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

Near Optimal Subdivision Algorithms for Real Root Isolation

Vikram Sharma1 and Prashant Batra2

1Institute of Mathematical Sciences, Chennai, India. 2Technische 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 I0. 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 I0. Output: Disjoint intervals with endpoints in Q containing roots of f.

f(x) I0

• 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 I0. Output: Disjoint intervals with endpoints in Q containing roots of f.

f(x) I0

• 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 I0. Output: Disjoint intervals with endpoints in Q containing roots of f.

f(x) I0

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 C0(I): if true then I has no roots. Inclusion C1(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 C0(I): if true then I has no roots. Inclusion C1(I): if true then I has exactly one root. E.g., Sturm sequences, Descartes’s rule of signs, Interval arithmetic,... The Binary Tree TI0

f(x) I0

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 3 / 14

Subdivision Methods

Predicates Exclusion C0(I): if true then I has no roots. Inclusion C1(I): if true then I has exactly one root. E.g., Sturm sequences, Descartes’s rule of signs, Interval arithmetic,... The Binary Tree TI0

f(x) I0

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 3 / 14

Subdivision Methods

Predicates Exclusion C0(I): if true then I has no roots. Inclusion C1(I): if true then I has exactly one root. E.g., Sturm sequences, Descartes’s rule of signs, Interval arithmetic,... The Binary Tree TI0

f(x) I0

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 3 / 14

Subdivision Methods

Predicates Exclusion C0(I): if true then I has no roots. Inclusion C1(I): if true then I has exactly one root. E.g., Sturm sequences, Descartes’s rule of signs, Interval arithmetic,... The Binary Tree TI0

f(x) I0

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 3 / 14

Subdivision Methods

Predicates Exclusion C0(I): if true then I has no roots. Inclusion C1(I): if true then I has exactly one root. E.g., Sturm sequences, Descartes’s rule of signs, Interval arithmetic,... The Binary Tree TI0

f(x) I0

• 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 #TI0 of the tree TI0. 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 #TI0 of the tree TI0. 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 #TI0 of the tree TI0. The worst case arithmetic complexity at a node. Usually O(n). Bounds on #TI0 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 The size #TI0 of the tree TI0. The worst case arithmetic complexity at a node. Usually O(n). Bounds on #TI0 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. Optimal for pure subdivision

I0

• 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 #TI0 of the tree TI0. The worst case arithmetic complexity at a node. Usually O(n). Bounds on #TI0 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. Optimal for pure subdivision

I0

• 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 #TI0 of the tree TI0. The worst case arithmetic complexity at a node. Usually O(n). Bounds on #TI0 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. Optimal for pure subdivision

I0

• 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 #TI0 of the tree TI0. The worst case arithmetic complexity at a node. Usually O(n). Bounds on #TI0 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. Optimal for pure subdivision

I0

• 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 #TI0 of the tree TI0. The worst case arithmetic complexity at a node. Usually O(n). Bounds on #TI0 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. Optimal for pure subdivision

I0

• 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 #TI0 of the tree TI0. The worst case arithmetic complexity at a node. Usually O(n). Bounds on #TI0 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. Optimal for pure subdivision

I0

• 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 #TI0 of the tree TI0. The worst case arithmetic complexity at a node. Usually O(n). Bounds on #TI0 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. Optimal for pure subdivision

I0

Ω(log 1/σ)

• 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 #TI0 of the tree TI0. The worst case arithmetic complexity at a node. Usually O(n). Bounds on #TI0 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. Optimal for pure subdivision

I0

Ω(log 1/σ)

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

C a subset of roots.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 5 / 14

The Remedy – Subdivision + Newton Iteration

Cluster

rC C mC

C a subset of roots.

mC is the centroid. rC the cluster radius.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 5 / 14

The Remedy – Subdivision + Newton Iteration

Cluster

rC C 3rC mC

C a subset of roots.

mC is the centroid. rC the cluster radius.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 5 / 14

The Remedy – Subdivision + Newton Iteration

Cluster

rC C 3rC mC RC

C a subset of roots.

mC is the centroid. rC the cluster radius. RC the isolation radius.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 5 / 14

The Remedy – Subdivision + Newton Iteration

Cluster

rC C 3rC mC RC

C a subset of roots.

mC is the centroid. rC the cluster radius. RC the isolation radius. Z(f) and single roots are trivial clusters.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 5 / 14

The Remedy – Subdivision + Newton Iteration

Cluster

rC C mC RC

C a subset of roots.

mC is the centroid. rC the cluster radius. RC the isolation radius. Z(f) and single roots are trivial clusters. Quadratic convergence of Newton iteration If RC n2rC then

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 5 / 14

The Remedy – Subdivision + Newton Iteration

Cluster

rC C mC RC

C a subset of roots.

mC is the centroid. rC the cluster radius. RC the isolation radius. Z(f) and single roots are trivial clusters. Quadratic convergence of Newton iteration If RC n2rC then ∃ an annulus in which Newton iteration converges to mC.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 5 / 14

Resulting Improvements

Relevant Results Pan’00 – Predicates based on distance to nearest root, Newton + Graeffe iteration for cluster detection, complex roots isolation.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 6 / 14

Resulting Improvements

Relevant Results Pan’00 – Predicates based on distance to nearest root, Newton + Graeffe iteration for cluster detection, complex roots isolation. Sagraloff’12, Sagraloff-Mehlhorn’13 – Predicates based on Pellet’s test

• r Descartes’s rule of Signs, Quadratic Interval Refinement+Newton

iteration.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 6 / 14

Resulting Improvements

Relevant Results Pan’00 – Predicates based on distance to nearest root, Newton + Graeffe iteration for cluster detection, complex roots isolation. Sagraloff’12, Sagraloff-Mehlhorn’13 – Predicates based on Pellet’s test

• r Descartes’s rule of Signs, Quadratic Interval Refinement+Newton

iteration. Bounds on Treesize – n degree, σ root separation O(nlog(nlog1/σ)))

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 6 / 14

Resulting Improvements

Relevant Results Pan’00 – Predicates based on distance to nearest root, Newton + Graeffe iteration for cluster detection, complex roots isolation. Sagraloff’12, Sagraloff-Mehlhorn’13 – Predicates based on Pellet’s test

• r Descartes’s rule of Signs, Quadratic Interval Refinement+Newton

iteration. Bounds on Treesize – n degree, σ root separation O(nlog(nlog1/σ))) Our result Choice of any inclusion-exclusion predicate, Newton iteration. O(nlogn), for Descartes’s rule of signs or Sturm sequences. Analysis is independent of root bounds, uses geometry of clusters. Use the framework of continuous amortization.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 6 / 14

Basic Idea

Cluster Tree

C C′

Claim If C ∩C′ = /

0 then either C ⊆ C′ or

vice versa.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 7 / 14

Basic Idea

Cluster Tree

C C′

Claim If C ∩C′ = /

0 then either C ⊆ C′ or

vice versa.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 7 / 14

Basic Idea

Cluster Tree

Not possible Possible

Claim If C ∩C′ = /

0 then either C ⊆ C′ or

vice versa.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 7 / 14

Basic Idea

Cluster Tree

Z(f) Z(f)

Clusters Roots

Claim If C ∩C′ = /

0 then either C ⊆ C′ or

vice versa. Lemma (Sagraloff-S.-Yap’13) The set of clusters form a tree with root node as the set of all zeros Z(f).

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 7 / 14

Basic Idea

Number of Bisections in Cluster Tree

Bisections O

• log

RC (RC/n2)

• Bisections O
• log nrC

rC

• C

Newton itrns. O

• log log RC

rC

• Strongly Separated Clusters

RC n3rC. Bisection O(logRC/rC) steps. Detect and converge using O(loglogRC/rC) Newton iterations.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 7 / 14

Basic Idea

Number of Bisections in Cluster Tree

C Bisections O

• log RC

rC

• Strongly Separated Clusters

RC n3rC. Bisection O(logRC/rC) steps. Detect and converge using O(loglogRC/rC) Newton iterations. Ordinary Clusters RC n3rC. Bisection steps O(log RC

rC ) = O(logn).

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 7 / 14

Basic Idea

Number of Bisections in Cluster Tree

Z(f) Z(f)

Clusters Roots

Strongly Separated Clusters RC n3rC. Bisection O(logRC/rC) steps. Detect and converge using O(loglogRC/rC) Newton iterations. Ordinary Clusters RC n3rC. Bisection steps O(log RC

rC ) = O(logn).

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 7 / 14

Basic Idea

Number of Bisections in Cluster Tree

Z(f) Z(f)

Clusters Roots

Strongly Separated Clusters RC n3rC. Bisection O(logRC/rC) steps. Detect and converge using O(loglogRC/rC) Newton iterations. Ordinary Clusters RC n3rC. Bisection steps O(log RC

rC ) = O(logn).

The number of bisection steps in TI0 is O(nlogn).

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 7 / 14

Basic Idea

Number of Bisections in Cluster Tree

Z(f) Z(f)

Clusters Roots

Strongly Separated Clusters RC n3rC. Bisection O(logRC/rC) steps. Detect and converge using O(loglogRC/rC) Newton iterations. Ordinary Clusters RC n3rC. Bisection steps O(log RC

rC ) = O(logn).

The number of bisection steps in TI0 is O(nlogn).

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 7 / 14

Detecting and Approximating Clusters

Ostrowski’34, S.-Batra’15 Let f(x + z) := ∑j fj(z)xj, i.e., fj(z) is the jth Taylor coefficient at z.

For k ∈ [n], ρk(z) := maxj<k

• fj(z)

fk(z)

• 1

(k−j) , ρk+1(z) := minj>k

• fk(z)

fj(z)

• 1

(j−k) .

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 8 / 14

Detecting and Approximating Clusters

Ostrowski’34, S.-Batra’15 Let f(x + z) := ∑j fj(z)xj, i.e., fj(z) is the jth Taylor coefficient at z.

For k ∈ [n], ρk(z) := maxj<k

• fj(z)

fk(z)

• 1

(k−j) , ρk+1(z) := minj>k

• fk(z)

fj(z)

• 1

(j−k) .

If ρk+1(z)/3 > 3· 3ρk(z) then k roots in D(z,3ρk(z)) ⊆ D(z, ρk+1(z)

3

).

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 8 / 14

Detecting and Approximating Clusters

Ostrowski’34, S.-Batra’15 Let f(x + z) := ∑j fj(z)xj, i.e., fj(z) is the jth Taylor coefficient at z.

For k ∈ [n], ρk(z) := maxj<k

• fj(z)

fk(z)

• 1

(k−j) , ρk+1(z) := minj>k

• fk(z)

fj(z)

• 1

(j−k) .

If ρk+1(z)/3 > 3· 3ρk(z) then k roots in D(z,3ρk(z)) ⊆ D(z, ρk+1(z)

3

). z 3ρk(z) z∗

ρk+1(z) 3 3ρk(z) k

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 8 / 14

Detecting and Approximating Clusters

Ostrowski’34, S.-Batra’15 Let f(x + z) := ∑j fj(z)xj, i.e., fj(z) is the jth Taylor coefficient at z.

For k ∈ [n], ρk(z) := maxj<k

• fj(z)

fk(z)

• 1

(k−j) , ρk+1(z) := minj>k

• fk(z)

fj(z)

• 1

(j−k) .

If ρk+1(z)/3 > 3· 3ρk(z) then k roots in D(z,3ρk(z)) ⊆ D(z, ρk+1(z)

3

).

Newton iteration starting from z converges to a unique root z∗ of f (k−1). z 3ρk(z) z∗

ρk+1(z) 3 3ρk(z) k

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 8 / 14

Detecting and Approximating Clusters

Ostrowski’34, S.-Batra’15 Let f(x + z) := ∑j fj(z)xj, i.e., fj(z) is the jth Taylor coefficient at z.

For k ∈ [n], ρk(z) := maxj<k

• fj(z)

fk(z)

• 1

(k−j) , ρk+1(z) := minj>k

• fk(z)

fj(z)

• 1

(j−k) .

If ρk+1(z)/3 > 3· 3ρk(z) then k roots in D(z,3ρk(z)) ⊆ D(z, ρk+1(z)

3

).

Newton iteration starting from z converges to a unique root z∗ of f (k−1). For a ssc C all points in an annulus AC satisfy ρk+1(z)/3 > 3· 3ρk(z).

C mC rC |C|rC RC z∗ z

RC n2

AC

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 8 / 14

The algorithm NewtonIsol

NewtonIsol(f,I0) 1. Initialize Q ← {I0}. While Q is not empty Remove an interval I = (a,b) from Q. m ← (a+ b)/2. If C0(I)∨ C1(I) then add I to P. else Push (a,m) and (m,b) into Q. 3. Output P.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 9 / 14

The algorithm NewtonIsol

NewtonIsol(f,I0) 1. Initialize Q ← {I0}. While Q is not empty Remove an interval I = (a,b) from Q. m ← (a+ b)/2. If C0(I)∨ C1(I) then add I to P. else if there is a k s.t. ρk+1(m) ρk(m) and else Push (a,m) and (m,b) into Q. 3. Output P.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 9 / 14

The algorithm NewtonIsol

NewtonIsol(f,I0) 1. Initialize Q ← {I0}. While Q is not empty Remove an interval I = (a,b) from Q. m ← (a+ b)/2. If C0(I)∨ C1(I) then add I to P. else if there is a k s.t. ρk+1(m) ρk(m) and for the smallest such k, I ⊆ D(m,ρk+1(m)/3) then else Push (a,m) and (m,b) into Q. 3. Output P.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 9 / 14

The algorithm NewtonIsol

NewtonIsol(f,I0) 1. Initialize Q ← {I0}. While Q is not empty Remove an interval I = (a,b) from Q. m ← (a+ b)/2. If C0(I)∨ C1(I) then add I to P. else if there is a k s.t. ρk+1(m) ρk(m) and for the smallest such k, I ⊆ D(m,ρk+1(m)/3) then z0 := m, g := f (k−1), i := 0. else Push (a,m) and (m,b) into Q. 3. Output P.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 9 / 14

The algorithm NewtonIsol

NewtonIsol(f,I0) 1. Initialize Q ← {I0}. While Q is not empty Remove an interval I = (a,b) from Q. m ← (a+ b)/2. If C0(I)∨ C1(I) then add I to P. else if there is a k s.t. ρk+1(m) ρk(m) and for the smallest such k, I ⊆ D(m,ρk+1(m)/3) then z0 := m, g := f (k−1), i := 0. While ρk(zi) ≤ 21−2iρk(z0) else Push (a,m) and (m,b) into Q. 3. Output P.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 9 / 14

The algorithm NewtonIsol

NewtonIsol(f,I0) 1. Initialize Q ← {I0}. While Q is not empty Remove an interval I = (a,b) from Q. m ← (a+ b)/2. If C0(I)∨ C1(I) then add I to P. else if there is a k s.t. ρk+1(m) ρk(m) and for the smallest such k, I ⊆ D(m,ρk+1(m)/3) then z0 := m, g := f (k−1), i := 0. While ρk(zi) ≤ 21−2iρk(z0) zi+1 := zi − g(zi)/g′(zi); i := i + 1. else Push (a,m) and (m,b) into Q. 3. Output P.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 9 / 14

The algorithm NewtonIsol

NewtonIsol(f,I0) 1. Initialize Q ← {I0}. While Q is not empty Remove an interval I = (a,b) from Q. m ← (a+ b)/2. If C0(I)∨ C1(I) then add I to P. else if there is a k s.t. ρk+1(m) ρk(m) and for the smallest such k, I ⊆ D(m,ρk+1(m)/3) then z0 := m, g := f (k−1), i := 0. While ρk(zi) ≤ 21−2iρk(z0) zi+1 := zi − g(zi)/g′(zi); i := i + 1. J := [zi−1 ± 3ρk(zi−1)]. else Push (a,m) and (m,b) into Q. 3. Output P.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 9 / 14

The algorithm NewtonIsol

NewtonIsol(f,I0) 1. Initialize Q ← {I0}. While Q is not empty Remove an interval I = (a,b) from Q. m ← (a+ b)/2. If C0(I)∨ C1(I) then add I to P. else if there is a k s.t. ρk+1(m) ρk(m) and for the smallest such k, I ⊆ D(m,ρk+1(m)/3) then z0 := m, g := f (k−1), i := 0. While ρk(zi) ≤ 21−2iρk(z0) zi+1 := zi − g(zi)/g′(zi); i := i + 1. J := [zi−1 ± 3ρk(zi−1)]. If w(J) < w(I)/2 and J ∩ I0 = /

0 then

else Push (a,m) and (m,b) into Q. 3. Output P.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 9 / 14

The algorithm NewtonIsol

NewtonIsol(f,I0) 1. Initialize Q ← {I0}. While Q is not empty Remove an interval I = (a,b) from Q. m ← (a+ b)/2. If C0(I)∨ C1(I) then add I to P. else if there is a k s.t. ρk+1(m) ρk(m) and for the smallest such k, I ⊆ D(m,ρk+1(m)/3) then z0 := m, g := f (k−1), i := 0. While ρk(zi) ≤ 21−2iρk(z0) zi+1 := zi − g(zi)/g′(zi); i := i + 1. J := [zi−1 ± 3ρk(zi−1)]. If w(J) < w(I)/2 and J ∩ I0 = /

0 then

Add J to Q. Remove all intervals in Q that intersect D(m,ρk+1(m)/3)

else Push (a,m) and (m,b) into Q. 3. Output P.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 9 / 14

The algorithm NewtonIsol

NewtonIsol(f,I0) 1. Initialize Q ← {I0}. While Q is not empty Remove an interval I = (a,b) from Q. m ← (a+ b)/2. If C0(I)∨ C1(I) then add I to P. else if there is a k s.t. ρk+1(m) ρk(m) and for the smallest such k, I ⊆ D(m,ρk+1(m)/3) then z0 := m, g := f (k−1), i := 0. While ρk(zi) ≤ 21−2iρk(z0) zi+1 := zi − g(zi)/g′(zi); i := i + 1. J := [zi−1 ± 3ρk(zi−1)]. If w(J) < w(I)/2 and J ∩ I0 = /

0 then

Add J to Q. Remove all intervals in Q that intersect D(m,ρk+1(m)/3)

else Push (a,m) and (m,b) into Q. 3. Output P.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 9 / 14

Bounding the size of the tree TI0

TI0 is a binary tree. Let L(TI0) be its set of leaves. Bound |L(TI0)|. Intervals I in L(TI0) – Two Types

1

Where C0(I) or C1(I) holds.

2

I was discarded by the Newton-step. Observation: If J is the parent of I then both C0(J) and C1(J) failed.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 10 / 14

Bounding the size of the tree TI0

TI0 is a binary tree. Let L(TI0) be its set of leaves. Bound |L(TI0)|. Intervals I in L(TI0) – Two Types

1

Where C0(I) or C1(I) holds.

2

I was discarded by the Newton-step. Observation: If J is the parent of I then both C0(J) and C1(J) failed. Continuous Amortization (Burr-Krahmer-Yap’09, Burr’13)

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 10 / 14

Bounding the size of the tree TI0

TI0 is a binary tree. Let L(TI0) be its set of leaves. Bound |L(TI0)|. Intervals I in L(TI0) – Two Types

1

Where C0(I) or C1(I) holds.

2

I was discarded by the Newton-step. Observation: If J is the parent of I then both C0(J) and C1(J) failed. Continuous Amortization (Burr-Krahmer-Yap’09, Burr’13)

1

Stopping Function G : R → R≥0 wrt the predicates C0, C1. Given J, if ∃x ∈ J such that w(J)G(x) ≤ 1 then C0(J) or C1(J) holds.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 10 / 14

Bounding the size of the tree TI0

TI0 is a binary tree. Let L(TI0) be its set of leaves. Bound |L(TI0)|. Intervals I in L(TI0) – Two Types

1

Where C0(I) or C1(I) holds.

2

I was discarded by the Newton-step. Observation: If J is the parent of I then both C0(J) and C1(J) failed. Continuous Amortization (Burr-Krahmer-Yap’09, Burr’13)

1

Stopping Function G : R → R≥0 wrt the predicates C0, C1. Given J, if ∃x ∈ J such that w(J)G(x) ≤ 1 then C0(J) or C1(J) holds.

2

If C0(J) and C1(J) failed then ∀ I ⊆ J, 2w(I) ≥ w(J), 2

• I G(x)dx ≥ 1.
• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 10 / 14

Bounding the size of the tree TI0

TI0 is a binary tree. Let L(TI0) be its set of leaves. Bound |L(TI0)|. Intervals I in L(TI0) – Two Types

1

Where C0(I) or C1(I) holds.

2

I was discarded by the Newton-step. Observation: If J is the parent of I then both C0(J) and C1(J) failed. Continuous Amortization (Burr-Krahmer-Yap’09, Burr’13)

1

Stopping Function G : R → R≥0 wrt the predicates C0, C1. Given J, if ∃x ∈ J such that w(J)G(x) ≤ 1 then C0(J) or C1(J) holds.

2

If C0(J) and C1(J) failed then ∀ I ⊆ J, 2w(I) ≥ w(J), 2

• I G(x)dx ≥ 1.

Proof: 2

• I G(x)dx ≥ 2w(I)minx∈I G(x) ≥ w(J)minx∈I G(x) ≥ 1.
• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 10 / 14

Bounding the size of the tree TI0

TI0 is a binary tree. Let L(TI0) be its set of leaves. Bound |L(TI0)|. Intervals I in L(TI0) – Two Types

1

Where C0(I) or C1(I) holds.

2

I was discarded by the Newton-step. Observation: If J is the parent of I then both C0(J) and C1(J) failed. Continuous Amortization (Burr-Krahmer-Yap’09, Burr’13)

1

Stopping Function G : R → R≥0 wrt the predicates C0, C1. Given J, if ∃x ∈ J such that w(J)G(x) ≤ 1 then C0(J) or C1(J) holds.

2

If C0(J) and C1(J) failed then ∀ I ⊆ J, 2w(I) ≥ w(J), 2

• I G(x)dx ≥ 1.

Proof: 2

• I G(x)dx ≥ 2w(I)minx∈I G(x) ≥ w(J)minx∈I G(x) ≥ 1.

Lemma (S.-Batra’15)

|L(TI0)| ≤ O(n)+ 2

• I0\∪CAC G(x)dx, where C are ssc.
• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 10 / 14

Bounding treesize for Descartes’s rule of signs

Theorem (S.-Batra’15)

|L(TI0)| ≤ O(n)+ 2

• I0\∪CAC G(x)dx = O(nlogn), where C are ssc.
• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 11 / 14

Bounding treesize for Descartes’s rule of signs

Theorem (S.-Batra’15)

|L(TI0)| ≤ O(n)+ 2

• I0\∪CAC G(x)dx = O(nlogn), where C are ssc.

Stopping function for Descartes’s rule of signs

1

d(x,Z(f)) distance to nearest root.

2

d2(x,Z(f)) distance to second nearest root.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 11 / 14

Bounding treesize for Descartes’s rule of signs

Theorem (S.-Batra’15)

|L(TI0)| ≤ O(n)+ 2

• I0\∪CAC G(x)dx = O(nlogn), where C are ssc.

Stopping function for Descartes’s rule of signs

1

d(x,Z(f)) distance to nearest root.

2

d2(x,Z(f)) distance to second nearest root.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 11 / 14

Bounding treesize for Descartes’s rule of signs

Theorem (S.-Batra’15)

|L(TI0)| ≤ O(n)+ 2

• I0\∪CAC G(x)dx = O(nlogn), where C are ssc.

Stopping function for Descartes’s rule of signs

1

d(x,Z(f)) distance to nearest root.

2

d2(x,Z(f)) distance to second nearest root.

3

G(x) = 1/d2(x,Z(f)) near real roots.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 11 / 14

Bounding treesize for Descartes’s rule of signs

Theorem (S.-Batra’15)

|L(TI0)| ≤ O(n)+ 2

• I0\∪CAC G(x)dx = O(nlogn), where C are ssc.

Stopping function for Descartes’s rule of signs

1

d(x,Z(f)) distance to nearest root.

2

d2(x,Z(f)) distance to second nearest root.

3

G(x) = 1/d2(x,Z(f)) near real roots.

4

G(x) = 1/d(x,Z(f)) everywhere else.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 11 / 14

Bounding treesize for Descartes’s rule of signs

Theorem (S.-Batra’15)

|L(TI0)| ≤ O(n)+ 2

• I0\∪CAC G(x)dx = O(nlogn), where C are ssc.

Stopping function for Descartes’s rule of signs

1

d(x,Z(f)) distance to nearest root.

2

d2(x,Z(f)) distance to second nearest root.

3

G(x) = 1/d2(x,Z(f)) near real roots.

4

G(x) = 1/d(x,Z(f)) everywhere else.

5

If ∃x ∈ I s.t. w(I)G(x) ≤ 1, i.e., w(I) ≤ d(x,Z(f)) then C0(I) holds.

x I

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 11 / 14

Bounding treesize for Descartes’s rule of signs

Claim

• I0\∪CAC G(x)dx = O(|T|logn), where C are ssc, T is cluster tree?
• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 12 / 14

Bounding treesize for Descartes’s rule of signs

Claim

• I0\∪CAC G(x)dx = O(|T|logn), where C are ssc, T is cluster tree?

Z(f) Z(f)

Clusters Roots

I0 M

– Cluster C, IC := [mC ± 2rC].

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 12 / 14

Bounding treesize for Descartes’s rule of signs

Claim

• I0\∪CAC G(x)dx = O(|T|logn), where C are ssc, T is cluster tree?

Z(f) Z(f)

Clusters Roots

I0 M

– Cluster C, IC := [mC ± 2rC]. –

• IC\∪C′AC′ G(x)dx = O(|TC|logn).
• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 12 / 14

Bounding treesize for Descartes’s rule of signs

Claim

• I0\∪CAC G(x)dx = O(|T|logn), where C are ssc, T is cluster tree?

C

mC

d(x, Z(f)) = |x − mC|

mC − |C|rC mC + |C|rC mC − RC/n2 mC + RC/n2 mC − RC

2

mC + RC

2

mC− RC

n2

mC− RC

2

dx |x−mC| = O(log n)

mC+|C|rC

mC+2rC dx |x−mC| = O(log n)

Strongly Separated Cluster

– Cluster C, IC := [mC ± 2rC]. –

• IC\∪C′AC′ G(x)dx = O(|TC|logn).

– Near C, integral is O(logn).

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 12 / 14

Bounding treesize for Descartes’s rule of signs

Claim

• I0\∪CAC G(x)dx = O(|T|logn), where C are ssc, T is cluster tree?

C

mC

d(x, Z(f)) = |x − mC|

mC − |C|rC mC + |C|rC mC − RC

2

mC + RC

2

mC−2rC

mC− RC

2

dx |x−mC| = O(log RC rC ) = O(log n)

RC n3rC Ordinary Cluster

– Cluster C, IC := [mC ± 2rC]. –

• IC\∪C′AC′ G(x)dx = O(|TC|logn).

– Near C, integral is O(logn).

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 12 / 14

Bounding treesize for Descartes’s rule of signs

Claim

• I0\∪CAC G(x)dx = O(|T|logn), where C are ssc, T is cluster tree?

Z(f) Z(f)

Clusters Roots

I0 M

– Cluster C, IC := [mC ± 2rC]. –

• IC\∪C′AC′ G(x)dx = O(|TC|logn).

– Near C, integral is O(logn). – |M|+∑C∈M |TC| = O(|T|).

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 12 / 14

Bounding treesize for Descartes’s rule of signs

Claim

• I0\∪CAC G(x)dx = O(|T|logn), where C are ssc, T is cluster tree?

Z(f) Z(f)

Clusters Roots

I0 M P

– Cluster C, IC := [mC ± 2rC]. –

• IC\∪C′AC′ G(x)dx = O(|TC|logn).

– Near C, integral is O(logn). – |M|+∑C∈M |TC| = O(|T|). – Shrink clusters to points to get P.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 12 / 14

Bounding treesize for Descartes’s rule of signs

Claim

• I0\∪CAC G(x)dx = O(|T|logn), where C are ssc, T is cluster tree?

Z(f) Z(f)

Clusters Roots

I0 M P

– Cluster C, IC := [mC ± 2rC]. –

• IC\∪C′AC′ G(x)dx = O(|TC|logn).

– Near C, integral is O(logn). – |M|+∑C∈M |TC| = O(|T|). – Shrink clusters to points to get P. – Resulting pointset is dense.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 12 / 14

Bounding treesize for Descartes’s rule of signs

Claim

• I0\∪CAC G(x)dx = O(|T|logn), where C are ssc, T is cluster tree?

Z(f) Z(f)

Clusters Roots

I0 M P

– Cluster C, IC := [mC ± 2rC]. –

• IC\∪C′AC′ G(x)dx = O(|TC|logn).

– Near C, integral is O(logn). – |M|+∑C∈M |TC| = O(|T|). – Shrink clusters to points to get P. – Resulting pointset is dense.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 12 / 14

The Main Lemma

Bound for dense pointset P

1

For a q ∈ P ∩R, Jq := [q ± d2(q,P)/2].

2

IP := [mP ± rP]

3

• IP\∪q∈P∩RJq

1 d(x,P) = O(|P|log|P|). IP P q Jq

d2(q, P)

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 13 / 14

The Main Lemma

Bound for dense pointset P

1

For a q ∈ P ∩R, Jq := [q ± d2(q,P)/2].

2

IP := [mP ± rP]

3

• IP\∪q∈P∩RJq

1 d(x,P) = O(|P|log|P|).

1

Vq be the real Voronoi region of q.

2

• IP\∪q∈P∩RJq

dx d(x,P)

.

IP P q Jq

d2(q, P)

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 13 / 14

The Main Lemma

Bound for dense pointset P

1

For a q ∈ P ∩R, Jq := [q ± d2(q,P)/2].

2

IP := [mP ± rP]

3

• IP\∪q∈P∩RJq

1 d(x,P) = O(|P|log|P|).

1

Vq be the real Voronoi region of q.

2

• IP\∪q∈P∩RJq

dx d(x,P) =

∑q∈P

• Vq

dx

|x−q|

.

IP P q Jq

d2(q, P)

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 13 / 14

The Main Lemma

Bound for dense pointset P

1

For a q ∈ P ∩R, Jq := [q ± d2(q,P)/2].

2

IP := [mP ± rP]

3

• IP\∪q∈P∩RJq

1 d(x,P) = O(|P|log|P|).

1

Vq be the real Voronoi region of q.

2

• IP\∪q∈P∩RJq

dx d(x,P) =

∑q∈P

• Vq

dx

|x−q| =

∑q∈P O(log

w(IP) d2(q,P))

.

IP P q Jq

d2(q, P)

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 13 / 14

The Main Lemma

Bound for dense pointset P

1

For a q ∈ P ∩R, Jq := [q ± d2(q,P)/2].

2

IP := [mP ± rP]

3

• IP\∪q∈P∩RJq

1 d(x,P) = O(|P|log|P|).

1

Vq be the real Voronoi region of q.

2

• IP\∪q∈P∩RJq

dx d(x,P) =

∑q∈P

• Vq

dx

|x−q| =

∑q∈P O(log

w(IP) d2(q,P))

.

3

for all q ∈ P, w(IP) ≤ 3|P|d2(q,P).

IP P q Jq

d2(q, P)

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 13 / 14

The Main Lemma

Bound for dense pointset P

1

For a q ∈ P ∩R, Jq := [q ± d2(q,P)/2].

2

IP := [mP ± rP]

3

• IP\∪q∈P∩RJq

1 d(x,P) = O(|P|log|P|).

1

Vq be the real Voronoi region of q.

2

• IP\∪q∈P∩RJq

dx d(x,P) =

∑q∈P

• Vq

dx

|x−q| =

∑q∈P O(log

w(IP) d2(q,P))

.

3

for all q ∈ P, w(IP) ≤ 3|P|d2(q,P).

IP P q Jq

d2(q, P) 3d2(q, P)

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 13 / 14

The Main Lemma

Bound for dense pointset P

1

For a q ∈ P ∩R, Jq := [q ± d2(q,P)/2].

2

IP := [mP ± rP]

3

• IP\∪q∈P∩RJq

1 d(x,P) = O(|P|log|P|).

1

Vq be the real Voronoi region of q.

2

• IP\∪q∈P∩RJq

dx d(x,P) =

∑q∈P

• Vq

dx

|x−q| =

∑q∈P O(log

w(IP) d2(q,P)) = O(|P|2).

3

for all q ∈ P, w(IP) ≤ 3|P|d2(q,P).

IP P q Jq

d2(q, P) 3d2(q, P)

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 13 / 14

Conclusion and Further directions

Summary

1

Provide a subroutine that speedsup any subdivision algorithm for real root isolation.

2

Independent of the inclusion-exclusion predicates.

3

Number of bisections for Descartes’s rule of signs and sturms method is O(nlogn).

4

Highlights the geometry of root clusters.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 14 / 14

Conclusion and Further directions

Summary

1

Provide a subroutine that speedsup any subdivision algorithm for real root isolation.

2

Independent of the inclusion-exclusion predicates.

3

Number of bisections for Descartes’s rule of signs and sturms method is O(nlogn).

4

Highlights the geometry of root clusters. Further Directions

1

Root isolation in the complex plane.

2

Bit complexity of the algorithm where coefficients are in R.

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 14 / 14

Conclusion and Further directions

Summary

1

Provide a subroutine that speedsup any subdivision algorithm for real root isolation.

2

Independent of the inclusion-exclusion predicates.

3

Number of bisections for Descartes’s rule of signs and sturms method is O(nlogn).

4

Highlights the geometry of root clusters. Further Directions

1

Root isolation in the complex plane.

2

Bit complexity of the algorithm where coefficients are in R.

Thank You!

• V. Sharma and P

. Batra (IMSc, TuHH) Optimal Subdivision Algorithms for Real Roots ISSAC, Bath, 2015 14 / 14