Real Solving on Bivariate Systems with Sturm Sequences and SLV Maple TM library

Transcription

1 Real Solving on Bivariate Systems with Sturm Sequences and SLV Maple TM library Dimitris Diochnos University of Illinois at Chicago Dept. of Mathematics, Statistics, and Computer Science September 27, 2007 D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep 07 1 / 65

2 Outline 1 Introduction 2 Results in Univariate Polynomials 3 Results in Bivariate Polynomials 4 Real Solving on Bivariate Systems 5 Implementation D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep 07 2 / 65

3 Outline 1 Introduction 2 Results in Univariate Polynomials 3 Results in Bivariate Polynomials 4 Real Solving on Bivariate Systems 5 Implementation D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep 07 3 / 65

4 Notation and Conventions Complexity: Õ B implies that we are ignoring (poly-)logarithmic factors. Length function: L() Given ν Z, L(ν) implies the bitsize of integer ν. Given A Z[x] L(A) implies the maximum bitsize of the coefficients. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep 07 4 / 65

5 Operations on Lists. Sign Variations: Given a list of signs compute sign-swaps. Ignore zeros. Example VAR([+, +,, 0,, 0, 0, +]) = 2 Intermediate Points: Given a list of (sorted) rational numbers compute rationals in between. Compute two more bounding rationals for the entire sequence. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep 07 5 / 65

6 Polynomial GCD Computation Euclid s Algorithm. Works fine when F, G Q[x]. What if we want to work in Z[x]? Pseudo-divisions are required. k F = Q G + λ R where F, G, Q, R Z[x] and k, λ Z. Pseudo-Euclidean: (k, λ) = (lead (G) δ+1, 1) R = rem (F, G) Q = quo (F, G) D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep 07 6 / 65

7 Polynomial GCD Computation Euclid s Algorithm. Works fine when F, G Q[x]. What if we want to work in Z[x]? Pseudo-divisions are required. k F = Q G + λ R where F, G, Q, R Z[x] and k, λ Z. Pseudo-Euclidean: (k, λ) = (lead (G) δ+1, 1) R = rem (F, G) Q = quo (F, G) D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep 07 6 / 65

8 Polynomial GCD Computation Euclid s Algorithm. Works fine when F, G Q[x]. What if we want to work in Z[x]? Pseudo-divisions are required. k F = Q G + λ R where F, G, Q, R Z[x] and k, λ Z. Pseudo-Euclidean: (k, λ) = (lead (G) δ+1, 1) R = rem (F, G) Q = quo (F, G) D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep 07 6 / 65

9 Sturm Sequences and Signed PRSs. Corollary Every sequence (A i ) = (A, A,...) where λ i A i = k i A i 2 + A i 1 Q i 1 where k i, λ i R, k i λ i < 0 and A 1 = A square-free is a Sturm sequence in [I l, I R ] where A(I L )A(I R ) 0. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep 07 7 / 65

10 Definitions on Signed PRSs. λ i A i = k i A i 2 + A i 1 Q i 1 Pseudo-Euclidean: L(A i ) = O((1 + 2) i ) Impractical. Primitive-Part: 8 < SubResultant: : (k i, λ i ) = (lead (A i 1 ) δ i +1, content (prem (A i, A i 1 ))) A i = rem (λ i A i 2, A i 1 ) /k i Q i = quo (λ i A i 2, A i 1 ) Time Bound: e OB (p 2 q 2 τ). Output Bound: L(A i ) = O(pτ). Ai = prem (A i 2, A i 1 ) / lead (A i 2 ) Time Bound: e OB (p 2 qτ). Output Bound: L(A i ) = O(pτ). Sturm-Habicht: Similar to SubResultant sequences. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep 07 8 / 65

11 Example on PRSs. Given: f = x 8 + x 6 3x 4 3x 3 + 8x 2 + 2x 5 g = 3x 6 + 5x 4 4x 2 9x + 21 Euclidean: 15x 4 3x x x x Primitive Part: 5x 4 x x x x SubResultant: 15x 4 3x x x x D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep 07 9 / 65

12 Outline 1 Introduction 2 Results in Univariate Polynomials Bounds Real Algebraic Numbers Solving in One Variable 3 Results in Bivariate Polynomials 4 Real Solving on Bivariate Systems 5 Implementation D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

13 Bounding Roots Let ϱ C be a root of the polynomial A(x) = a d x d + + a 1 x + a 0. Cauchy, Mignotte: ϱ 1 + max{ a 0, a 1,..., a d 1 } a d { d a ϱ max d 1 d a, d 2, 3 a d a d d a d 3,..., d a d } d a 0 a d Zassenhaus: ϱ 2 max k { d k a k a d } Complexity: Õ B (dτ). D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

14 Real Algebraic Numbers Definition: Let A Z[x]. Each ϱ R such that A(ϱ) = 0 is a and we will write ϱ R alg. Representation: Isolating Intervals. Real Algebraic Number ϱ = [ square-free(a), [ I L, I R ] ] such that I L, I R Q, I L ϱ I R and ϱ unique root in interval [I L, I R ]. Basic Operations: Sign-At. Comparison. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

15 Real Algebraic Numbers Definition: Let A Z[x]. Each ϱ R such that A(ϱ) = 0 is a and we will write ϱ R alg. Representation: Isolating Intervals. Real Algebraic Number ϱ = [ square-free(a), [ I L, I R ] ] such that I L, I R Q, I L ϱ I R and ϱ unique root in interval [I L, I R ]. Basic Operations: Sign-At. Comparison. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

16 Real Algebraic Numbers Definition: Let A Z[x]. Each ϱ R such that A(ϱ) = 0 is a and we will write ϱ R alg. Representation: Isolating Intervals. Real Algebraic Number ϱ = [ square-free(a), [ I L, I R ] ] such that I L, I R Q, I L ϱ I R and ϱ unique root in interval [I L, I R ]. Basic Operations: Sign-At. Comparison. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

17 Polynomial Evaluation at a R alg point. Input: α = [A, [L, R]] and f R[x]; A, f square-free Output: sign(f (α)) 1 Compute SubResultant sequence. 2 Evaluate on endpoints. 3 Yield result with Sign-Variations. Complexity: Õ B (pqτ + p min{p, q} 2 τ). D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

18 Comparison between R alg numbers. Input: α = [A, [L, R]] and β = [B, [L, R]] Output: Decide α β. Idea: Compute SIGN-AT(A(β)). Note: We know the sign of A (α). Complexity: Õ B (pqτ + p min{p, q} 2 τ). D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

19 Solving in One Variable Input: A Z[x], square-free. Output: A list S of intervals that contain the real roots of A. Subdivision Method. Complexity: [Emiris,Mourrain,Tsigaridas ] Time: Õ B (p 6 + p 4 τ 2 ) (+ multiplicities) Output: O(pτ) D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

20 Outline 1 Introduction 2 Results in Univariate Polynomials 3 Results in Bivariate Polynomials Extensions. Resultant 4 Real Solving on Bivariate Systems 5 Implementation D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

21 Bivariate SignAt. Input: α = [A, [A L, A R ]], β = [B, [B L, B R ]] and f Z[x, y]. Output: Compute the sign of f (α, β). 1 Compute a Sturm sequence of (A, f ). 2 Evaluate the sequence on each of endpoints A L, A R of α. 3 Perform SIGN-AT at y = β on each polynomial in sequence. 4 Count sign-variations on sequences. 5 Yield result with (V L V R ) A (α) Complexity: Sign can be computed in ÕB(n1 2n3 2 τ). D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

22 Working in Extension-Fields. GCD Computation [Hoeij and Monagan ] Other required operations are reduced to univariate SIGN-AT. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

23 Resultant Theorem Given f, g Z[x] such that f = n i=1 a ix x and g = m j=1 b jx j with a n b m 0, there is a unique (up to sign) irreducible polynomial res(f, g) Z[a n,..., a 0, b m,..., b 0 ] which is zero iff f, g have a common factor. It is homogeneous and deg(res(f, g)) = deg(f ) + deg(g) = n + m. This polynomial is called resultant. Theorem Given f, g K[x] we can compute the resultant res(f, g) via the Sylvester matrix Syl(f, g). More specifically, we have: Complexity: Õ B (nmτ). res(f, g) = det(syl(f, g)). D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

24 Resultant Definition (Sylvester Matrix) The Sylvester matrix of f and g is the (n + m) (n + m) matrix defined as follows: a n a n 1 a n a 0 a n a n 1 a n a a n a n 1 a n a 0 Syl(f, g) = b m b m 1 b m 2... b 0 b m b m 1 b m 2... b b m b m 1 b m 2... b 0 D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

25 Resultant D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

26 Outline 1 Introduction 2 Results in Univariate Polynomials 3 Results in Bivariate Polynomials 4 Real Solving on Bivariate Systems Introduction GRID M RUR G RUR 5 Implementation D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

27 The Problem Given f, g Z[x, y] we want to compute all real solutions of the system f = g = 0. Previous Work Assuming d f, d g N and L(f ), L(g) N previous complexity bounds were: Isolating Intervals: Õ B (N 30 ) [Arnon, McCallum ] Thom s Encoding: Õ B (N 16 ) [González-Vega, El Kahoui ] Our Work [Diochnos, Emiris, Tsigaridas ] 3 Algorithms using projection and Isolating Intervals GRID, M RUR, G RUR Algorithms differentiate on matching Running Times: Õ B (N 14 ) and ÕB(N 12 ) Input Size: e OB (N 3 ) Output Size: e OB (N 4 ) Projection Complexity: e OB (N 12 ) D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

28 Outline 1 Introduction 2 Results in Univariate Polynomials 3 Results in Bivariate Polynomials 4 Real Solving on Bivariate Systems Introduction GRID M RUR G RUR 5 Implementation D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

29 GRID Algorithm 1 Project solutions on axes and find roots. 2 Generate all candidate pairs. 3 Check with BIVARIATESIGNAT for solutions. Theorem (GRID Complexity) Isolating all real roots of the system using GRID has complexity Õ B (n 14 + n 13 τ), provided that τ = O(n 3 ) D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

30 GRID Example Assume we want to solve the system: { f (x, y) = y x 3 + 2x 1 g(x, y) = y x 2 The system has solutions: (+0.45, +0.20) ( 1.25, +1.55) (+1.80, +3.25) D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

31 GRID Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

32 GRID Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

33 GRID Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

34 Outline 1 Introduction 2 Results in Univariate Polynomials 3 Results in Bivariate Polynomials 4 Real Solving on Bivariate Systems Introduction GRID M RUR G RUR 5 Implementation D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

35 Generic Position [González-Vega, El Kahoui and Basu, Pollack, Roy] Definition (Generic Position) Two polynomials f, g R[x, y] are in generic position if (f (ϱ, β 1 ) = g(ϱ, β 1 ) = 0) (f (ϱ, β 2 ) = g(ϱ, β 2 ) = 0) β 1 = β 2. Leading coefficients w.r.t. y are not zero when specializing x. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

36 Solutions in Generic Position Theorem (Solution Existence) Let A, B square-free and co-prime polynomials in generic position. If SR j (x, y) = sr j (x)y j + sr j,j 1 (x)y j sr j,0 (x), then if (ϱ, β) is a real solution of the system A = B = 0, then there exists k, such that sr 0 (ϱ) =... = sr k 1 (ϱ) = 0, sr k (ϱ) 0 and β = 1 k sr k,k 1 (ϱ) sr k (ϱ) What we need is: The minimum k such that sr k (ϱ) = psc k (ϱ) 0. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

37 How to Compute K? Naive: Evaluate all psc j (x) at x = ϱ and pick k = min j (psc j (ϱ)) 0. Better: Assume that it works for j = k > 0. It follows that psc j (ϱ) = 0 j {0, 1,..., k 1}. Hence ϱ appears on the gcd(psc 0,..., psc k 1 ). We define the polynomials: as well as Φ 0 (x) = psc 0 (x) gcd(psc 0 (x), psc 0 (x)) Φ j (x) = gcd(φ j 1 (x), psc j (x)) Γ j = Φ j 1(x), j {1,..., n 1} Φ j (x) k = j + 1 Complexity: Index k can be computed in time Õ B (n 8 τ). D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

38 How to Compute K? Naive: Evaluate all psc j (x) at x = ϱ and pick k = min j (psc j (ϱ)) 0. Better: Assume that it works for j = k > 0. It follows that psc j (ϱ) = 0 j {0, 1,..., k 1}. Hence ϱ appears on the gcd(psc 0,..., psc k 1 ). We define the polynomials: as well as Φ 0 (x) = psc 0 (x) gcd(psc 0 (x), psc 0 (x)) Φ j (x) = gcd(φ j 1 (x), psc j (x)) Γ j = Φ j 1(x), j {1,..., n 1} Φ j (x) k = j + 1 Complexity: Index k can be computed in time Õ B (n 8 τ). D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

39 How to Compute K? Naive: Evaluate all psc j (x) at x = ϱ and pick k = min j (psc j (ϱ)) 0. Better: Assume that it works for j = k > 0. It follows that psc j (ϱ) = 0 j {0, 1,..., k 1}. Hence ϱ appears on the gcd(psc 0,..., psc k 1 ). We define the polynomials: as well as Φ 0 (x) = psc 0 (x) gcd(psc 0 (x), psc 0 (x)) Φ j (x) = gcd(φ j 1 (x), psc j (x)) Γ j = Φ j 1(x), j {1,..., n 1} Φ j (x) k = j + 1 Complexity: Index k can be computed in time Õ B (n 8 τ). D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

40 How to Compute K? Naive: Evaluate all psc j (x) at x = ϱ and pick k = min j (psc j (ϱ)) 0. Better: Assume that it works for j = k > 0. It follows that psc j (ϱ) = 0 j {0, 1,..., k 1}. Hence ϱ appears on the gcd(psc 0,..., psc k 1 ). We define the polynomials: as well as Φ 0 (x) = psc 0 (x) gcd(psc 0 (x), psc 0 (x)) Φ j (x) = gcd(φ j 1 (x), psc j (x)) Γ j = Φ j 1(x), j {1,..., n 1} Φ j (x) k = j + 1 Complexity: Index k can be computed in time Õ B (n 8 τ). D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

41 How to Compute K? Naive: Evaluate all psc j (x) at x = ϱ and pick k = min j (psc j (ϱ)) 0. Better: Assume that it works for j = k > 0. It follows that psc j (ϱ) = 0 j {0, 1,..., k 1}. Hence ϱ appears on the gcd(psc 0,..., psc k 1 ). We define the polynomials: as well as Φ 0 (x) = psc 0 (x) gcd(psc 0 (x), psc 0 (x)) Φ j (x) = gcd(φ j 1 (x), psc j (x)) Γ j = Φ j 1(x), j {1,..., n 1} Φ j (x) k = j + 1 Complexity: Index k can be computed in time Õ B (n 8 τ). D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

42 Matching Solutions in M RUR - FIND. k computation yields first k such that psc k (x) 0 at x = ϱ. SR k (x = ϱ) (Z[ϱ])[y] has k roots. Generic Position At most 1 root. Hence 1 root with multiplicity k of the form: R(ϱ) = 1 k sr k,k 1 (ϱ) psc k (ϱ). Binary search for the unique interval such that: Q j < R(ϱ) < Q j+1 Complexity: The matching procedure takes time ÕB(n 7 τ) for each candidate on x-axis. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

43 Matching Solutions in M RUR - FIND. k computation yields first k such that psc k (x) 0 at x = ϱ. SR k (x = ϱ) (Z[ϱ])[y] has k roots. Generic Position At most 1 root. Hence 1 root with multiplicity k of the form: R(ϱ) = 1 k sr k,k 1 (ϱ) psc k (ϱ). Binary search for the unique interval such that: Q j < R(ϱ) < Q j+1 Complexity: The matching procedure takes time ÕB(n 7 τ) for each candidate on x-axis. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

44 Matching Solutions in M RUR - FIND. k computation yields first k such that psc k (x) 0 at x = ϱ. SR k (x = ϱ) (Z[ϱ])[y] has k roots. Generic Position At most 1 root. Hence 1 root with multiplicity k of the form: R(ϱ) = 1 k sr k,k 1 (ϱ) psc k (ϱ). Binary search for the unique interval such that: Q j < R(ϱ) < Q j+1 Complexity: The matching procedure takes time ÕB(n 7 τ) for each candidate on x-axis. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

45 Matching Solutions in M RUR - FIND. k computation yields first k such that psc k (x) 0 at x = ϱ. SR k (x = ϱ) (Z[ϱ])[y] has k roots. Generic Position At most 1 root. Hence 1 root with multiplicity k of the form: R(ϱ) = 1 k sr k,k 1 (ϱ) psc k (ϱ). Binary search for the unique interval such that: Q j < R(ϱ) < Q j+1 Complexity: The matching procedure takes time ÕB(n 7 τ) for each candidate on x-axis. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

46 Regarding Genericity But is the algorithm general? YES! D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

47 Regarding Genericity But is the algorithm general? YES! D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

48 Performing Shear. We are performing the transformation (X, Y ) (X + α Y, Y ) for some α Z. Complexity: The deterministic computation of α has complexity Õ B (n 9 τ). Bad practical performance. Random shear is the choice. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

49 M RUR Algorithm 1 Project solutions on axes and find roots. 2 Find Intermediate Points on y-axis. 3 For each x solution, compute k x. 4 Find suitable interval on y-axis to match solutions. Theorem (M RUR Complexity) Isolating all real roots of the system using M RUR has complexity Õ B (n 12 + n 10 τ 2 ). D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

50 M RUR Example Assume we want to solve the system: { f (x, y) = y x 3 + 2x 1 g(x, y) = y x 2 The system is not in generic position. The shear (x, y) (x + 3y, y) makes it generic: { f (x, y) = 27y 3 27xy 2 + (7 9x 2 )y x 3 + 2x 1 g(x, y) = 9y 2 + (1 6x)y x 2 The sheared system has solutions: ( 0.15, +0.20) ( 5.91, +1.55) ( 7.94, +3.25) D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

51 M RUR Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

52 M RUR Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

53 M RUR Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

54 M RUR Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

55 M RUR Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

56 Outline 1 Introduction 2 Results in Univariate Polynomials 3 Results in Bivariate Polynomials 4 Real Solving on Bivariate Systems Introduction GRID M RUR G RUR 5 Implementation D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

57 G RUR Algorithm 1 Project solutions on axes and find roots. 2 Find Intermediate Points on y-axis. 3 For each candidate solution ϱ on x-axis: Compute H(y) = gcd( f (ϱ, y), g(ϱ, y)) (Z[ϱ])[y]. Check for solutions of H(y) on candidate intervals along the y-axis. Theorem (G RUR Complexity) Isolating all real roots of the system using G RUR has complexity Õ B (n 12 + n 10 τ 2 ). D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

58 G RUR Example Assume we want to solve the system: { f (x, y) = y x 3 + 2x 1 g(x, y) = y x 2 The system has solutions: (+0.45, +0.20) ( 1.25, +1.55) (+1.80, +3.25) D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

59 G RUR Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

60 G RUR Example Assume ϱ R alg such that res y (f, g)(ϱ) = 0. We want to compute H = gcd(f (ϱ, y), g(ϱ, y)) (Z[ϱ])[y]. Since res y (f, g)(α) = 0 we have ϱ 3 ϱ 2 2ϱ + 1 = 0. Or equivalently: ϱ 3 = ϱ 2 + 2ϱ 1 Hence f (ϱ, y) = y ϱ 3 + 2ϱ 1 = y ϱ 2 = g(ϱ, y). As a result gcd(f (ϱ, y), g(ϱ, y)) = y ϱ 2. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

61 G RUR Example Assume ϱ R alg such that res y (f, g)(ϱ) = 0. We want to compute H = gcd(f (ϱ, y), g(ϱ, y)) (Z[ϱ])[y]. Since res y (f, g)(α) = 0 we have ϱ 3 ϱ 2 2ϱ + 1 = 0. Or equivalently: ϱ 3 = ϱ 2 + 2ϱ 1 Hence f (ϱ, y) = y ϱ 3 + 2ϱ 1 = y ϱ 2 = g(ϱ, y). As a result gcd(f (ϱ, y), g(ϱ, y)) = y ϱ 2. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

62 G RUR Example Assume ϱ R alg such that res y (f, g)(ϱ) = 0. We want to compute H = gcd(f (ϱ, y), g(ϱ, y)) (Z[ϱ])[y]. Since res y (f, g)(α) = 0 we have ϱ 3 ϱ 2 2ϱ + 1 = 0. Or equivalently: ϱ 3 = ϱ 2 + 2ϱ 1 Hence f (ϱ, y) = y ϱ 3 + 2ϱ 1 = y ϱ 2 = g(ϱ, y). As a result gcd(f (ϱ, y), g(ϱ, y)) = y ϱ 2. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

63 G RUR Example Assume ϱ R alg such that res y (f, g)(ϱ) = 0. We want to compute H = gcd(f (ϱ, y), g(ϱ, y)) (Z[ϱ])[y]. Since res y (f, g)(α) = 0 we have ϱ 3 ϱ 2 2ϱ + 1 = 0. Or equivalently: ϱ 3 = ϱ 2 + 2ϱ 1 Hence f (ϱ, y) = y ϱ 3 + 2ϱ 1 = y ϱ 2 = g(ϱ, y). As a result gcd(f (ϱ, y), g(ϱ, y)) = y ϱ 2. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

64 G RUR Example The solutions of the system come from the H = gcd (Z[ϱ])[y]. Consequently, by substituting y with Intermediate Points H must change sign at x = ϱ. The Intermediate Points are: Q 1 = 1 Q 2 = Q 3 = Q 4 = 5 Checking interval [Q 1, Q 2 ] : { H1 (x) = 1 x 2 H 2 (x) = x 2 Note that H 1 (x) < 0 x R. Hence, we need to find a root ϱ R alg such that H 2 (ϱ) > 0. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

65 G RUR Example The solutions of the system come from the H = gcd (Z[ϱ])[y]. Consequently, by substituting y with Intermediate Points H must change sign at x = ϱ. The Intermediate Points are: Q 1 = 1 Q 2 = Q 3 = Q 4 = 5 Checking interval [Q 1, Q 2 ] : { H1 (x) = 1 x 2 H 2 (x) = x 2 Note that H 1 (x) < 0 x R. Hence, we need to find a root ϱ R alg such that H 2 (ϱ) > 0. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

66 G RUR Example The solutions of the system come from the H = gcd (Z[ϱ])[y]. Consequently, by substituting y with Intermediate Points H must change sign at x = ϱ. The Intermediate Points are: Q 1 = 1 Q 2 = Q 3 = Q 4 = 5 Checking interval [Q 1, Q 2 ] : { H1 (x) = 1 x 2 H 2 (x) = x 2 Note that H 1 (x) < 0 x R. Hence, we need to find a root ϱ R alg such that H 2 (ϱ) > 0. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

67 G RUR Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

68 G RUR Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

69 G RUR Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

70 G RUR Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

71 G RUR Example Working similarly... D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

72 G RUR Example D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

73 Outline 1 Introduction 2 Results in Univariate Polynomials 3 Results in Bivariate Polynomials 4 Real Solving on Bivariate Systems 5 Implementation SLV Library and Sample Usage In-Depth View D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

74 Sturm solver and Sample Usage Open Source MAPLE Code. Homepage: erga/soft/slv index.html 2700 lines of source code. Real Algebraic Numbers Isolating Intervals Representation. Univariate Sign-At - Sturm Sequences Bivariate Sign-At - Sturm Sequences Real Solving on Univariate Polynomials Real Solving on Bivariate Systems Sample Usage: f := 1 + 2*x + xˆ2*y - 5*x*y + xˆ2: g := 2*x + y - 3: bivsols := SLV:-solveGRID ( f, g ): SLV:-display_2 ( bivsols ); < 2*xˆ2-12*x+1, [ 3, 7], >, < xˆ2+6*x-25, [ -2263/256, -35/4], > < x-1, [ 1, 1], 1 >, < x-1, [ 1, 1], 1 > < 2*xˆ2-12*x+1, [3/64, 3/32], e-1 >, < xˆ2+6*x-25, [23179/8192, 2899/1024], > D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

75 GCD in Extension Field, Filtering and Speedup. GCD in Extension Field. [Hoeij and Monagan ] Filtering Interval and floating point arithmetic. Quadratic Interval Refinement [Abbott ] Iterations based on total degree of input polynomials. GCD computation. Finally exact algorithms and computation. M RUR pre-computation filtering. Speedup Real Solving: + multiplicities: GRID 4 M RUR 10 G RUR 1.1 GRID and M RUR 10 G RUR 2 D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

76 Analyzing SLV algorithms deg 5 similar performance in most cases As the degree increases: Real Solving: + multiplicities: GRID GRUR GRID GRUR = 7-10 and MRUR GRUR 38 = 60 and MRUR GRUR 12 D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

77 SLV: Performance Fragments GRID MRUR GRUR phase of the algorithm median mean projections univ. solving biv. solving sorting projection univ. solving StHa seq inter. points filter x-cand compute K biv. solving projections univ. solving inter. points rational biv R alg biv sorting Table: Procedures at a glance. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

78 SLV: Running Times system deg R Average Time (msecs) f g sols GRID MRUR GRUR R R R M M M M D D C , C C , C , , C > 20 60, 832 3, 877 W , 293 2, W W , W , , Table: Running Times are averages over 10 runs. D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

79 Comparative Performance Real Solving Bivariate Systems vs. GBRS Performance: G RUR Similar performance with GBRS on average SLV increased fluctuation vs. uniform treatment in GBRS Robustness: SLV reliable kernel on loops rs isolate may cause problems with MAPLE kernel vs. SYNAPS (STURM, SUBDIV and NEWMAC) Performance: deg 5 SYNAPS slightly faster deg SYNAPS slower or incomplete solution set STURM determinants compute square-free part SUBDIV and NEWMAC double precision Precision is limited May lose solutions May generate false solutions D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

80 Comparative Performance Topology vs. INSULATE [R. Seidel, N. Wolpert] Performance: deg 4 INSULATE faster. No more than 2 deg SLV faster. G RUR 40 faster when deg 16 M RUR 2-3 faster when deg 16 vs. TOP [L. González-Vega, I. Necula] Performance: No theory for good settings on the extra required parameter of TOP Critical Points G RUR faster even for good settings for TOP G RUR 7-20 faster than TOP (param. = 60, 500) Quality: TOP may produce incorrect results under bad initialization May lose solutions May generate false solutions D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

81 Summary Three algorithms for real solving of bivariate polynomial systems: GRID ÕB(N 14 ). M RUR and G RUR ÕB(N 12 ). The topology of a real plane algebraic curve can be computed in Õ B (N 12 ). Novel and robust MAPLE implementation with promising results. Outlook Express solutions of the sheared system in the original coordinate system. Better implementation on PRSs. Fan-In / Fan-Out D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65

82 THANK YOU! D. I. Diochnos (UIC - MSCS) Real Solving on Bivariate Algebraic Systems Sep / 65