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Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Real Root Isolation of Polynomial Equations Based on Hybrid Computation

Fei Shen1 Wenyuan Wu2 Bican Xia1

LMAM & School of Mathematical Sciences, Peking University Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences Beijing, Oct. 27, 2012

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Outline

1 Introduction

The problem Related work

2 Interval analysis 3 Homotopy Continuation Method 4 Our contribution

Construct initial boxes An empirical estimation

5 Examples 6 Conclusion and future work

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work The problem Related work

The problem

Let F = (f1, . . . , fn)T be a polynomial system defined on Rn where fi ∈ R[x1, . . . , xn]. 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 S1, . . . , Sm, Sj ⊂ Rn(1 ≤ j ≤ m), such that ξ(j) ∈ Sj and Si ∩ Sj = ∅ (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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work The problem Related work

The problem

Let F = (f1, . . . , fn)T be a polynomial system defined on Rn where fi ∈ R[x1, . . . , xn]. 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 S1, . . . , Sm, Sj ⊂ Rn(1 ≤ j ≤ m), such that ξ(j) ∈ Sj and Si ∩ Sj = ∅ (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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work 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), John- son(1998), Rouillier-Zimmermann(2004), Johnson-Krandick-

• et. al.(2005, 2006), Sagraloff(2012), ...

Based on Vincent’s Theorem and continued fractions: Akritas(1980), Akritas-Strzebo´ nski-et. al.(2005, 2006, 2008), S ¸tef˘ anescu(2005), Sharma(2008), ...

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work The problem Related work

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 form. Or, the system has to be transformed into such forms symbolically. Motivation Can we develop a method which avoids pre-processing the system symbolically?

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work The problem Related work

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 form. Or, the system has to be transformed into such forms symbolically. Motivation Can we develop a method which avoids pre-processing the system symbolically?

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work The problem Related work

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 form. Or, the system has to be transformed into such forms symbolically. Motivation Can we develop a method which avoids pre-processing the system symbolically?

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

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

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Matlab package Intlab (Rump(1999, 2010)) Interval arithmetic operations, Jacobian matrix and Hessian ma- trix calculations, etc. Using floating-point arithmetic but the results are rigorous. Theoretically, the width of intervals for some special prob- lems 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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Matlab package Intlab (Rump(1999, 2010)) Interval arithmetic operations, Jacobian matrix and Hessian ma- trix calculations, etc. Using floating-point arithmetic but the results are rigorous. Theoretically, the width of intervals for some special prob- lems 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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Krawczyk operator

Definition (Moore(1966))

Suppose f : D ⊆ Rn → Rn 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))−1f (mid(X)) +(I − mid(F′(X))−1F′(X))rad(X)[−1, 1] (2)

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Krawczyk operator

Definition (Moore(1966))

Suppose f : D ⊆ Rn → Rn 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))−1f (mid(X)) +(I − mid(F′(X))−1F′(X))rad(X)[−1, 1] (2)

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Homotopy continuation method is an important numerical com- putation 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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Homotopy continuation method is an important numerical com- putation 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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Homotopy continuation method is an important numerical com- putation 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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work 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?

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work 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?

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work 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 con- taining 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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work 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 con- taining 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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work 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 con- taining 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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work 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 Fr´ echet differentiable on an open convex set D0 ⊆ D. For equation F(x) = 0, if the given approximate zero x0 ∈ D0 meets the following three conditions: F ′(x0)−1 exists, and there are real numbers B and η such that F ′(x0)−1 ≤ B, F ′(x0)−1F(x0) ≤ η, F ′ satisfies the Lipschitz condition on D0: F ′(x) − F ′(y) ≤ Kx − y, ∀x, y ∈ D0, h = BKη ≤ 1

2, O(x0, 1− √ 1−2h h

η) ⊂ D0,

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work 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(x0, 1−

√ 1−2h h

η) ⊂ D0, and the sequence {xk : xk+1 = xk − F ′(xk)−1F(xk)} of Newton method converges to x∗; For the convergence of x∗, we have: x∗ − xk+1 ≤ θ2k+1(1−θ2)

θ(1−θ2k+1) η

where θ = 1−

√ 1−2h 1+ √ 1−2h ;

The root x∗ is unique in D0 ∩ O(x0, 1+

√ 1−2h h

η). The proof can be found in Gragg-Tapia(1974).

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work Construct initial boxes An empirical estimation

Construct initial boxes

Proposition

Let F = (f1, . . . , fn)T be a polynomial system, where fi ∈ R[x1, . . . , xn]. De- note by J the Jacobian matrix of F. For an approximation x0 ∈ C n, if the following conditions hold:

1

J −1(x0) exists, and there are real numbers B and η such that J −1(x0) ≤ B, J −1(x0)F(x0) ≤ η,

2

There exists a neighborhood O(x0, ω) such that J(x) satisfies the Lips- chitz condition on it: J(x) − J(y) ≤ Kx − 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(x0, ω) ∩ O(x0, 1+√1−2h

h

η).

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work Construct initial boxes An empirical estimation

Construct initial boxes

Proof. We consider F as an operator on Cn → Cn, obviously it is Fr´ echet differentiable, and from F(x +h) = F(x)+J(x)h +o(h) we can get lim

h→0

F(x + h) − F(x) − J(x)h h = 0. Thus the first order Fr´ echet 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

• f x∗ − x0.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work 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 neigh-

borhood O(x0, 2η). Even though the initial x0 does not satisfy the conditions, we can still find a proper xk after several New- ton 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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work Construct initial boxes An empirical estimation

Construct initial boxes

Bound for the Lipschitz constant K

Let Jij = ∂fi/∂xj, apply mean value inequality (Mujica(1986)) to each ele- ment of J on O(x0, ω) to get Jij(y) − Jij(x) ≤ sup

κij∈line(x,y)

∇Jij(κij) · y − x, ∀x, y ∈ O(x0, ω) (3) where ∇ = (∂/∂x1, ∂/∂x2, . . . , ∂/∂xn) 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 (X0)|max,

(4) where h(i)

j

are the column vectors of the Hessian matrix of fi.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work Construct initial boxes An empirical estimation

Construct initial boxes

Bound for the Lipschitz constant K

Let Jij = ∂fi/∂xj, apply mean value inequality (Mujica(1986)) to each ele- ment of J on O(x0, ω) to get Jij(y) − Jij(x) ≤ sup

κij∈line(x,y)

∇Jij(κij) · y − x, ∀x, y ∈ O(x0, ω) (3) where ∇ = (∂/∂x1, ∂/∂x2, . . . , ∂/∂xn) 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 (X0)|max,

(4) where h(i)

j

are the column vectors of the Hessian matrix of fi.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work Construct initial boxes An empirical estimation

Algorithm: init width

Input: Equation F; Approximation x0; Number of variables n Output: Initial interval’s radius r

1:

repeat

2:

x0 = x0 − J−1(x0)F(x0);

3:

B = J−1(x0)∞; η = J−1(x0)F(x0)∞;

4:

ω = 2η;

5:

X0 = midrad(x0, ω);

6:

K = 0;

7:

for i = 1 to n do

8:

Compute the Hessian matrix Hi = (h(i)

1

, h(i)

2

, . . . , h(i)

n

) of F on X0;

9:

if n

j=1 |h(i) j

(X0)|max > K then

10:

K = n

j=1 |h(i) j

(X0)|max;

11:

end if

12:

end for

13:

h = BKη;

14:

until h ≤ 1/2

15:

return r = 1−√

1−2h h

η Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work Construct initial boxes An empirical estimation

An empirical estimation

For F = 0, let x∗ be an accurate root and x0 be its approximation. we

• btain the empirical estimation

x∗ − x0∞ ≈ J −1(ξ)F(x0)∞ ≤ J −1(x0)∞F(x0)∞ 1 − λJ −1(x0)2

∞F(x0)∞

, where λ = max1≤i≤n

n

j=1 |h(i) j (x0)|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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work Construct initial boxes An empirical estimation

An empirical estimation

For F = 0, let x∗ be an accurate root and x0 be its approximation. we

• btain the empirical estimation

x∗ − x0∞ ≈ J −1(ξ)F(x0)∞ ≤ J −1(x0)∞F(x0)∞ 1 − λJ −1(x0)2

∞F(x0)∞

, where λ = max1≤i≤n

n

j=1 |h(i) j (x0)|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.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Our algorithms have been implemented as a Matlab pro-

• gram. We use Intlab of Version 6 for interval calculation and

Hom4ps-2.0 as our homotopy continuation tool. All the exper- iments are undertaken in Matlab2008b on a PC (OS: Windows Vista, CPU: Inter R Core 2 Duo T6500 2.10GHz, Memory: 2G.). For arbitrarily high accuracy, we can call Matlab’s vpa (variable precision arithmetic), but in fact all the real roots of the exam- ples below are isolated by using Matlab’s default double-precision floating point.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Our algorithms have been implemented as a Matlab pro-

• gram. We use Intlab of Version 6 for interval calculation and

Hom4ps-2.0 as our homotopy continuation tool. All the exper- iments are undertaken in Matlab2008b on a PC (OS: Windows Vista, CPU: Inter R Core 2 Duo T6500 2.10GHz, Memory: 2G.). For arbitrarily high accuracy, we can call Matlab’s vpa (variable precision arithmetic), but in fact all the real roots of the exam- ples below are isolated by using Matlab’s default double-precision floating point.

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Equation # roots # real roots

DISCOVERER complex roots detected

barry 20 2 2 18 cyclic5 70 10 10 60 cyclic6 156 24

• 132

des18 3 46 6

• 40

eco7 32 8 8 24 eco8 64 8

• 56

geneig 10 10

• kinema

40 8

• 32

reimer4 36 8 8 28 reimer5 144 24

• 120

virasoro 256 224

• 32

Table: Real root isolation results comparison

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Equations Total time Homotopy time Interval time DISCOVERER barry 0.421203 0.093601 0.327602 0.063 cyclic5 2.948419 0.218401 2.652017 0.624 cyclic6 9.984064 0.639604 9.063658 > 1000 des18 3 4.180827 0.702004 3.385222 > 1000 eco7 2.371215 0.265202 2.012413 15.881 eco8 3.946825 0.499203 3.354022 > 1000 geneig 4.243227 0.249602 3.868825 > 1000 kinema 3.946825 1.014006 2.808018 > 1000 reimer4 2.480416 0.374402 2.059213 24.711 reimer5 12.963683 3.073220 9.578461 > 1000 virasoro 137.124879 4.570829 109.996305 > 1000

Table: Execution time comparison, unit:s

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Equation avg.rad.of intv avg.rad.of init. avg.time of iter max time of iter barry 3.552714e-015 1.377800e-014 0.054600 0.062400 cyclic5 1.614703e-009 7.142857e-007 0.113881 0.140401 cyclic6 4.440892e-016 2.137195e-015 0.183951 0.234002 des18 3 3.768247e-007 9.737288e-007 0.241802 0.296402 eco7 1.998401e-015 1.483754e-013 0.122851 0.156001 eco8 2.109424e-015 3.283379e-013 0.183301 0.218401 geneig 2.664535e-016 5.721530e-014 0.315122 0.436803 kinema 1.998401e-015 6.784427e-011 0.157951 0.218401 reimer4 1.110223e-016 1.258465e-014 0.122851 0.156001 reimer5 1.110223e-016 4.754080e-014 0.195001 0.421203 virasoro 9.472120e-009 2.265625e-006 0.387844 0.624004

Table: Detail data for each iteration, unit:s

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

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...

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

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...

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

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...

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

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...

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

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...

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

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...

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based

Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work

Thanks!

Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based