Quadratic Interval Refinement Nikolaos Arvanitopoulos Seminar on - - PowerPoint PPT Presentation
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
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work
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
1
Introduction Root Isolation in Short Main Goal of this Work
2
Root Isolation
3
Quadratic Interval Refinement (QIR) Bisection Method Quadratic Interval Refinement Algorithm
4
Analysis of Quadratic Interval Refinement Cost of the Initial Sequence Cost of the Quadratic Sequence
5
Conclusions and Further Work
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Root Isolation in Short Main Goal of this Work
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
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Root Isolation in Short Main Goal of this Work
α1 α2 α3 I1 I2 I3
Output of Root Isolation Algorithm
b b bα1 α2 α3 I′
1
I′
2
I′
3
Output of Root Refinement Algorithm Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Root Isolation in Short Main Goal of this Work
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
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Root Isolation in Short Main Goal of this Work
Input Square-free polynomial f := p
i=0 aixi ∈ Z[x] of degree p and
|ai| < 2σ User-specified threshold ǫ Worst-case complexity bound: ˜ O(p4σ2 + p3 log ǫ−1)
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Root Isolation in Short Main Goal of this Work
f := p
i=0 aixi ∈ Z[x] polynomial of degree p and |ai| < 2σ
First s roots α1, . . . , αs 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
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(p4σ2) bit operations in the worst
2ℓ , a+1 2ℓ ), a, ℓ ∈ Z and
log | a
2ℓ | = O(σ)
Bound proved for root isolation based on Sturm sequences and Descartes’ rule of sign.
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
Most naive way to refine isolating intervals
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
Most naive way to refine isolating intervals Evaluate f at the midpoint m = 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 Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
Most naive way to refine isolating intervals Evaluate f at the midpoint m = 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 Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
Most naive way to refine isolating intervals Evaluate f at the midpoint m = 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 + pτ)) For intervals smaller than ǫ perform up to σ +L bisection steps Total complexity: O(p(σ + L)M(p(σ + L))) = ˜ O(p2(σ + L)2)
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
b
m α
b b b b
Bisection Method
(c, f(c)) (d, f(d))
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
Proposed by Abbott
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
Proposed by Abbott ℓ the secant through the points (c, f (c)) and (d, f (d)) ∈ R2 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 Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
Proposed by Abbott ℓ the secant through the points (c, f (c)) and (d, f (d)) ∈ R2 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 Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
Proposed by Abbott ℓ the secant through the points (c, f (c)) and (d, f (d)) ∈ R2 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 ← N2 Otherwise, keep I as isolating interval and set N ← √ N
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
Proposed by Abbott ℓ the secant through the points (c, f (c)) and (d, f (d)) ∈ R2 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 ← N2 Otherwise, keep I as isolating interval and set N ← √ N If N = 2, the method reduces to bisection step
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
Algorithm 1 Quadratic Interval Refinement
Input: (f , I = (c, d), N) Output: (J, Nnew ) // J is the refined interval if N = 2 then return (BISECTION(f , I),4) end if w ← d−c
N
m ← c +
f (c) f (c)−f (d) (d − c)
m′ ← c + round(N
f (c) f (c)−f (d) )w
s ← sgn(f (m′)) if s = 0 then return ([m′, m′], ∞) end if if (s = sgn(f (c)) and sgn(f (m′ + w)) = sgn(f (d))) then return ((m′, m′ + w), N2) end if if (s = sgn(f (d)) and sgn(f (m′ − w)) = sgn(f (c))) then return ((m′ − w, m′), N2) end if return (I, √ N) Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
b b
m′ m α m′ + w
b b b b b
ℓ
Successful QIR instance for N = 4
(c, f(c)) (d, f(d))
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
Definition A QIR call (J, N2) ← QIR(f , I, N1) succeeds if J I, and it fails if J = I. Equivalently, the QIR call succeeds, if and only if N2 > N1.
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
Definition A QIR call (J, N2) ← QIR(f , I, N1) succeeds if J I, and it fails if J = I. Equivalently, the QIR call succeeds, if and only if N2 > N1. O(p) arithmetic operations to evaluate f at m′ and m′ ± w Bitsize of m′ and m′ ± w is bounded by O(log N + τ) Cost of one QIR call is bounded by O(pM(σ + pτ))
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Bisection Method Quadratic Interval Refinement Algorithm
Definition A QIR call (J, N2) ← QIR(f , I, N1) succeeds if J I, and it fails if J = I. Equivalently, the QIR call succeeds, if and only if N2 > N1. O(p) arithmetic operations to evaluate f at m′ and m′ ± w Bitsize of m′ and m′ ± w is bounded by O(log N + τ) Cost of one QIR call is bounded by O(pM(σ + pτ)) Equal to the cost of one bisection step One successful QIR step yields the same result as log N bisections
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Definition Let α be a root of f that corresponds to the isolating interval I0. The QIR sequence (s0, . . . , sn) for α is defined as s0 := (I0, 4) si := (Ii, Ni) := QIR(f , Ii−1, Ni−1) for i ≥ 1 where n is the first index such that w(In) ≤ ǫ. We say that si−1 → si succeeds if QIR(f , Ii−1, Ni−1) succeeds, and that si−1 → si fails otherwise.
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Definition Let α be a root of f that corresponds to the isolating interval I0. The QIR sequence (s0, . . . , sn) for α is defined as s0 := (I0, 4) si := (Ii, Ni) := QIR(f , Ii−1, Ni−1) for i ≥ 1 where n is the first index such that w(In) ≤ ǫ. We say that si−1 → si succeeds if QIR(f , Ii−1, Ni−1) succeeds, and that si−1 → si fails otherwise. QIR sequence split according to the value Mα Mα upper bound for w(I) that ensures quadratic convergence
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Lemma Let α ∈ C be a root of f . We define Mα := |f ′(α)| 2ep32σ max{|α|, 1}p−1 with e ≈ 2.718. It holds that 0 < Mα < 1
p
Let µ ∈ C be such that |α − µ| < Mα. Then Mα < |f ′(α)| 2|f ′′(µ)|
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Definition Let (s0, . . . , sn) be the QIR sequence for α. Let k be the minimal index such that sk = (Ik, Nk) → sk+1 succeeds, and w(Ik) ≤ Mα. We call the sequence (s0, . . . , sk) the initial sequence and (sk, . . . , sn) the quadratic sequence. Quadratic Sequence: Maximal sequence that only contains intervals of width at most Mα and starts with a successful QIR step.
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Lemma Let I be an isolating interval for α. The cost of the initial sequence
˜ O(p2(σ + log 1 Mα )2)
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Lemma Let I be an isolating interval for α. The cost of the initial sequence
˜ O(p2(σ + log 1 Mα )2) Number of QIR calls until w(I) < Mα is in O(σ + log
1 Mα )
Bitsizes of QIR calls are in O(p(σ + log
1 Mα ))
Total complexity O(p(σ + log
1 Mα )M(p(σ + log 1 Mα )))
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Theorem Let α1, . . . , αs be the real roots of f . Then
s
log 1 Mα = O(p(σ + log p)) = ˜ O(pσ)
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Theorem Let α1, . . . , αs be the real roots of f . Then
s
log 1 Mα = O(p(σ + log p)) = ˜ O(pσ) Corollary The total computational cost for all initial sequences is ˜ O(p4σ2).
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Theorem Let α1, . . . , αs be the real roots of f . Then
s
log 1 Mα = O(p(σ + log p)) = ˜ O(pσ) Corollary The total computational cost for all initial sequences is ˜ O(p4σ2). Proof.
˜ O(s
i=1 p2(σ + log 1 Mαi )2) = ˜
O(p3σ2 + p2(s
i=1 log 1 Mαi )2) = ˜
O(p4σ2)
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Lemma Let I = (c, d) be an isolating interval of α, with w(I) = δ, and consider the QIR call QIR(f , I, N) for some N. Let m be defined as in the QIR method. If |m − α| <
δ 2N , the QIR call succeeds.
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Lemma Let I = (c, d) be an isolating interval of α, with w(I) = δ, and consider the QIR call QIR(f , I, N) for some N. Let m be defined as in the QIR method. If |m − α| <
δ 2N , the QIR call succeeds.
Theorem Let (c, d) be an isolating interval for α of width δ < Mα. Then |m − α| <
δ2 2Mα .
The above distance depends quadratically on the width of the isolating interval.
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
We apply the previous theorem on the quadratic sequence: Corollary Let Ij be an isolating interval for α of width δj ≤
1 Nj Mα. Then,
each call of the QIR sequence (Ij, Nj) → (Ij+1, Nj+1) → . . . succeeds.
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
We apply the previous theorem on the quadratic sequence: Corollary Let Ij be an isolating interval for α of width δj ≤
1 Nj Mα. Then,
each call of the QIR sequence (Ij, Nj) → (Ij+1, Nj+1) → . . . succeeds. Corollary In the quadratic sequence there is at most one failing QIR call.
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
We apply the previous theorem on the quadratic sequence: Corollary Let Ij be an isolating interval for α of width δj ≤
1 Nj Mα. Then,
each call of the QIR sequence (Ij, Nj) → (Ij+1, Nj+1) → . . . succeeds. Corollary In the quadratic sequence there is at most one failing QIR call. From the above, it follows that w(Ii+k+1) = w(Ii+k)2 Nkw(Ik) Quadratic decrease of interval width at each step.
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Lemma The number of bit operations in the quadratic sequence of a root α is bounded by ˜ O(p2 log L(σ + log 1 Mα ) + p2L)
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Lemma The number of bit operations in the quadratic sequence of a root α is bounded by ˜ O(p2 log L(σ + log 1 Mα ) + p2L) Corollary The total cost of all quadratic sequences for the real roots α1, . . . , αs of f is bounded by ˜ O(p3σ log L + p3L)
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work Cost of the Initial Sequence Cost of the Quadratic Sequence
Theorem Isolating the real roots of f and computing an isolating interval of width at most ǫ for each root requires ˜ O(p4σ2 + p3(L + σ log L)) = ˜ O(p4σ2 + p3L) bit operations. Complexity of ˜ O(p4σ2 + p2L) for only one isolating interval.
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work
Same computational complexity between refinement and root isolation if log ǫ−1 = ˜ O(pσ2) Great interest when dealing with real algebraic numbers Prevention of the asymptotic bottleneck in topology computation Comparison of QIR with other hybrid approaches, such as Brent’s method is missing
Nikolaos Arvanitopoulos Quadratic Interval Refinement
Introduction Root Isolation Quadratic Interval Refinement (QIR) Analysis of Quadratic Interval Refinement Conclusions and Further Work
Nikolaos Arvanitopoulos Quadratic Interval Refinement