Certified predictor-corrector tracking for Newton homotopies

Transcription

1 Certified predictor-corrector tracking for Newton homotopies Jonathan D. Hauenstein Alan C. Liddell, Jr. March 11, 2014 Abstract We develop certified tracking procedures for Newton homotopies, which are homotopies for which only the constant terms are changed. For these homotopies, our certified procedures include using a constant predictor with Newton corrections, an Euler predictor with no corrections, and an Euler predictor with Newton corrections. In each case, the predictor is guaranteed to produce a point in the quadratic convergence basin of Newton s method. The complexity of a tracking procedure using a constant predictor with Newton corrections is analyzed with the number of steps bounded above by a constant multiple of the length of the path in the γ-metric. Details about using exact rational arithmetic are also included using continued fractions to control the growth of the heights of the rational numbers. Key words and phrases. certified tracking, alpha theory, numerical algebraic geometry, homotopy continuation, Newton s method Mathematics Subject Classification. Primary 65G20; Secondary 65H10,14Q99. Introduction The ability to compute solutions to systems of polynomial equations has applications in many areas of science and engineering. Software for numerically solving polynomial systems based on homotopy continuation, such as Bertini [1], HOM4PS-2 [10], and PHCpack [19], utilize heuristic computations in their solving algorithms which employ various predictor-corrector tracking methods [2], such as an RKF45 predictor and several Newton corrections. For NAG4M2 [11], scripts described in [3] can certifiably track paths using a constant predictor and one Newton correction. Some examples of other certified tracking methods using similar strategies include [4, 5, 14, 15]. To reduce the gap between the predictor-corrector strategies used in practice and the certifiable tracking methods, we develop a certified predictor-corrector strategy using an Euler predictor and Newton corrections for so-called Newton homotopies. The path tracking certificate is developed via Smale s α-theory [17]. For an analytic system f : C N C N, a fundamental question is to compute a solution of f(x) = 0. Here, compute means to produce a point which quadratically converges under Newton s method to a solution. This definition provides that a true solution can be efficiently approximated via Newton s method to any desired level of accuracy. One approach for computing Department of Mathematics, North Carolina State University, Raleigh, NC (hauenstein@ncsu.edu, www. math.ncsu.edu/~jdhauens). This author was supported in part by NSF grant DMS and DARPA Young Faculty Award. Department of Mathematics, North Carolina State University, Raleigh, NC (acliddel@ncsu.edu, www. math.ncsu.edu/~acliddel). This author was supported in part by NSF grant DMS

2 a solution of f(x) = 0 is via homotopy continuation. The homotopies of current interest are Newton homotopies of the form H(x, t) = f(x) + t v (1) for a fixed v C N. For a given point x 0 C N, one can take v = f(x 0 ) and considers the solution path of H(x, t) = 0 starting at t = 1 with x 0. Newton homotopies can also be used when performing monodromy loops. Such loops are used to compute numerical irreducible decompositions [18] and monodromy groups of parameterized problems [12]. Moreover, paths for Newton homotopies can be certified a posteriori using the scheme presented in [7]. For N 1 and d (Z 1 ) N, let H N,d be the space of polynomial systems f : C N C N such that deg f i = d i. For any reasonable probability distribution on H N,d, the path φ : [0, 1] C N defined by φ(1) = 0 and H(φ(t), t) = f(φ(t)) t f(0) 0 is a smooth continuous path such that Df(φ(t)) is invertible for all t [0, 1] with probability one. In particular, φ(0) is a nonsingular solution of f(x) = 0. Therefore, one obtains a numerical approximation of a solution of f(x) = 0 by approximately following the curve φ(t) from t = 1 to t = 0. Such an approach is a deterministic algorithm for computing one solution of f(x) = 0 which, in the univariate case, i.e., N = 1, was used in [16]. Our contributions include the certified tracking procedure NewtonTracker presented Section 2 and procedures EulerTracker and EulerNewtonTracker presented in Section 4. On the complexity side, our main contribution is Theorem 5 which shows that the number of predictor-corrector steps used by a modification of NewtonTracker presented in Section 3 is bounded above by 352L where L is the length of the path in the γ-metric. In [13], the condition metric (or µ norm -metric) was used to describe the complexity of path tracking for polynomial systems. Since, for nonlinear polynomial systems, µ norm is an upper bound on γ, the γ-metric path length is shorter than the condition metric path length. In Section 5, we describe how to implement these certified tracking procedures for polynomial systems with coefficients in Q[ 1] using exact rational arithmetic. To control the heights of the rational numbers arising in the computations, continued fractions together with robust results in α-theory are used. Examples are presented in Section 6 demonstrating the practicality of these procedures. To illustrate, consider the Newton homotopy H(x, t) = x 2 (1 + m) + mt starting at x = 1 when t = 1 and tracking to t = 0 that was considered in [3, 9.1] for various m 10. In [3, Table 3], it is reported that the certified tracking approach of [3] took 184 steps for m = 10. Table 1 shows that NewtonTracker, EulerTracker, and EulerNewtonTracker all require 5 steps. The next section summarizes the necessary items of α-theory. 1 Smale s α-theory For an analytic system f : C N C N, α-theory provides sufficient conditions to prove that a point x C N is in the quadratic convergence basin of some solution of f = 0. The following provides the necessary background information needed to develop our certified tracking algorithms with expanded details provided in [6, Ch. 8]. Let Df(x) be the Jacobian matrix of f evaluated at x and consider the map N f : C N C N defined by { x Df(x) N f (x) = 1 f(x) Df(x) is invertible x otherwise. 2

3 The map N f defines a Newton iteration of f and, for k N, the map defines the k th Newton iteration of f. N k f = N f N f }{{} k times Definition 1 For an analytic system f : C N C N, a point x C N is an approximate solution of f = 0 if there exists ξ C N such that f(ξ) = 0 and ( ) 2 1 k 1 Nf k (x) ξ x ξ 2 for each k N. In this case, the point ξ is called the associated solution of x and the sequence {Nf k (x) k N} converges quadratically to ξ. If Df(x) is invertible, the α-theoretic sufficient condition for certifying x is an approximate solution of f = 0 is based on the following constants: α(f, x) = β(f, x) γ(f, x), β(f, x) = x N f (x), γ(f, x) = sup Df(x) 1 D k f(x) k! k 2 If Df(x) is not invertible, we take β(f, x) = 0 and γ(f, x) =. In this case, the product 0 is defined based on f(x), namely α(f, x) = 0 if f(x) = 0 and α(f, x) = if f(x) 0. The following version of Theorem 2 from [6, pg. 160] certifies that x is an approximate solution. 1 k 1 Theorem 2 If f : C N C N is an analytic system and x C N such that α(f, x) α 0 = , (2) then x is an approximate solution of f = 0. Moreover, x ξ 2β(f, x) where ξ is the associated solution of x. In Section 5, we use the following theorem derived from Theorem 4 and Remark 6 of [6, Ch. 8] to replace an approximate solution x with another point y that is also an approximate solution with the same corresponding associated solution. Theorem 3 If f : C N C N is an analytic system and x, y C N such that α(f, x) < and x y < 1 20γ(f, x), then x and y are approximate solutions of f = 0 with the same associated solution. 2 Newton tracking Traditional α-theoretic certified tracking schemes, such as [3, 4, 14], use a constant predictor with a corrector consisting of one Newton iteration. In Section 2.1, we compare this type of predictorcorrector scheme with simply taking an Euler predictor and no corrector. Section 2.2 derives a certified tracking scheme that uses a constant predictor and a corrector consisting of potentially multiple Newton iterations per step. 3

4 2.1 Equivalence of Newton and Euler Let f : C N C N be an analytic system, v C N, and H(x, t) = f(x)+t v. Suppose that x C N and t [0, 1] such that H(x, t ) = 0 and Df(x ) is invertible. For t 0, the resulting point using a constant predictor, moving from t to t + t, followed by a Newton iteration at t + t is N H(,t + t)(x ) = x Df(x ) 1 H(x, t + t) = N H(,t )(x ) Df(x ) 1 v t. (3) Now, we consider the same setup using a Euler predictor without a corrector. To that end, we first define an Euler predictor step which arises from the following Taylor series approximation: H(x + x, t + t) H(x, t) + Df(x) x + v t. Upon setting the right-hand side equal to zero and solving for x, we have that the Euler prediction of H from t to t + t is E H,t, t(x ) = x Df(x ) 1 (H(x, t )+v t) = x Df(x ) 1 H(x, t + t) = N H(,t + t)(x ). That is, for Newton homotopies, a constant predictor with one Newton iteration is exactly the same as using an Euler predictor with no corrections. The key distinction between the certified tracker presented next and the one in Section 4 is the requirement on the prediction step. The following imposes that the constant predictor yield a point in the quadratic convergence basin and then applies Newton iterations to move closer to the corresponding solution. In Section 4, the Euler predictor is constrained to certifiably produce a point in the quadratic convergence basin followed by Newton iterations to move closer to the corresponding solution. Thus, by the equivalence provided above and the results presented in Section 4, we show how to use α-theory to certify that the result of a Newton iteration applied to a point x is an approximate solution where x need not a priori be an approximate solution. 2.2 Tracking using only Newton iterations The following derives a certified tracking approach such that the constant predictor lands within a quadratic convergence basin of Newton s method. We then apply K 0 Newton iterations based on local information to attempt to minimize the total number of Newton iterations used throughout the total tracking process. In particular, more Newton iterations will result in the next stepsize to increase but there is a diminishing return on the computational investment. Before deciding the number of Newton iterations, we first provide an upper bound on the stepsize for the constant predictor to certifiably yield an approximate solution. Theorem 4 Let f : C N C N be an analytic system, v C N, and H(x, t) = f(x) + t v. Suppose that x C N and t [0, 1] such that Df(x ) is invertible and α(h(, t ), x ) α 0 where α 0 is defined in (2). If t α 0 α(h(, t ), x ) γ(f, x ) Df(x ) 1 v, (4) then α(h(, t + t), x ) α 0, that is, x is an approximate solution of H(x, t + t) = 0. Proof. Clearly, γ(h(, t + t), x ) = γ(h(, t )) = γ(f, x ). Moreover, β(h(, t + t), x ) = Df(x ) 1 H(x, t + t) = Df(x ) 1 H(x, t ) + Df(x ) 1 v t β(h(, t ), x ) + Df(x ) 1 v t. 4

5 Therefore, α(h(, t + t), x ) α(h(, t ), x ) + γ(f, x ) Df(x ) 1 v t so that α(h(, t + t), x ) α 0 if (4) holds. After using the constant predictor with t based on (4), one then applies K 0 Newton iterations with the system H(x, t + t) = 0 starting at x. Intuitively, one expects that α 0 α(h(, t + t), Nf K(x )) will increase and converge quadratically to α 0 as K increases whereas γ(h(, t ), Nf K(x )) Df(Nf K(x )) 1 v remains roughly constant. Taking these as assumptions along with assuming that α(h(, t + t), Nf K (x )) = α(h(, t + t), x ), 2 2K 1 we want to determine K 0 to minimize the total number of Newton iterations per length of the step. For simplicity, we abbreviate α = α(h(, t + t), x ) and want to maximize K such that K + 1 α 0 α/2 2K 1 < j + 1 α 0 α/2 2j 1 for all j = 0, 1..., K 1. We note the additional one in the numerator arises since one Newton iteration must be performed to compute α. When j = K 1, we need K + 1 α 0 α/2 < K 2K 1 α 0 α/2. 2K 1 1 This is equivalent to ( K + 1 α 0 < α 2 K ). 2K K 1 Since α α 0, we must have K = 1 or K = 2. With this, our analysis suggests 2 if 4α 0 /5 < α(h(, t + t), x ) α 0, K = 1 if 2α 0 /3 < α(h(, t + t), x ) 4α 0 /5, 0 otherwise. (5) That is, the number of Newton corrections performed depends on α(h(, t + t), x ). This suggests the following tracking procedure, with each step yielding an approximate solution of H(x, t + t) = 0 via Theorem 4. Procedure x 0 = NewtonTracker(H, x 1 ) Input A Newton homotopy H(x, t) = f(x) + t v where f : C N C N is analytic and v C N, and a point x 1 C N with α(h(, 1), x 1 ) α 0. Output A point x 0 C N with α(f, x 0 ) α 0. Begin 1. Initialize x := x 1 and t := Do the following while t > 0: 5

6 Return x 0 := x. (a) If α(h(, t ), x ) > 4α 0 /5, set x := N 2 H(,t ) (x ). Otherwise, set x := N H(,t )(x ) if α(h(, t ), x ) > 2α 0 /3. (b) Compute t := α 0 α(h(, t ), x ) γ(f, x ) Df(x ) 1 v. (c) If t > t, set t := 0. Otherwise, set t := t t. 3 Complexity analysis for Newton tracking For a Newton homotopy H(x, t) = f(x) + t v, we have ẋ(t) = Df(x(t)) 1 v along a solution path x(t). Since the stepsizes for NewtonTracker are inversely proportional to γ(f, x ) ẋ(t), the length of the path in the γ-metric, namely 1 0 γ(f, x(t)) ẋ(t) dt = 1 0 γ(f, x(t)) Df(x(t)) 1 v dt, (6) is directly related to the number of predictor-corrector steps used when tracking the path. In Theorem 5, we make this statement precise by bounding the number of predictor-corrector steps in terms of the length in the γ-metric for the following modification of NewtonTracker. Procedure x 0 = ModifiedNewtonTracker(H, x 1 ) Input A Newton homotopy H(x, t) = f(x) + t v where f : C N C N is analytic and v C N, and a point x 1 C N with α(h(, 1), x 1 ) α 0. Output A point x 0 C N with α(f, x 0 ) α 0. Begin 1. Initialize x := x 1 and t := Do the following while t > 0: Return x 0 := x. (a) While α(h(, t ), x ) > α 0 /6, update x := N H(,t )(x ). (b) Compute t := α 0/3 α(h(, t ), x ) γ(f, x ) Df(x ) 1 v. (c) If t > t, set t := 0. Otherwise, set t := t t. There are two differences between NewtonTracker and ModifiedNewtonTracker. The first is that ModifiedNewtonTracker enforces an upper bound, α 0 /6 in this case, on the value of α(h(, t ), x ) before one performs a step. If z is the associated solution of x with respect to H(x, t ) = 0 such that Df(z ) is invertible, then one could simply replace this while loop with a fixed number of Newton iterations to enforce this upper bound. Thus, we can still consider the complexity of ModifiedNewtonTracker in terms of the number of predictor-corrector steps. The second change is that the stepsize t is selected so that the proof of Theorem 4 yields α(h(, t + r), x ) α 0 /3 for r t. 6

7 These changes imply that which produces the following complexity result. t γ(f, x ) Df(x ) 1 v α 0 /6, Theorem 5 Let f : C N C N be analytic, v C N, and H(x, t) = f(x) + t v. Suppose that the ModifiedNewtonTracker procedure terminates when tracking a homotopy path x(t) for 0 t 1 where Df is invertible on the path. If L is the length of the path in the γ-metric given in (6), then the number of predictor-corrector steps needed to obtain an approximate solution x 0 with associated solution x(0) is bounded above by 352L. Proof. Suppose that ModifiedNewtonTracker took P steps defined by 1 = t 0 > t 1 > > t P 1 > t P = 0. For i = 0,..., P 1, define t i = t i t i+1 > 0 and let y i be the approximate solution of H(x, t i ) = 0 computed after Step 2a. We know L = P 1 i=0 ti t i+1 γ(f, x(t)) Df(x(t)) 1 v dt P 1 i=0 Fix 0 i P 1 and suppose that t i+1 t t i. Define t i min γ(f, x(t)) Df(x(t)) 1 v. t i+1 t t i v i (t) = y i x(t) γ(f, x(t)) and u i (t) = y i x(t) γ(f, y i ). If v i (t) and u i (t) are less than 1 2/2, Proposition 3 of [6, Ch. 8] provides γ(f, y i ) γ(f, x(t)) (1 v i (t))(1 4v i (t) + 2v i (t) 2 ) Since y i x(t) 2β(H(, t), y i ), we know which implies Therefore, v i (t) and γ(f, x(t)) u i (t) 2α(H(, t), y i ) 2α 0 /3 < 1 2/2 2α 0 /3 (1 u i (t))(1 4u i (t) + 2u i (t) 2 ) < 1 2/2. γ(f, x(t)) γ(f, y i )(1 v i (t))(1 4v i (t) + 2v i (t) 2 ). Similarly, Lemma 2 of [6, Ch. 8] provides that Df(y i ) 1 v Df(y i ) 1 Df(x(t)) Df(x(t)) 1 v Hence, γ(f, y i ) (1 u i (t))(1 4u i (t) + 2u i (t) 2 ). (1 v i (t)) 2 1 4v i (t) + 2v i (t) 2 Df(x(t)) 1 v. γ(f, x(t)) Df(x(t)) 1 v (1 4v i(t) + 2v i (t) 2 ) 2 γ(f, y i ) Df(y i ) 1 v 1 v i (t) 4 37 γ(f, y i) Df(y i ) 1 v. Therefore, L 4 37 P 1 i=0 t i γ(f, y i ) Df(y i ) 1 v α 0 P which shows that P 111L 2α 0 352L. 7

8 4 Euler-Newton tracking In NewtonTracker, the stepsize was determined so that the constant predictor would yield an approximate solution. In this section, we determine a stepsize to guarantee that the Euler predictor will yield an approximate solution. Due to (3), this is equivalent to having the point arising after using a constant predictor and performing one Newton iteration is an approximate solution. This immediately yields that the certifiable stepsize for an Euler predictor can be at least as large as the constant predictor. Even though the following theorem shows a small increase in stepsize over the constant predictor, it implies that one can certifiably track a path using only Euler predictions. Theorem 6 Let f : C N C N be an analytic system, v C N, and H(x, t) = f(x) + t v. Suppose that x C N and t [0, 1] such that Df(x ) is invertible and α(h(, t ), x ) α 0 where α 0 is defined in (2). If t e 0 α(h(, t ), x ) γ(f, x ) Df(x ) 1 v where e 0 = α0 8α α (7) α 0 then α(h(, t + t), E H,t, t(x )) α 0, that is, E H,t, t(x ) = N H(,t + t)(x ) is an approximate solution of H(x, t + t) = 0. Proof. Define y( t) = E H,t, t(x ) = x Df(x ) 1 (H(x, t )+v t). First, we need to ensure that Df(y( t)) is invertible. By Lemma 2(a) of [6, Ch. 8], Df(y( t)) is invertible provided that x y( t) γ(f, x ) < c 0 = 1 2/2. Since α(h(, t ), x ) α 0, we enforce so that t < c 0 α 0 γ(f, x ) Df(x ) 1 v x y( t) γ(f, x ) γ(f, x )( Df(x ) 1 H(x, t ) + t Df(x ) 1 v ) = α(h(, t ), x ) + t γ(f, x ) Df(x ) 1 v < α 0 + (c 0 α 0 ) = c 0. Define u = x y( t) γ(f, x ) < c 0 < 1. By Lemma 2(b) of [6, Ch. 8], Df(y( t)) 1 Df(x ) (1 u)2 1 4u + 2u 2. Since the constant and linear terms of a Taylor series expansion of H(y( t), t + t) centered at (x, t ) vanish, we have β(h(, t + t), y( t)) = Df(y( t)) 1 H(y( t), t + t) Df(y( t)) 1 Df(x ) By Proposition 3 of [6, Ch. 8], (1 u)2 x y( t) 1 4u + 2u 2 k=2 uk 1 = γ(h(, t + t), y( t)) = γ(f, y( t)) k=2 Df(x ) 1 D k f(x ) k! x y( t) k (1 u)u 2 (1 4u + 2u 2 )γ(f, x ). γ(f, x ) (1 4u + 2u 2 )(1 u). 8

9 Thus, we have ( ) 2 α(h(, t u + t), y( t)) 1 4u + 2u 2. This shows that if u (1 4u + 2u 2 ) α 0, then α(h(, t + t), y( t)) α 0. Consider the univariate polynomial p(z) = (1 4z + 2z 2 ) α 0 z. Since p(0) > 0 and e 0, as defined in (7), is the smallest positive solution of p(z) = 0, the result follows if u e 0 < c 0. Since u α(h(, t ), x ) + t γ(f, x ) Df(x ) 1 v, we know u e 0 provided that (7) holds. Remark 7 This proof depended on Lemma 2(a) of [6, Ch. 8] to show that Df(y( t)) is invertible. This places a constraint on the stepsize that would also limit the use of higher-order predictors provided that one follows a similar proof strategy. In particular, this constraint is x y( t) γ(f, x ) < c 0 = 1 2/2 < 2e 0. Thus, if one would instead use a higher-order method to determine y( t), one has Up to first order, this constraint yields t < x y( t) = t Df(x ) 1 v + O( t 2 ). c 0 γ(f, x ) Df(x ) 1 v < 2e 0 γ(f, x ) Df(x ) 1 v. Theorem 6 suggests the following certified tracker using only Euler predictions. Procedure x 0 = EulerTracker(H, x 1 ) Input A Newton homotopy H(x, t) = f(x) + t v where f : C N C N is analytic and v C N, and a point x 1 C N with α(h(, 1), x 1 ) α 0. Output A point x 0 C N with α(f, x 0 ) α 0. Begin 1. Initialize x := x 1 and t := Do the following while t > 0: e 0 α(h(, t ), x ) (a) Compute t := γ(f, x ) Df(x ) 1 v. (b) If t > t, set t := t. (c) Update x := E H,t, t(x ) and then t := t t. Return x 0 := x. Following similar simplifying assumptions as posed in Section 2.2, the following Euler-Newton tracking algorithm uses the same Newton correction strategy described in (5). 9

10 Procedure x 0 = EulerNewtonTracker(H, x 1 ) Input A Newton homotopy H(x, t) = f(x) + t v where f : C N C N is analytic and v C N, and a point x 1 C N with α(h(, 1), x 1 ) α 0. Output A point x 0 C N with α(f, x 0 ) α 0. Begin 1. Initialize x := x 1 and t := Do the following while t > 0: Return x 0 := x. (a) If α(h(, t ), x ) > 4α 0 /5, set x := NH(,t 2 ) (x ). Otherwise, set x := N H(,t )(x ) if α(h(, t ), x ) > 2α 0 /3. e 0 α(h(, t ), x ) (b) Compute t := γ(f, x ) Df(x ) 1 v. (c) If t > t, set t := t. (d) Update x := E H,t, t(x ) and then t := t t. 5 Exact computations using continued fractions Suppose that H(x, t) is a Newton homotopy that is polynomial in x with coefficients in Q[ 1]. Given a starting point with coordinates in Q[ 1], one could adjust the tracking procedures to approximate t from below by a rational number so that all computations can be performed exactly in Q[ 1]. With such an approach, the heights of the rational numbers could grow extremely large quickly thereby making such an exact tracking approach computationally infeasible. Here, the height of a rational number p/q with relatively prime integers p and q is log max{ p, q }. By using the robustness presented in Theorem 3, we briefly describe how one can replace the computed points with points of small heights using continued fractions. Given an approximate solution x C N of H(x, t ) = f(x) + t v = 0 with associated solution ξ such that Df(ξ ) is invertible, we can always keep applying Newton iterations with respect to H(x, t ) starting at x to compute a point x C N such that α(h(, t ), x ) Since x = N k H(,t ) (x ) for some k 0, we know that x and x have the same associated solution ξ. For j = 1,..., N, consider writing x j = a j + b j 1 where aj, b j R. Thus, using a continued fraction expansion of a j and b j, for example, one can compute integers p j, q j, r j, and s j such that a j p j q j, b j r j s j < 1 20γ(f, x ) 2N. For y Q[ 1] N defined by y j = p j/q j + r j /s j 1, then x y 1 20γ(f, x ). Hence, x, x, and y are all approximate solutions of H(x, t ) = 0 associated with ξ by Theorem 3. Since α 0 /6 < 0.03, this height-reducing robust strategy can be incorporated directly into ModifiedNewtonTracker. Since Newton iterations are not performed in EulerTracker, one can 10

11 Number of steps (Total number of Newton and Euler iterations) m Newton Euler EulerNewton ModifiedNewton Robust [3] 10 5 (8) 5 (5) 5 (5) 13 (12) 184 (183) (18) 10 (10) 10 (10) 25 (24) 292 (291) (28) 15 (15) 15 (15) 38 (37) 395 (394) (36) 19 (19) 19 (19) 50 (49) 499 (498) (40) 21 (21) 21 (21) 54 (53) 530 (529) (42) 22 (22) 22 (22) 56 (55) 547 (546) Table 1: Number of predictor-corrector steps and total number of Newton and Euler iterations using various tracking methods for selected m simply decrease the stepsize t to ensure that the resulting prediction has α less than 0.03 and then use this strategy after each predictor step is performed. For NewtonTracker and EulerNewtonTracker, one can simply adjust the correction strategy to ensure that the corresponding α is less than 0.03 and then perform this height-reducing robust strategy. 6 Examples 6.1 Univariate quadratics A collection of quadratic homotopies in P 1 was considered in [3, 9.1]. After dehomogenizing and replacing t with 1 t to track from t = 1 to t = 0, these are univariate quadratic Newton homotopies of the form H(x, t) = x 2 (1 + m) + m t, for m > 1 starting at x = 1 for t = 1. Table 1 presents the number of steps taken by NewtonTracker, ModifiedNewtonTracker, EulerTracker, and EulerNewtonTracker for selected values of m 10 that are also presented in Table 3 of [3]. In each test, NewtonTracker, EulerTracker, and EulerNewtonTracker all took the same number of steps, which is due to the small relatively difference between the constants α 0 and e 0. However, the cost for NewtonTracker to do this was due to the use of two Newton iterations after each predictor step (except at the end). The Euler predictions were all relatively close to the path so that EulerTracker could take large steps and EulerNewtonTracker did not need to perform any Newton iterations. Thus, in terms of the total number of Newton and Euler iterations, the two Euler-based methods were identical and roughly one-half of the cost of NewtonTracker. For ModifiedNewtonTracker, only one Newton correction was used after each prediction, but the stepsize restriction needed to simplify the complexity analysis made the approach take more steps than the other three new methods, but still considerably less than the robust certified tracking approach of [3]. The key measure for the number of steps in our tracking methods is the length of the path in the γ-metric presented in (6). For m > 1, since f(x) = x 2 (1 + m) and v = m, we have γ(f, x) = 1 2 x and Df(x) 1 v = m 2 x. Thus, the length of the the path x(t) = 1 + m m t in the γ-metric is L(m) = 1 0 m m 1 dt = 4x(t) m m t ln(m + 1) dt =. 4

12 Steps/L(m) m Newton, Euler & EulerNewton ModifiedNewton Table 2: Number of predictor-corrector steps per unit length of the path in the γ-metric Number of steps (Total number of Newton and Euler iterations) m Newton Euler EulerNewton ModifiedNewton (14) 9 (9) 9 (9) 32 (31) (22) 15 (15) 15 (15) 52 (51) (32) 21 (21) 21 (21) 73 (72) (42) 27 (27) 27 (27) 94 (93) (50) 33 (33) 33 (33) 114 (113) (60) 39 (39) 39 (39) 135 (134) Table 3: Number of predictor-corrector steps and total number of Newton and Euler iterations using various tracking methods for selected m In particular, since L(m) is logarithmic in m+1, the number of steps for ModifiedNewtonTracker is also logarithmic in m+1 by Theorem 5. Due to the relationship between ModifiedNewtonTracker and the other three tracking methods, one expects this relationship to hold for all of them. Table 2 demonstrates this by showing that the number of steps taken per unit length of the path in the γ-metric in roughly constant as m increases. As m increase without bound, the path in both the Euclidean metric and γ-metric increases without bound. However, as m decreases to 1, the path heads towards the singular solution at the origin. The length of the path in the Euclidean metric is bounded whereas it is unbounded in the γ-metric. Table 3 shows the number of steps and total number of Newton and Euler iterations for selected values of m approaching 1 with Table 4 showing that the number of steps per unit length in the γ-metric is roughly constant. In this case, due to the curvature of the path approaching the singularity, the values of α after the Euler predictions were larger than the above examples, but not large enough for EulerNewtonTracker to need to perform additional Newton iterations. However, just as above, NewtonTracker used two Newton iterations at every step except the last. The added cost of these extra Newton iterations allowed for overall slightly fewer steps than the Euler prediction methods, but still more total iterations. 6.2 Random polynomial systems For random dense polynomial systems f : C N C N, we applied our certified tracking approaches to track the solution path defined by the Newton homotopy H(x, t) = f(x) t f(0) starting at the origin. For any reasonable probability measure, this yields a deterministic approach that, with probability one, computes a solution of f = 0 in finitely many steps. In our experiment for selected N 1 and d 2, we tracked the corresponding solution path of the Newton homotopy starting at the origin for 100 random systems of N dense polynomials of 12

13 Steps/L(m) m Newton Euler & EulerNewton ModifiedNewton Table 4: Number of predictor-corrector steps per unit length of the path in the γ-metric degree d in N variables where the coefficients were independent complex Gaussian with mean 0 and variance 1. Table 5 shows the average number of steps as well as average number of Newton and Euler iterations used our four certified strategies for tracking the corresponding 100 solution paths. In our experiments, NewtonTracker and ModifiedNewtonTracker performed two and one Newton iterations per step, respectively, except for the last step. Moreover, EulerNewtonTracker did not utilize any Newton iterations so that it was equivalent to EulerTracker. Table 6 shows the logarithm of average number of steps where the base is the input size, namely the number of coefficients N (N+d ) d. This data provides initial computational evidence that our certified algorithms could be polynomial in the input size and we are currently exploring analytically techniques for proving such a statement for various probability distributions. References [1] D.J. Bates, J.D. Hauenstein, A.J. Sommese, and C.W. Wampler. Bertini: Software for Numerical Algebraic Geometry. Available at bertini.nd.edu. [2] D.J. Bates, J.D. Hauenstein, and A.J. Sommese. Efficient path tracking methods. Numer. Algorithms, 58(4), , [3] C. Beltrán and A. Leykin. Robust certified numerical homotopy tracking. Found. Comput. Math., 13(2), , [4] C. Beltrán and A. Leykin. Certified numerical homotopy tracking. Exp. Math., 21, 69 83, [5] C. Beltrán and L.M. Pardo. On Smale s 17th problem: a probabilistic positive solution. Found. Comput. Math., 8, 1 43, [6] L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation. Springer- Verlag, New York, [7] J.D. Hauenstein, I. Haywood, and A.C. Liddell, Jr. An a posteriori certification algorithm for Newton homotopies. Preprint available at [8] J.D. Hauenstein and F. Sottile. Algorithm 921: alphacertified: certifying solutions to polynomial systems. ACM TOMS, 38(4), 28, [9] J.D. Hauenstein and F. Sottile. alphacertified: software for certifying solutions to polynomial systems. Available at 13

14 Average number of steps (Average number of Newton and Euler iterations) N d Newton Euler & EulerNewton ModifiedNewton (22.4) 11.5 (11.5) 33.1 (32.1) (68.8) 33.8 (33.8) (99.7) (101.0) 49.5 (49.5) (148.1) (185.6) 90.5 (90.5) (273.4) (136.4) 66.8 (66.8) (200.2) (233.0) (113.9) (344.6) (472.8) (230.8) (703.3) (770.0) (375.8) (1148.8) (262.6) (128.5) (389.1) (621.0) (303.2) (924.7) (1033.8) (504.8) (1544.8) (1845.2) (901.0) (2761.0) (461.8) (225.6) (686.6) (1308.4) (638.9) (1954.4) (2259.6) (1103.4) (3381.4) (3968.2) (1937.8) (5943.9) (817.4) (399.3) (1219.1) (1742.8) (850.9) (2605.8) (3913.8) (1911.3) (5861.4) (6721.4) (3282.6) ( ) (1101.2) (538.0) (1644.4) (2942.6) (1436.9) (4404.1) (5751.0) (2808.6) (8616.9) (1400.2) (683.8) (2091.9) (4085.8) (1995.3) (6118.2) (7797.0) (3808.0) ( ) (1737.2) (848.4) (2597.4) (5705.2) (2786.3) (8546.9) (2218.8) (1083.7) (3318.9) (8031.8) (3922.7) ( ) (2983.0) (1457.0) (4464.6) (8692.0) (4377.1) ( ) Table 5: Average number of predictor-corrector steps and average number of Newton and Euler iterations using various tracking methods for 100 random systems for selected N and d. 14

15 Log input size (Average number of steps) N d Newton Euler & EulerNewton ModifiedNewton Table 6: Logarithm of the average number of predictor-corrector steps with the base being the input size N (N+d ) d. 15

16 [10] T.-L. Lee, T.-Y. Li, and C.-H. Tsai. HOM4PS-2.0: A software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing, 83, , Software available at [11] A. Leykin. Numerical algebraic geometry for Macaulay2. J. Softw. Algebr. Geom. 3, 5-10, [12] A. Leykin, and F. Sottile. Galois groups of Schubert problems via homotopy computation. Math. Comp., 78, , [13] M. Shub. Complexity of Bezout s theorem. VI. Geodesics in the condition (number) metric. Found. Comput. Math., 9(2), , [14] M. Shub and S. Smale. Complexity of Bézout s theorem. I. Geometric aspects. J. Amer. Math. Soc., 6(2), , [15] M. Shub and S. Smale. Complexity of Bezout s theorem. V. Polynomial time. Theoret. Comput. Sci., 133, , [16] S. Smale. The fundamental theorem of algebra and complexity theory. Bull. Amer. Math. Soc. (N.S.), 4(1), 1 36, [17] S. Smale. Newton s method estimates from data at one point. The merging of disciplines: new directions in pure, applied, and computational mathematics (Laramie, Wyo., 1985), Springer, New York, 1986, pp [18] A.J. Sommese, J. Verschelde, and C.W. Wampler. Using monodromy to decompose solution sets of polynomial systems into irreducible components. In Applications of algebraic geometry to coding theory, physics and computation, Vol. 36 of NATO Sci. Ser. II Math. Phys. Chem., Kluwer Acad. Publ., Dordrecht, 2001, pp [19] J. Verschelde. Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw., 25(2), , Software available at 16