Introduction Interval analysis Homotopy Continuation Method Our contribution Examples Conclusion and future work 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 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 The problem Our contribution Related work Examples Conclusion and future 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. Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based
Introduction Interval analysis Homotopy Continuation Method The problem Our contribution Related work Examples Conclusion and future 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. Fei Shen, Wenyuan Wu, Bican Xia Real Root Isolation of Polynomial Equations Based
Introduction Interval analysis Homotopy Continuation Method The problem Our contribution Related work Examples Conclusion and future 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 The problem Our contribution Related work Examples Conclusion and future 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 The problem Our contribution Related work Examples Conclusion and future 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 The problem Our contribution Related work Examples Conclusion and future 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 ⊆ 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. mid( X ) − mid( F ′ ( X )) − 1 f (mid( X )) K ( X ) = +( I − mid( F ′ ( X )) − 1 F ′ ( 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 ⊆ 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. mid( X ) − mid( F ′ ( X )) − 1 f (mid( X )) K ( X ) = +( I − mid( F ′ ( X )) − 1 F ′ ( 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
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