Near Optimal Subdivision Algorithms for Real Root Isolation Vikram - PowerPoint PPT Presentation

Near Optimal Subdivision Algorithms for Real Root Isolation Vikram Sharma 1 and Prashant Batra 2 1 Institute of Mathematical Sciences, Chennai, India. 2 Technische Universität Hamburg-Harburg, Hamburg, Germany. ISSAC, Bath, 2015 V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 1 / 14

Real Root Isolation The Problem Input: f ( x ) ∈ R [ x ] , degree n , and an interval I 0 . Output: Disjoint intervals with endpoints in Q containing roots of f . f ( x ) V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 2 / 14

Real Root Isolation The Problem Input: f ( x ) ∈ R [ x ] , degree n , and an interval I 0 . Output: Disjoint intervals with endpoints in Q containing roots of f . f ( x ) I 0 V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 2 / 14

Real Root Isolation The Problem Input: f ( x ) ∈ R [ x ] , degree n , and an interval I 0 . Output: Disjoint intervals with endpoints in Q containing roots of f . f ( x ) I 0 V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 2 / 14

Real Root Isolation The Problem Input: f ( x ) ∈ R [ x ] , degree n , and an interval I 0 . Output: Disjoint intervals with endpoints in Q containing roots of f . f ( x ) I 0 Assumption All the roots are of multiplicity one, i.e., f is square-free. V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 2 / 14

Subdivision Methods Predicates Exclusion C 0 ( I ) : if true then I has no roots. Inclusion C 1 ( I ) : if true then I has exactly one root. E.g., Sturm sequences, Descartes’s rule of signs, Interval arithmetic,... V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 3 / 14

Subdivision Methods Predicates Exclusion C 0 ( I ) : if true then I has no roots. Inclusion C 1 ( I ) : if true then I has exactly one root. E.g., Sturm sequences, Descartes’s rule of signs, Interval arithmetic,... The Binary Tree T I 0 f ( x ) I 0 V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 3 / 14

Subdivision Methods Predicates Exclusion C 0 ( I ) : if true then I has no roots. Inclusion C 1 ( I ) : if true then I has exactly one root. E.g., Sturm sequences, Descartes’s rule of signs, Interval arithmetic,... The Binary Tree T I 0 f ( x ) I 0 V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 3 / 14

Subdivision Methods Predicates Exclusion C 0 ( I ) : if true then I has no roots. Inclusion C 1 ( I ) : if true then I has exactly one root. E.g., Sturm sequences, Descartes’s rule of signs, Interval arithmetic,... The Binary Tree T I 0 f ( x ) I 0 V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 3 / 14

Subdivision Methods Predicates Exclusion C 0 ( I ) : if true then I has no roots. Inclusion C 1 ( I ) : if true then I has exactly one root. E.g., Sturm sequences, Descartes’s rule of signs, Interval arithmetic,... The Binary Tree T I 0 f ( x ) I 0 V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 3 / 14

Subdivision Methods Predicates Exclusion C 0 ( I ) : if true then I has no roots. Inclusion C 1 ( I ) : if true then I has exactly one root. E.g., Sturm sequences, Descartes’s rule of signs, Interval arithmetic,... The Binary Tree T I 0 f ( x ) I 0 V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 3 / 14

Subdivision Methods – Complexity Analysis Two Components The size # T I 0 of the tree T I 0 . The worst case arithmetic complexity at a node. Usually � O ( n ) . V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 4 / 14

Subdivision Methods – Complexity Analysis Two Components The size # T I 0 of the tree T I 0 . The worst case arithmetic complexity at a node. Usually � O ( n ) . V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 4 / 14

Subdivision Methods – Complexity Analysis Two Components The size # T I 0 of the tree T I 0 . The worst case arithmetic complexity at a node. Usually � O ( n ) . Bounds on # T I 0 O ( log1 / σ ) , σ is the root separation bound for f . Sturm’s method, Descartes’s rule of signs, Interval arithmetic. Dav.’85, Eig.-S.-Yap’06, S.-Yap’12. V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 4 / 14

Subdivision Methods – Complexity Analysis Two Components Optimal for pure subdivision The size # T I 0 of the tree T I 0 . The worst case arithmetic complexity at a node. Usually � O ( n ) . Bounds on # T I 0 O ( log1 / σ ) , σ is the root separation bound for f . Sturm’s method, Descartes’s rule of I 0 signs, Interval arithmetic. Dav.’85, Eig.-S.-Yap’06, S.-Yap’12. V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 4 / 14

Subdivision Methods – Complexity Analysis Two Components Optimal for pure subdivision The size # T I 0 of the tree T I 0 . The worst case arithmetic complexity at a node. Usually � O ( n ) . Bounds on # T I 0 O ( log1 / σ ) , σ is the root separation bound for f . Sturm’s method, Descartes’s rule of I 0 signs, Interval arithmetic. Dav.’85, Eig.-S.-Yap’06, S.-Yap’12. V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 4 / 14

Subdivision Methods – Complexity Analysis Two Components Optimal for pure subdivision The size # T I 0 of the tree T I 0 . The worst case arithmetic complexity at a node. Usually � O ( n ) . Bounds on # T I 0 O ( log1 / σ ) , σ is the root separation bound for f . Sturm’s method, Descartes’s rule of I 0 signs, Interval arithmetic. Dav.’85, Eig.-S.-Yap’06, S.-Yap’12. V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 4 / 14

Subdivision Methods – Complexity Analysis Two Components Optimal for pure subdivision The size # T I 0 of the tree T I 0 . The worst case arithmetic complexity at a node. Usually � O ( n ) . Bounds on # T I 0 O ( log1 / σ ) , σ is the root separation bound for f . Sturm’s method, Descartes’s rule of I 0 signs, Interval arithmetic. Dav.’85, Eig.-S.-Yap’06, S.-Yap’12. V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 4 / 14

Subdivision Methods – Complexity Analysis Two Components Optimal for pure subdivision The size # T I 0 of the tree T I 0 . The worst case arithmetic complexity at a node. Usually � O ( n ) . Bounds on # T I 0 O ( log1 / σ ) , σ is the root separation bound for f . Sturm’s method, Descartes’s rule of I 0 signs, Interval arithmetic. Dav.’85, Eig.-S.-Yap’06, S.-Yap’12. V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 4 / 14

Subdivision Methods – Complexity Analysis Two Components Optimal for pure subdivision The size # T I 0 of the tree T I 0 . The worst case arithmetic complexity at a node. Usually � O ( n ) . Bounds on # T I 0 O ( log1 / σ ) , σ is the root separation bound for f . Sturm’s method, Descartes’s rule of I 0 signs, Interval arithmetic. Dav.’85, Eig.-S.-Yap’06, S.-Yap’12. V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 4 / 14

Subdivision Methods – Complexity Analysis Two Components Optimal for pure subdivision The size # T I 0 of the tree T I 0 . The worst case arithmetic complexity at a node. Usually � O ( n ) . Ω(log 1 /σ ) Bounds on # T I 0 O ( log1 / σ ) , σ is the root separation bound for f . Sturm’s method, Descartes’s rule of I 0 signs, Interval arithmetic. Dav.’85, Eig.-S.-Yap’06, S.-Yap’12. V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 4 / 14

Subdivision Methods – Complexity Analysis Two Components Optimal for pure subdivision The size # T I 0 of the tree T I 0 . The worst case arithmetic complexity at a node. Usually � O ( n ) . Ω(log 1 /σ ) Bounds on # T I 0 O ( log1 / σ ) , σ is the root separation bound for f . Sturm’s method, Descartes’s rule of I 0 signs, Interval arithmetic. Dav.’85, Eig.-S.-Yap’06, S.-Yap’12. The Problem Subdivision only gives linear convergence to clusters of roots. V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 4 / 14

The Remedy – Subdivision + Newton Iteration Cluster C a subset of roots. C V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 5 / 14

The Remedy – Subdivision + Newton Iteration Cluster C a subset of roots. m C is the centroid. r C the cluster radius. m C r C C V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 5 / 14

The Remedy – Subdivision + Newton Iteration Cluster C a subset of roots. m C is the centroid. r C the cluster radius. m C 3 r C r C C V. Sharma and P . Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 5 / 14