Practical Advances in Complex Root Clustering Collaborative and - PowerPoint PPT Presentation

Practical Advances in Complex Root Clustering Collaborative and ongoing works R. Imbach 1 , V. Pan 2 , M. Pouget 3 , C. Yap 1 1 Courant Institute of Mathematical Sciences, New York University, USA 2 Lehman College, City University of New York, USA 3 INRIA Nancy - Grand Est, France

Introduction Triangular systems Univariate case 1/ 29 Example System: Let σ ≥ 3 and f ( z ) = 0 be: � ( z 1 − 2 − σ ) ( z 1 + 2 − σ ) = 0 ( z 2 + 2 σ z 2 1 ) ( z 2 − 1) z 2 = 0 Solutions: f ( z ) = 0 has 6 solutions, all real: a 1 = (2 − σ , 0) 1 a 2 = (2 − σ , 1) a 3 a 2 a 3 = ( − 2 − σ , 1) a 4 = ( − 2 − σ , 0) a 5 = ( − 2 − σ − 2 − σ ) , 2 − σ 2 − σ a 6 = (2 − σ − 2 − σ ) , a 4 a 1 2 − σ a 5 a 6 − 1 R. Imbach Seminar in Symbolic-Numeric Computing

Introduction Triangular systems Univariate case 1/ 29 Example System: Let σ ≥ 3 and f ( z ) = 0 be: � ( z 1 − 2 − σ ) ( z 1 + 2 − σ ) = 0 ( z 2 + 2 σ z 2 1 ) ( z 2 − 1) z 2 = 0 Solutions: f ( z ) = 0 has 6 solutions, all real: a 1 = (2 − σ ∆ 2 ← m ( a 1 , f ) = 1 , 0) 1 a 2 = (2 − σ ← m ( a 2 , f ) = 1 , 1) a 3 a 2 a 3 = ( − 2 − σ ← m ( a 3 , f ) = 1 , 1) a 4 = ( − 2 − σ ← m ( a 4 , f ) = 1 , 0) a 5 = ( − 2 − σ − 2 − σ ) ← m ( a 5 , f ) = 1 , 2 − σ 2 − σ a 6 = (2 − σ − 2 − σ ) ← m ( a 6 , f ) = 1 , a 4 a 1 Natural clusters: 2 − σ ∆ 1 ( ∆ 1 , 4) ( ∆ 2 , 2) a 5 a 6 − 1 Notations: m ( a , f ): multiplicity of a as a sol. of f R. Imbach Seminar in Symbolic-Numeric Computing

Introduction Triangular systems Univariate case 1/ 29 Example System: Let σ ≥ 3 and f ( z ) = 0 be: � ( z 1 − 2 − σ ) 2 ( z 1 + 2 − σ ) = 0 ( z 2 + 2 σ z 2 1 ) 2 ( z 2 − 1) z 2 = 0 Solutions: f ( z ) = 0 has 6 solutions, all real: a 1 = (2 − σ ∆ 2 ← m ( a 1 , f ) = 2 , 0) 1 a 2 = (2 − σ ← m ( a 2 , f ) = 2 , 1) a 3 a 2 a 3 = ( − 2 − σ ← m ( a 3 , f ) = 1 , 1) a 4 = ( − 2 − σ ← m ( a 4 , f ) = 1 , 0) a 5 = ( − 2 − σ − 2 − σ ) ← m ( a 5 , f ) = 2 , 2 − σ 2 − σ a 6 = (2 − σ − 2 − σ ) ← m ( a 6 , f ) = 4 , a 4 a 1 Natural clusters: 2 − σ ∆ 1 ( ∆ 1 , 9) ( ∆ 2 , 3) a 5 a 6 − 1 Notations: m ( a , f ): multiplicity of a as a sol. of f R. Imbach Seminar in Symbolic-Numeric Computing

Introduction Triangular systems Univariate case 2/ 29 Local solution Clustering Problem (LCP) a polynomial map f : C n → C n (assume f ( z ) = 0 is 0-dim), Input: a polybox B ⊂ C n , the Region of Interest (RoI), ǫ > 0 Output: Notations: f = ( f 1 , . . . , f n ), B = ( B 1 , . . . , B n ) where the B i ’s are square complex boxes R. Imbach Seminar in Symbolic-Numeric Computing

Introduction Triangular systems Univariate case 2/ 29 Local solution Clustering Problem (LCP) a polynomial map f : C n → C n (assume f ( z ) = 0 is 0-dim), Input: a polybox B ⊂ C n , the Region of Interest (RoI), ǫ > 0 Output: a set of pairs { ( ∆ 1 , m 1 ) , . . . , ( ∆ ℓ , m ℓ ) } where: • the ∆ j s are pairwise disjoint polydiscs of radius r ( ∆ j ) ≤ ǫ , Notations: f = ( f 1 , . . . , f n ), B = ( B 1 , . . . , B n ) where the B i ’s are square complex boxes ∆ j = (∆ j n ) where the ∆ j 1 , . . . , ∆ j i ’s are complex discs r (∆ j r ( ∆ j ) = max i ) i R. Imbach Seminar in Symbolic-Numeric Computing

Introduction Triangular systems Univariate case 2/ 29 Local solution Clustering Problem (LCP) a polynomial map f : C n → C n (assume f ( z ) = 0 is 0-dim), Input: a polybox B ⊂ C n , the Region of Interest (RoI), ǫ > 0 Output: a set of pairs { ( ∆ 1 , m 1 ) , . . . , ( ∆ ℓ , m ℓ ) } where: • the ∆ j s are pairwise disjoint polydiscs of radius r ( ∆ j ) ≤ ǫ , • m j = #( ∆ j , f ) = #(3 ∆ j , f ) for all 1 ≤ j ≤ ℓ , and Notations: f = ( f 1 , . . . , f n ), B = ( B 1 , . . . , B n ) where the B i ’s are square complex boxes ∆ j = (∆ j n ) where the ∆ j 1 , . . . , ∆ j i ’s are complex discs r (∆ j r ( ∆ j ) = max i ) i #( S , f ): nb. of sols (with mult.) of f ( z ) = 0 in S R. Imbach Seminar in Symbolic-Numeric Computing

Introduction Triangular systems Univariate case 2/ 29 Local solution Clustering Problem (LCP) a polynomial map f : C n → C n (assume f ( z ) = 0 is 0-dim), Input: a polybox B ⊂ C n , the Region of Interest (RoI), ǫ > 0 Output: a set of pairs { ( ∆ 1 , m 1 ) , . . . , ( ∆ ℓ , m ℓ ) } where: • the ∆ j s are pairwise disjoint polydiscs of radius r ( ∆ j ) ≤ ǫ , • m j = #( ∆ j , f ) = #(3 ∆ j , f ) for all 1 ≤ j ≤ ℓ , and • Z ( B , f ) ⊆ � ℓ j =1 Z ( ∆ j , f ) ⊆ Z ((1 + δ ) B , f ) for a small δ Notations: f = ( f 1 , . . . , f n ), B = ( B 1 , . . . , B n ) where the B i ’s are square complex boxes ∆ j = (∆ j n ) where the ∆ j 1 , . . . , ∆ j i ’s are complex discs r (∆ j r ( ∆ j ) = max i ) i #( S , f ): nb. of sols (with mult.) of f ( z ) = 0 in S Z ( S , f ): sols of f ( z ) = 0 in S R. Imbach Seminar in Symbolic-Numeric Computing

Introduction Triangular systems Univariate case 2/ 29 Local solution Clustering Problem (LCP) a polynomial map f : C n → C n (assume f ( z ) = 0 is 0-dim), Input: a polybox B ⊂ C n , the Region of Interest (RoI), ǫ > 0 Output: a set of pairs { ( ∆ 1 , m 1 ) , . . . , ( ∆ ℓ , m ℓ ) } where: • the ∆ j s are pairwise disjoint polydiscs of radius r ( ∆ j ) ≤ ǫ , • m j = #( ∆ j , f ) = #(3 ∆ j , f ) for all 1 ≤ j ≤ ℓ , and • Z ( B , f ) ⊆ � ℓ j =1 Z ( ∆ j , f ) ⊆ Z ((1 + δ ) B , f ) for a small δ Notations: f = ( f 1 , . . . , f n ), B = ( B 1 , . . . , B n ) where the B i ’s are square complex boxes ∆ j = (∆ j n ) where the ∆ j 1 , . . . , ∆ j i ’s are complex discs r (∆ j r ( ∆ j ) = max i ) i #( S , f ): nb. of sols (with mult.) of f ( z ) = 0 in S Z ( S , f ): sols of f ( z ) = 0 in S R. Imbach Seminar in Symbolic-Numeric Computing

Introduction Triangular systems Univariate case 2/ 29 Local solution Clustering Problem (LCP) a polynomial map f : C n → C n (assume f ( z ) = 0 is 0-dim), Input: a polybox B ⊂ C n , the Region of Interest (RoI), ǫ > 0 Output: a set of pairs { ( ∆ 1 , m 1 ) , . . . , ( ∆ ℓ , m ℓ ) } where: • the ∆ j s are pairwise disjoint polydiscs of radius r ( ∆ j ) ≤ ǫ , • m j = #( ∆ j , f ) = #(3 ∆ j , f ) for all 1 ≤ j ≤ ℓ , and • Z ( B , f ) ⊆ � ℓ j =1 Z ( ∆ j , f ) ⊆ Z ((1 + δ ) B , f ) for a small δ Notations: f = ( f 1 , . . . , f n ), B = ( B 1 , . . . , B n ) where the B i ’s are square complex boxes ∆ j = (∆ j n ) where the ∆ j 1 , . . . , ∆ j i ’s are complex discs r (∆ j r ( ∆ j ) = max i ) i #( S , f ): nb. of sols (with mult.) of f ( z ) = 0 in S Z ( S , f ): sols of f ( z ) = 0 in S R. Imbach Seminar in Symbolic-Numeric Computing

Introduction Triangular systems Univariate case 2/ 29 Local solution Clustering Problem (LCP) a polynomial map f : C n → C n (assume f ( z ) = 0 is 0-dim), Input: a polybox B ⊂ C n , the Region of Interest (RoI), ǫ > 0 Output: a set of pairs { ( ∆ 1 , m 1 ) , . . . , ( ∆ ℓ , m ℓ ) } where: • the ∆ j s are pairwise disjoint polydiscs of radius r ( ∆ j ) ≤ ǫ , • m j = #( ∆ j , f ) = #(3 ∆ j , f ) for all 1 ≤ j ≤ ℓ , and • Z ( B , f ) ⊆ � ℓ j =1 Z ( ∆ j , f ) ⊆ Z ((1 + δ ) B , f ) for a small δ Definition: a pair ( ∆ , m ) is called natural cluster (relative to f ) when it satisfies: m = #( ∆ , f ) = #(3 ∆ , f ) ≥ 1 if r ( ∆ ) ≤ ǫ , it is a natural ǫ -cluster R. Imbach Seminar in Symbolic-Numeric Computing

Introduction Triangular systems Univariate case 2/ 29 Example System: Let σ ≥ 3 and f ( z ) = 0 be: � ( z 1 − 2 − σ ) 2 ( z 1 + 2 − σ ) = 0 ( z 2 + 2 σ z 2 1 ) 2 ( z 2 − 1) z 2 = 0 Solutions: f ( z ) = 0 has 6 solutions, all real: a 1 = (2 − σ ∆ 2 ← m ( a 1 , f ) = 2 , 0) 1 a 2 = (2 − σ ← m ( a 2 , f ) = 2 , 1) a 3 a 2 a 3 = ( − 2 − σ ← m ( a 3 , f ) = 1 , 1) a 4 = ( − 2 − σ ← m ( a 4 , f ) = 1 , 0) a 5 = ( − 2 − σ − 2 − σ ) ← m ( a 5 , f ) = 2 , 2 − σ 2 − σ a 6 = (2 − σ − 2 − σ ) ← m ( a 6 , f ) = 4 , a 4 a 1 Natural clusters: 2 − σ ∆ 1 ( ∆ 1 , 9) ( ∆ 2 , 3) a 5 a 6 − 1 Notations: m ( a , f ): multiplicity of a as a sol. of f R. Imbach Seminar in Symbolic-Numeric Computing

Introduction Triangular systems Univariate case 2/ 29 Example System: Let σ ≥ 3 and f ( z ) = 0 be: � ( z 1 − 2 − σ ) 2 ( z 1 + 2 − σ ) = 0 ( z 2 + 2 σ z 2 1 ) 2 ( z 2 − 1) z 2 = 0 Solutions: f ( z ) = 0 has 6 solutions, all real: a 1 = (2 − σ ∆ 2 ← m ( a 1 , f ) = 2 , 0) 1 a 2 = (2 − σ ← m ( a 2 , f ) = 2 , 1) a 3 a 2 a 3 = ( − 2 − σ ← m ( a 3 , f ) = 1 , 1) a 4 = ( − 2 − σ ← m ( a 4 , f ) = 1 , 0) a 5 = ( − 2 − σ − 2 − σ ) ← m ( a 5 , f ) = 2 , 2 − σ 2 − σ a 6 = (2 − σ − 2 − σ ) ← m ( a 6 , f ) = 4 ∆ 3 , a 4 a 1 Natural clusters: 2 − σ ( ∆ 1 , 9) ∆ 4 ( ∆ 2 , 3) a 5 a 6 − 1 ( ∆ 3 , 3), ( ∆ 4 , 6) are not natural clusters R. Imbach Seminar in Symbolic-Numeric Computing

Introduction Triangular systems Univariate case 3/ 29 Why root clustering instead of root isolation? Root isolation: • input polynomials with Z or Q coefficients, or • input polynomials squarefree Root clustering: • input polynomials with any C coefficients • robust to multiple roots R. Imbach Seminar in Symbolic-Numeric Computing