Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Quadratic Interval Refinement Nikolaos Arvanitopoulos Seminar on Computational Geometry and Geometric Computing Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Outline Introduction 1 Root Isolation in Short Main Goal of this Work Root Isolation 2 Quadratic Interval Refinement (QIR) 3 Bisection Method Quadratic Interval Refinement Algorithm Analysis of Quadratic Interval Refinement 4 Cost of the Initial Sequence Cost of the Quadratic Sequence Conclusions and Further Work 5 Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Root Isolation in Short Quadratic Interval Refinement (QIR) Main Goal of this Work Analysis of Quadratic Interval Refinement Conclusions and Further Work Root Isolation in Short Several subdivision approaches Descartes’ rule of sign Sturm’s Theorem Output: Isolating Intervals Easy to implement Good performance in practice Applications in fields of Computational Geometry, such as Topology and Arrangement Computation Nikolaos Arvanitopoulos Quadratic Interval Refinement
b b b b b b Introduction Root Isolation Root Isolation in Short Quadratic Interval Refinement (QIR) Main Goal of this Work Analysis of Quadratic Interval Refinement Conclusions and Further Work Example of Root Isolation α 1 α 1 α 2 α 3 α 2 α 3 I 1 I ′ I 3 1 I ′ 3 I 2 I ′ 2 Output of Root Isolation Algorithm Output of Root Refinement Algorithm Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Root Isolation in Short Quadratic Interval Refinement (QIR) Main Goal of this Work Analysis of Quadratic Interval Refinement Conclusions and Further Work Problem Statement Investigation of the computational cost of Root Isolation, with intervals of user-specified width (below some threshold ǫ ) Use of the Quadratic Interval Refinement (QIR) algorithm to subsequently refine intervals Many concrete applications Comparison of roots of polynomials Evaluation of the sign of an algebraic expression that depends on a root Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Root Isolation in Short Quadratic Interval Refinement (QIR) Main Goal of this Work Analysis of Quadratic Interval Refinement Conclusions and Further Work Complexity Bound Input i =0 a i x i ∈ Z [ x ] of degree p and Square-free polynomial f := � p | a i | < 2 σ User-specified threshold ǫ Worst-case complexity bound : O ( p 4 σ 2 + p 3 log ǫ − 1 ) ˜ Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Root Isolation in Short Quadratic Interval Refinement (QIR) Main Goal of this Work Analysis of Quadratic Interval Refinement Conclusions and Further Work Notation i =0 a i x i ∈ Z [ x ] polynomial of degree p and | a i | < 2 σ f := � p First s roots α 1 , . . . , α s are assumed to be real 0 < ǫ < 1 and L := log 1 ǫ M ( n ) = O ( n log n log log n ) = ˜ O ( n ) cost of fast integer multiplication of bitsize n Interval I = ( c , d ) with width w ( I ) = d − c Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Root Isolation The worst-case complexity bound of root isolation is of great importance: Theorem Computing isolating intervals for the real roots of a square-free polynomial f requires at most ˜ O ( p 4 σ 2 ) bit operations in the worst 2 ℓ , a +1 case. Each isolating interval is of the form ( a 2 ℓ ) , a , ℓ ∈ Z and log | a 2 ℓ | = O ( σ ) Bound proved for root isolation based on Sturm sequences and Descartes’ rule of sign. Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Bisection Method Quadratic Interval Refinement (QIR) Quadratic Interval Refinement Algorithm Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Most naive way to refine isolating intervals Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Bisection Method Quadratic Interval Refinement (QIR) Quadratic Interval Refinement Algorithm Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Most naive way to refine isolating intervals Evaluate f at the midpoint m = c + d of the interval I = ( c , d ) 2 If f ( m ) = 0, the root is found exactly Otherwise, choose either ( c , m ) or ( m , d ) depending on the change of the sign Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Bisection Method Quadratic Interval Refinement (QIR) Quadratic Interval Refinement Algorithm Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Most naive way to refine isolating intervals Evaluate f at the midpoint m = c + d of the interval I = ( c , d ) 2 If f ( m ) = 0, the root is found exactly Otherwise, choose either ( c , m ) or ( m , d ) depending on the change of the sign The isolating interval is halved in every step Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Bisection Method Quadratic Interval Refinement (QIR) Quadratic Interval Refinement Algorithm Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Most naive way to refine isolating intervals Evaluate f at the midpoint m = c + d of the interval I = ( c , d ) 2 If f ( m ) = 0, the root is found exactly Otherwise, choose either ( c , m ) or ( m , d ) depending on the change of the sign The isolating interval is halved in every step Number of bit operations: O ( pM ( σ + p τ )) For intervals smaller than ǫ perform up to σ + L bisection steps Total complexity: O ( p ( σ + L ) M ( p ( σ + L ))) = ˜ O ( p 2 ( σ + L ) 2 ) Nikolaos Arvanitopoulos Quadratic Interval Refinement
b b b b b Introduction Root Isolation Bisection Method Quadratic Interval Refinement (QIR) Quadratic Interval Refinement Algorithm Analysis of Quadratic Interval Refinement Conclusions and Further Work Example of Bisection Method ( d, f ( d )) α m ( c , f ( c )) Bisection Method Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Bisection Method Quadratic Interval Refinement (QIR) Quadratic Interval Refinement Algorithm Analysis of Quadratic Interval Refinement Conclusions and Further Work Quadratic Interval Refinement Algorithm Proposed by Abbott Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Bisection Method Quadratic Interval Refinement (QIR) Quadratic Interval Refinement Algorithm Analysis of Quadratic Interval Refinement Conclusions and Further Work Quadratic Interval Refinement Algorithm Proposed by Abbott ℓ the secant through the points ( c , f ( c )) and ( d , f ( d )) ∈ R 2 Given an integer N , divide I in N + 1 equidistant grid points with distance w = w ( I ) N Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Bisection Method Quadratic Interval Refinement (QIR) Quadratic Interval Refinement Algorithm Analysis of Quadratic Interval Refinement Conclusions and Further Work Quadratic Interval Refinement Algorithm Proposed by Abbott ℓ the secant through the points ( c , f ( c )) and ( d , f ( d )) ∈ R 2 Given an integer N , divide I in N + 1 equidistant grid points with distance w = w ( I ) N Compute the closest grid point m ′ to m and evaluate f ( m ′ ) Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Bisection Method Quadratic Interval Refinement (QIR) Quadratic Interval Refinement Algorithm Analysis of Quadratic Interval Refinement Conclusions and Further Work Quadratic Interval Refinement Algorithm Proposed by Abbott ℓ the secant through the points ( c , f ( c )) and ( d , f ( d )) ∈ R 2 Given an integer N , divide I in N + 1 equidistant grid points with distance w = w ( I ) N Compute the closest grid point m ′ to m and evaluate f ( m ′ ) Choose ( w − m ′ , m ′ ) or ( m ′ , m ′ + w ) depending on the sign changes and set N ← N 2 √ Otherwise, keep I as isolating interval and set N ← N Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Bisection Method Quadratic Interval Refinement (QIR) Quadratic Interval Refinement Algorithm Analysis of Quadratic Interval Refinement Conclusions and Further Work Quadratic Interval Refinement Algorithm Proposed by Abbott ℓ the secant through the points ( c , f ( c )) and ( d , f ( d )) ∈ R 2 Given an integer N , divide I in N + 1 equidistant grid points with distance w = w ( I ) N Compute the closest grid point m ′ to m and evaluate f ( m ′ ) Choose ( w − m ′ , m ′ ) or ( m ′ , m ′ + w ) depending on the sign changes and set N ← N 2 √ Otherwise, keep I as isolating interval and set N ← N If N = 2, the method reduces to bisection step Nikolaos Arvanitopoulos Quadratic Interval Refinement
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