N ) 4 3 N 0 4k 3 ) 4 ( 2k k 3 ) 4 ( k < 5 4 = Wrong algorithms run faster The algorithm Solve is based on operations of the form g :a g :a g 0 h :b At a first glance, the operation count would be around 8(d + (d ) + ) This can be estimated to 4d 2 However, a more careful study proves that not all those arithmetic operations need to be performed One can save computer work at a cost of introducing some extra truncation error Indeed, if the norms of g and h above are bounded by N, we have just seen that g 0 2 a N

29 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 29 and g 0 h i 2 a i N 2 What about skipping the operation g i g i g 0 h i? The resulting truncation error (in the appropriate norm) will be bounded by 2 a i g 0 h i 2 2i N 2 If this is less than Nɛ m, the error analysis of Lemma 6 and of Section 5 will still hold Therefore, we will obtain a result within the error bound in Theorem 4 The same is true for divisions in lines 30 and 70 of the algorithm Thence, we may replace lines like by g :a g :a g 0 h :b g :j g :j g 0 h :j where j = log 2 and ɛ is our error bound ɛ The operation count is now 4d log floating point operations, where ɛ it is assumed that ɛ < 24d 7 Proof of Theorem 3 We can now finish off the proof of Theorem 3 All that remains is to check that, for each Newton iteration, the conditions of Theorem 4 are satisfied Namely, we need N 0 4d and N 0 0 Under the conditions of the Main Theorem and according to Theorem 2 f x a h, < 3 64d 2

30 30 GREGORIO MALAJOVICH AND JORGE P ZUBELLI Also, if f = g h is the exact factorization, Lemma 5 implies that g x a m, < 2a R < a 32d 3 h a, < 2b R < b 32d 3 Moreover, it is known that the distance of an approximate zero to the exact zero is bounded by 2α This bound can be further sharpened, see [9] So, Using d+ d 2 Hence, d + d((g, h), (g, h )) 2α 2 8 < 2, one obtains d 2α 3 28d 3 d + 64d 2 g x a m g g m + g x a m 3 28d + 32d 2 6d h a < h h a + h a 6d We still need an estimate of f gh h We write f gh = (f x a ) (gh x a ) = (f x a ) (g x a )(h ) (g x a ) x a (h ) Therefore, the norms satisfy f gh h f x a h + (g x a )(h ) h + g x a m + h a We will need the Lemma 9 Let g be monic and h antimonic Then, (g x a )(h ) h d 3 g xa m h a

31 have A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 3 Whence we derive that f gh h 32d d + 6d + 6d 4d If d 2, then /4d < /0 In the particular case d = 2, we also f gh h < 0, almost finishing the proof of Theorem 3 It remains to prove Lemma 9 Proof of Lemma 9: Set, g = g x a, h = h, and ϕ = g h We note that for 0 i < d Now we estimate, ϕ i = min(a,i) k=max(0,i b) g kh i k ϕ h, i = i min(a,i) k=max(0,i b) min(a,i) k=max(0,i b) 2 a i g kh i k 2 a i 2 a i+2k 2 a k g k 2 i k h i k Using the fact k < min(a, i), one obtains a i a i + 2k 2 min(a, i) + 2k

32 32 GREGORIO MALAJOVICH AND JORGE P ZUBELLI and hence min(a,i) ϕ h, 2 2 min(a,i)+2k g m h a i k=max(0,i b) g m h ( a ) g m h a i i<d d 3 g m h a, where in the chain of inequalities we have used the Cauchy-Schwartz inequality 3

33 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 33 So, ϕ h ϕ h, d 3 g m h a 8 Proof of remaining Lemmata 8 Proof of Lemma Notice that for each i with 0 i a+b+ we have that the i-th component of D 2 ϕ f (g, h) is given by ( D 2 ϕ f (g, h) ) : (ĝ, ĥ, g, h) (ĝ i j h i j ) + ( g jĥi j), (5) where the sum is taken over a range of subindices j such that the corresponding coefficients make sense and ĝ j and g j (resp ĥj and h j ) are elements of the tangent space to the affine linear manifold of monic (resp antimonic) polynomials To estimate the operator norm mentioned above we bound D 2 ϕ f (g, h)(ĝ, ĥ, g, h) with (ĝ, ĥ) ma = and ( g, h) ma = Now, we estimate each of the sums in equation (5) Recall that 2 a i To estimate 0 j<a 0<i j b ĝ j h i j ĝ j 2 a j 2 i j h i j 2 a i a+2j i M a,b def = max 0 j<a 0<i j b max 0 j<a 0<i j b h 2 a i a i+2j 2 a i a i+2j, ĝ m h a we consider the cases i a and i > a In the former, we get 2 a i a i+2j = 2 2j 2i 4,

34 34 GREGORIO MALAJOVICH AND JORGE P ZUBELLI since 0 < i j In the latter, since j < a A similar reasoning gives, 2 a i 2 a i a i+2j = 2 2(j a) 4, 0 j<a 0<i j b g jĥi j 4 g m ĥ a Now we recall that and ĝ 2 2 m + ĥ =, a g 2 2 m + h = a Hence, ( 2 a i D 2 ϕ f (g, h)(ĝ, ĥ, g, h) ) 2 i 0 i a+b Therefore, D 2 ϕ f (g, h) ma h 0 i a+b 6 ( ĝ m h + ĥ g a a m ) 2 (a + b + ) (ĝ, 6 ĥ) ma 2 ( g, h) (a + b + ) 6 d + 4 ma 2 82 Proof of Lemma 2 We have to estimate the norm of the operator Dϕ f (g, h) in the case g = x a and h = Notice that in this case Dϕ f (x a, ) = I a+ 0 0 I b where I k denotes the k k identity matrix,

35 A FAST AND STABLE ALGORITHM FOR SPLITTING POLYNOMIALS 35 Now, if ϕ = ϕ 0 x ϕ a+b x a+b, with ϕ h =, then 2 2 a i ϕ i 2 =, 0 i a+b+ and Hence, Dϕ f (x a, ) 2 ϕ = 2 ϕ h = ma Dϕ f (x a, ) ma h = 9 Acknowledgments We thank very helpful and stimulating conversations with D Bini, J-P Cardinal B Fux Svaiter P Kirrinis, V Pan and References [] Victor Y Pan, Solving a Polynomial Equation: Some History and Recent Progress Preprint, CUNY (995) [2] Alexandre Ostrowski, Recherches sur la Méthode de Graeffe et les Zéros des Polynomes et des Séries de Laurent Acta Mathematica 72, (940) [3] J V Uspensky Theory of equations Mc Graw Hill, New York (948) [4] Arnold Schönhage, Equation Solving in Terms of Computational Complexity Proceedings of the International Congress of Mathematicians, Berkeley, 986, 3-53 American Mathematical Society (987) [5] Peter Kirrinis, Partial Fraction Decomposition in C(Ϝ) and simultaneous Newton Iteration for Factorization in C[Ϝ] Preprint, Bonn, (995) [6] Victor Y Pan, Optimal and Nearly Optimal Algorithms for Approximating Polynomial Zeros Computers Math Applic), to appear [7] Victor Y Pan, Deterministic Improvement of Complex Polynomial Factorization Based on the Properties of the Associated Resultant Computers Math Applic, 30 (2), 7-94 (995)

36 36 GREGORIO MALAJOVICH AND JORGE P ZUBELLI [8] Steve Smale, Newton method estimates from data at one point, in R Erwing, K Gross and C Martin (editors) The merging of disciplines: New directions in Pure, Applied and Computational Mathematics Springer, New York (986) [9] Michael Shub and Steve Smale, On the Complexity of Bezout s Theorem I - Geometric aspects Journal of the AMS, 6(2), (993) [0] Michael Shub and Steve Smale, On the complexity of Bezout s Theorem II - Volumes and Probabilities in: F Eysette and A Galligo, eds: Computational Algebraic Geometry Progress in Mathematics , Birkhauser (993) [] Michael Shub and Steve Smale, Complexity of Bezout s Theorem III ; Condition number and packing Journal of Complexity 9, 4-4 (993) [2] Michael Shub and Steve Smale, Complexity of Bezout s Theorem IV ; Probability of success ; Extensions Preprint, Berkeley (993) [3] Michael Shub and Steve Smale, Complexity of Bezout s Theorem V: Polynomial time Theoretical Computer Science 33(), 4-64 (994) [4] Gregorio Malajovich, On generalized Newton algorithms: quadratic convergence, path-following and error analysis Theoretical Computer Science 33, (994) Departamento de Matematica Aplicada da UFRJ, Caixa Postal 68530, CEP 2945, Rio, RJ, Brasil address: gregorio@lyriclabmaufrjbr Instituto de Matematica Pura e Aplicada CNPq, Estrada Dona Castorina 0, Jardim Botanico, CEP , Rio de Janeiro, RJ, Brasil address: zubelli@impabr