Real Root Isolation of Polynomial Equations Ba. Equations Based on Hybrid Computation

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1 Real Root Isolation of Polynomial Equations Based on Hybrid Computation Fei Shen 1 Wenyuan Wu 2 Bican Xia 1 LMAM & School of Mathematical Sciences, Peking University Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences Beijing, Oct. 27, 2012

2 Outline 1 Introduction The problem Related work Construct initial boxes An empirical estimation 5 6

3 The problem Related work The problem Let F = (f 1,..., f n ) T be a polynomial system defined on R n where f i R[x 1,..., x n ]. Suppose F(x) = 0 has only finite many real roots, say ξ (1),..., ξ (m). The target of real root isolation is to compute a family of regions S 1,..., S m, S j R n (1 j m), such that ξ (j) S j and S i S j = (1 i, j m). Hypothesis on the problem. 1 The system is square. 2 The system has only finite many roots. 3 The Jacobian matrix of F is non-singular at each root of F(x) = 0.

4 The problem Related work The problem Let F = (f 1,..., f n ) T be a polynomial system defined on R n where f i R[x 1,..., x n ]. Suppose F(x) = 0 has only finite many real roots, say ξ (1),..., ξ (m). The target of real root isolation is to compute a family of regions S 1,..., S m, S j R n (1 j m), such that ξ (j) S j and S i S j = (1 i, j m). Hypothesis on the problem. 1 The system is square. 2 The system has only finite many roots. 3 The Jacobian matrix of F is non-singular at each root of F(x) = 0.

5 The problem Related work Related work Univariate case Based on Descartes rule and bisection: Collins-Akritas(1976), Collins-Loos(1983), Collins-Johnson(1989), Johnson-Krandick(1997), von zur Gathen-Gerhard(1997), Johnson(1998), Rouillier-Zimmermann(2004), Johnson-Krandicket. al.(2005, 2006), Sagraloff(2012),... Based on Vincent s Theorem and continued fractions: Akritas(1980), Akritas-Strzeboński-et. al.(2005, 2006, 2008), Ştefănescu(2005), Sharma(2008),...

6 The problem Related work form. Multivariate case Xia-Yang(2002), Zhang-Xiao-Xia(2005), Xia-Zhang(2006), Cheng- Gao-Yap(2007), Boulier-Chen-Lemaire-Moreno Maza(2009), Mourrain-Pavone(2009), Cheng-Gao-Guo(2012),... The system should be in some special shapes, e.g., triangular symbolically. Or, the system has to be transformed into such forms Motivation Can we develop a method which avoids pre-processing the system symbolically?

7 The problem Related work form. Multivariate case Xia-Yang(2002), Zhang-Xiao-Xia(2005), Xia-Zhang(2006), Cheng- Gao-Yap(2007), Boulier-Chen-Lemaire-Moreno Maza(2009), Mourrain-Pavone(2009), Cheng-Gao-Guo(2012),... The system should be in some special shapes, e.g., triangular symbolically. Or, the system has to be transformed into such forms Motivation Can we develop a method which avoids pre-processing the system symbolically?

8 The problem Related work form. Multivariate case Xia-Yang(2002), Zhang-Xiao-Xia(2005), Xia-Zhang(2006), Cheng- Gao-Yap(2007), Boulier-Chen-Lemaire-Moreno Maza(2009), Mourrain-Pavone(2009), Cheng-Gao-Guo(2012),... The system should be in some special shapes, e.g., triangular symbolically. Or, the system has to be transformed into such forms Motivation Can we develop a method which avoids pre-processing the system symbolically?

9 Interval arithmetic Let X = [x, x], Y = [y, y] I (R), - X + Y = [x + y, x + y] - X Y = [x y, x y] - X Y = [min(xy, xy, xy, xy), max(xy, xy, xy, xy)] - X/Y = [x, x] [1/y, 1/y], 0 Y

10 Matlab package Intlab (Rump(1999, 2010)) Interval arithmetic operations, Jacobian matrix and Hessian matrix calculations, etc. Using floating-point arithmetic but the results are rigorous. Theoretically, the width of intervals for some special problems can be very small. Hence, we assume that the accuracy of numerical computation in this paper can be arbitrarily high (For arbitrarily high accuracy, we can call Matlab s vpa). However, it is also important to point out that such case rarely happens and double-precision is usually enough to obtain very small intervals in practice.

11 Matlab package Intlab (Rump(1999, 2010)) Interval arithmetic operations, Jacobian matrix and Hessian matrix calculations, etc. Using floating-point arithmetic but the results are rigorous. Theoretically, the width of intervals for some special problems can be very small. Hence, we assume that the accuracy of numerical computation in this paper can be arbitrarily high (For arbitrarily high accuracy, we can call Matlab s vpa). However, it is also important to point out that such case rarely happens and double-precision is usually enough to obtain very small intervals in practice.

12 Krawczyk operator Definition (Moore(1966)) Suppose f : D R n R n is continuous differentiable on D. Consider the equation f (x) = 0. Let f be the Jacobi matrix of f, F and F be the interval expand of f and f with inclusive monotonicity, respectively. For X I (D) and any y X, define the Krawczyk operator (K-operator) as: K(y, X) = y Y f (y) + (I Y F (X))(X y) (1) where Y is any n n non-singular matrix. K(X) = mid(x) mid(f (X)) 1 f (mid(x)) +(I mid(f (X)) 1 F (X))rad(X)[ 1, 1] (2)

13 Krawczyk operator Definition (Moore(1966)) Suppose f : D R n R n is continuous differentiable on D. Consider the equation f (x) = 0. Let f be the Jacobi matrix of f, F and F be the interval expand of f and f with inclusive monotonicity, respectively. For X I (D) and any y X, define the Krawczyk operator (K-operator) as: K(y, X) = y Y f (y) + (I Y F (X))(X y) (1) where Y is any n n non-singular matrix. K(X) = mid(x) mid(f (X)) 1 f (mid(x)) +(I mid(f (X)) 1 F (X))rad(X)[ 1, 1] (2)

14 Properties of K-operator Proposition (Moore(1966)) Suppose K(y, X) is defined as Formula (1), then 1 If x X is a root of F(x) = 0, then for any y X, we have x K(y, X); 2 y X, if X K(y, X) =, there is no roots in X; 3 y X and any non-singular matrix Y, if K(y, X) X, F(x) = 0 has a root in X; 4 y X and any non-singular matrix Y, if K(y, X) X, F(x) = 0 has only one root in X.

15 For a bounded region, say a box (i.e., interval vector), if we apply K-operator and subdivision strategy, it is possible to isolate all the roots in the box of F(x) = 0. The problem is: if the box is not small, the efficiency is low. Especially, if we want to isolate all the real roots of F(x) = 0, the estimated box is usually too big.

16 For a bounded region, say a box (i.e., interval vector), if we apply K-operator and subdivision strategy, it is possible to isolate all the roots in the box of F(x) = 0. The problem is: if the box is not small, the efficiency is low. Especially, if we want to isolate all the real roots of F(x) = 0, the estimated box is usually too big.

17 Homotopy continuation method is an important numerical computation method, which is used in various fields. We only treat it as an algorithm black box here, where the input is a polynomial system, and the output is its approximate roots. The details about the theory can be found in Li(1997) and Sommese-Wampler(2005). Homotopy software Hom4ps-2.0 (Li(2008)), PHCpack (Verschelde(1999)), HomLab (Wampler(2005)), etc. In our implementation, we use Hom4ps-2.0, which could return all the approximate complex roots of a given polynomial system efficiently, along with residues and condition numbers.

18 Homotopy continuation method is an important numerical computation method, which is used in various fields. We only treat it as an algorithm black box here, where the input is a polynomial system, and the output is its approximate roots. The details about the theory can be found in Li(1997) and Sommese-Wampler(2005). Homotopy software Hom4ps-2.0 (Li(2008)), PHCpack (Verschelde(1999)), HomLab (Wampler(2005)), etc. In our implementation, we use Hom4ps-2.0, which could return all the approximate complex roots of a given polynomial system efficiently, along with residues and condition numbers.

19 Homotopy continuation method is an important numerical computation method, which is used in various fields. We only treat it as an algorithm black box here, where the input is a polynomial system, and the output is its approximate roots. The details about the theory can be found in Li(1997) and Sommese-Wampler(2005). Homotopy software Hom4ps-2.0 (Li(2008)), PHCpack (Verschelde(1999)), HomLab (Wampler(2005)), etc. In our implementation, we use Hom4ps-2.0, which could return all the approximate complex roots of a given polynomial system efficiently, along with residues and condition numbers.

20 Construct initial boxes An empirical estimation Basic idea By Homotopy continuation, we can obtain all the approximate (complex) roots of the given polynomial system efficiently. Then, it is possible to get very small initial boxes. For each small initial box, K-operator can serve as an efficient tool verifying whether or not there is a real root in it. Key problem How to construct those initial boxes?

21 Construct initial boxes An empirical estimation Basic idea By Homotopy continuation, we can obtain all the approximate (complex) roots of the given polynomial system efficiently. Then, it is possible to get very small initial boxes. For each small initial box, K-operator can serve as an efficient tool verifying whether or not there is a real root in it. Key problem How to construct those initial boxes?

22 Construct initial boxes An empirical estimation Suppose z is an approximate root corresponding to the accurate root z of the system. Let I be an initial box to be constructed containing z. We must guarantee that I contains z. We give a rigorous result to estimate the distance between z and z based on the Kantorovich Theorem. For the sake of efficiency, we also give an empirical estimation to detect those roots with big imaginary parts which are highly impossible to be real.

23 Construct initial boxes An empirical estimation Suppose z is an approximate root corresponding to the accurate root z of the system. Let I be an initial box to be constructed containing z. We must guarantee that I contains z. We give a rigorous result to estimate the distance between z and z based on the Kantorovich Theorem. For the sake of efficiency, we also give an empirical estimation to detect those roots with big imaginary parts which are highly impossible to be real.

24 Construct initial boxes An empirical estimation Suppose z is an approximate root corresponding to the accurate root z of the system. Let I be an initial box to be constructed containing z. We must guarantee that I contains z. We give a rigorous result to estimate the distance between z and z based on the Kantorovich Theorem. For the sake of efficiency, we also give an empirical estimation to detect those roots with big imaginary parts which are highly impossible to be real.

25 Construct initial boxes An empirical estimation Construct initial boxes Theorem (Kantorovich) Let X and Y be Banach spaces and F : D X Y be an operator, which is Fréchet differentiable on an open convex set D 0 D. equation F(x) = 0, if the given approximate zero x 0 D 0 meets the following three conditions: F (x 0 ) 1 For exists, and there are real numbers B and η such that F (x 0 ) 1 B, F (x 0 ) 1 F(x 0 ) η, F satisfies the Lipschitz condition on D 0 : F (x) F (y) K x y, x, y D 0, h = BKη 1 2, O(x 0, 1 1 2h h η) D 0,

26 Construct initial boxes An empirical estimation Construct initial boxes Theorem (Kantorovich(continued)) then we claim that: F(x) = 0 has a root x in O(x 0, 1 1 2h h η) D 0, and the sequence {x k : x k+1 = x k F (x k ) 1 F(x k )} of Newton method converges to x ; For the convergence of x, we have: x x k+1 θ2k+1 (1 θ 2 ) θ(1 θ 2k+1 ) η where θ = 1 1 2h h ; The root x is unique in D 0 O(x 0, h h η). The proof can be found in Gragg-Tapia(1974).

27 Construct initial boxes An empirical estimation Construct initial boxes Proposition Let F = (f 1,..., f n) T be a polynomial system, where f i R[x 1,..., x n]. Denote by J the Jacobian matrix of F. For an approximation x 0 C n, if the following conditions hold: 1 J 1 (x 0) exists, and there are real numbers B and η such that J 1 (x 0) B, J 1 (x 0)F(x 0) η, 2 There exists a neighborhood O(x 0, ω) such that J(x) satisfies the Lipschitz condition on it: J(x) J(y) K x y 3 Let h = BKη, h 1 2, and ω 1 1 2h h η, then F(x) = 0 has only one root x in O(x 0, ω) O(x 0, h η). h

28 Construct initial boxes An empirical estimation Construct initial boxes Proof. We consider F as an operator on C n C n, obviously it is Fréchet differentiable, and from F(x +h) = F(x)+J(x)h +o(h) we can get F(x + h) F(x) J(x)h lim = 0. h 0 h Thus the first order Fréchet derivative of F is just the Jacobian matrix J, i.e. F (x) = J(x). So by the Kantorovich Theorem, the proof is completed immediately after checking the situation of x x 0.

29 Construct initial boxes An empirical estimation Construct initial boxes It is easy to know that 1 1 2h h 2. So we can just assign ω = 2η. Then we need to check whether BKη 1 2 in the neighborhood O(x 0, 2η). Even though the initial x 0 does not satisfy the conditions, we can still find a proper x k after several Newton iterations, since B and K are bounded and η will approach zero. And we only need to find an upper bound for the Lipschitz constant K.

30 Construct initial boxes An empirical estimation Construct initial boxes Bound for the Lipschitz constant K Let J ij = f i/ x j, apply mean value inequality (Mujica(1986)) to each element of J on O(x 0, ω) to get J ij(y) J ij(x) sup J ij(κ ij) y x, x, y O(x 0, ω) (3) κ ij line(x,y) where = ( / x 1, / x 2,..., / x n) is the gradient operator and line(x, y) refers to the line connecting x with y. By estimating the inequality, we finally get that K = max 1 i n n j=1 h (i) j (X 0) max, (4) where h (i) j are the column vectors of the Hessian matrix of f i.

31 Construct initial boxes An empirical estimation Construct initial boxes Bound for the Lipschitz constant K Let J ij = f i/ x j, apply mean value inequality (Mujica(1986)) to each element of J on O(x 0, ω) to get J ij(y) J ij(x) sup J ij(κ ij) y x, x, y O(x 0, ω) (3) κ ij line(x,y) where = ( / x 1, / x 2,..., / x n) is the gradient operator and line(x, y) refers to the line connecting x with y. By estimating the inequality, we finally get that K = max 1 i n n j=1 h (i) j (X 0) max, (4) where h (i) j are the column vectors of the Hessian matrix of f i.

32 Construct initial boxes An empirical estimation Algorithm: init width Input: Output: Equation F; Approximation x 0 ; Number of variables n Initial interval s radius r 1: repeat 2: x 0 = x 0 J 1 (x 0 )F(x 0 ); 3: B = J 1 (x 0 ) ; η = J 1 (x 0 )F(x 0 ) ; 4: ω = 2η; 5: X 0 = midrad(x 0, ω); 6: K = 0; 7: for i = 1 to n do 8: Compute the Hessian matrix H i = (h (i) 1, h(i) 2,..., h(i) n ) of F on X 0; 9: n if j=1 h(i) (X j 0 ) max > K then n j=1 h(i) (X j 0 ) max; 10: K = 11: end if 12: end for 13: h = BKη; 14: until h 1/2 15: return r = 1 1 2h η h

33 Construct initial boxes An empirical estimation An empirical estimation For F = 0, let x be an accurate root and x 0 be its approximation. we obtain the empirical estimation x x 0 J 1 (ξ)f(x 0) J 1 (x 0) F(x 0) 1 λ J 1 (x 0) 2 F(x 0), n where λ = max 1 i n j=1 h(i) j (x 0) max. Notice that the inequality is only a non-rigorous estimation. All the computation in it are carried out in a point-wise way, so it is faster than the rigorous result. In the numerical experiments later we will see that this empirically estimated radius performs very well. So we can use it to detect those non-real roots rather than the interval arithmetic.

34 Construct initial boxes An empirical estimation An empirical estimation For F = 0, let x be an accurate root and x 0 be its approximation. we obtain the empirical estimation x x 0 J 1 (ξ)f(x 0) J 1 (x 0) F(x 0) 1 λ J 1 (x 0) 2 F(x 0), n where λ = max 1 i n j=1 h(i) j (x 0) max. Notice that the inequality is only a non-rigorous estimation. All the computation in it are carried out in a point-wise way, so it is faster than the rigorous result. In the numerical experiments later we will see that this empirically estimated radius performs very well. So we can use it to detect those non-real roots rather than the interval arithmetic.

35 Our algorithms have been implemented as a Matlab program. We use Intlab of Version 6 for interval calculation and Hom4ps-2.0 as our homotopy continuation tool. All the experiments are undertaken in Matlab2008b on a PC (OS: Windows Vista, CPU: Inter R Core 2 Duo T GHz, Memory: 2G.). For arbitrarily high accuracy, we can call Matlab s vpa (variable precision arithmetic), but in fact all the real roots of the examples below are isolated by using Matlab s default double-precision floating point.

36 Our algorithms have been implemented as a Matlab program. We use Intlab of Version 6 for interval calculation and Hom4ps-2.0 as our homotopy continuation tool. All the experiments are undertaken in Matlab2008b on a PC (OS: Windows Vista, CPU: Inter R Core 2 Duo T GHz, Memory: 2G.). For arbitrarily high accuracy, we can call Matlab s vpa (variable precision arithmetic), but in fact all the real roots of the examples below are isolated by using Matlab s default double-precision floating point.

37 Equation # roots # real roots DISCOVERER complex roots detected barry cyclic cyclic des eco eco geneig kinema reimer reimer virasoro Table: Real root isolation results comparison

38 Equations Total time Homotopy time Interval time DISCOVERER barry cyclic cyclic > 1000 des > 1000 eco eco > 1000 geneig > 1000 kinema > 1000 reimer reimer > 1000 virasoro > 1000 Table: Execution time comparison, unit:s

39 Equation avg.rad.of intv avg.rad.of init. avg.time of iter max time of iter barry e e cyclic e e cyclic e e des e e eco e e eco e e geneig e e kinema e e reimer e e reimer e e virasoro e e Table: Detail data for each iteration, unit:s

40 For non-singular and zero-dimensional polynomial systems we propose a new algorithm for real root isolation based on hybrid computation. Our experiments on many benchmarks show that the new hybrid method is more efficient, compared with the traditional symbolic approaches. Future work: Systems with multiple roots or positive dimensional solutions (ongoing); Better construction of initial intervals; Parallel computation; Subdivision strategy and other techniques; Better implementation...

41 For non-singular and zero-dimensional polynomial systems we propose a new algorithm for real root isolation based on hybrid computation. Our experiments on many benchmarks show that the new hybrid method is more efficient, compared with the traditional symbolic approaches. Future work: Systems with multiple roots or positive dimensional solutions (ongoing); Better construction of initial intervals; Parallel computation; Subdivision strategy and other techniques; Better implementation...

42 For non-singular and zero-dimensional polynomial systems we propose a new algorithm for real root isolation based on hybrid computation. Our experiments on many benchmarks show that the new hybrid method is more efficient, compared with the traditional symbolic approaches. Future work: Systems with multiple roots or positive dimensional solutions (ongoing); Better construction of initial intervals; Parallel computation; Subdivision strategy and other techniques; Better implementation...

43 For non-singular and zero-dimensional polynomial systems we propose a new algorithm for real root isolation based on hybrid computation. Our experiments on many benchmarks show that the new hybrid method is more efficient, compared with the traditional symbolic approaches. Future work: Systems with multiple roots or positive dimensional solutions (ongoing); Better construction of initial intervals; Parallel computation; Subdivision strategy and other techniques; Better implementation...

44 For non-singular and zero-dimensional polynomial systems we propose a new algorithm for real root isolation based on hybrid computation. Our experiments on many benchmarks show that the new hybrid method is more efficient, compared with the traditional symbolic approaches. Future work: Systems with multiple roots or positive dimensional solutions (ongoing); Better construction of initial intervals; Parallel computation; Subdivision strategy and other techniques; Better implementation...

45 For non-singular and zero-dimensional polynomial systems we propose a new algorithm for real root isolation based on hybrid computation. Our experiments on many benchmarks show that the new hybrid method is more efficient, compared with the traditional symbolic approaches. Future work: Systems with multiple roots or positive dimensional solutions (ongoing); Better construction of initial intervals; Parallel computation; Subdivision strategy and other techniques; Better implementation...

46 Thanks!

Real Root Isolation of Regular Chains.

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