Computing Real Roots of Real Polynomials

Transcription

1 Computing Real Roots of Real Polynomials and now For Real! Alexander Kobel Max-Planck-Institute for Informatics, Saarbrücken, Germany Fabrice Rouillier INRIA & Université Pierre et Marie Curie, Paris, France Michael Sagraloff Max-Planck-Institute for Informatics, Saarbrücken, Germany

2 Computing Real Roots A classical approach

3 Computing Real Roots of Real Polynomials Given a square-free polynomial P(x) = p n x n + p n 1 x n p 1 x + p 0 R[x] and an open interval I = (a 0, b 0 ) R, find isolating intervals I 1,..., I r I such that each I j contains exactly one real root of P, the I j are pairwise disjoint, and j I j covers all real roots of P in I. 1

4 Subdivision approach Queue {I} Isol while Queue is not empty: pop I = (a, b) from Queue if I contains no root: (exclusion predicate) discard I else if I contains exactly one root: (inclusion-and-isolation predicate) add I to Isol otherwise (or we-don t-know): split I at m = a+b (or some other point) if P(m) = 0: add {m} to Isol add (a, m) and (m, b) to Queue

5 Subdivision approach Queue {I} Isol while Queue is not empty: pop I = (a, b) from Queue if I contains no root: (exclusion predicate) discard I else if I contains exactly one root: (inclusion-and-isolation predicate) add I to Isol otherwise (or we-don t-know): split I at m = a+b (or some other point) if P(m) = 0: add {m} to Isol add (a, m) and (m, b) to Queue

6 Descartes method Descartes Rule of Signs r = # positive real roots of P v = # sign changes in coeffs of P r v and r v (mod ) 3

7 Descartes method Descartes Rule of Signs r = # positive real roots of P v = # sign changes in coeffs of P r v and r v (mod ) localized version r I = # real roots of P in I = (a, b) v I = # sign changes of (x + 1) n P ( ) ax+b x+1 r I v I and r I v I (mod ) 3

8 Descartes method Descartes Rule of Signs r = # positive real roots of P v = # sign changes in coeffs of P r v and r v (mod ) localized version r I = # real roots of P in I = (a, b) v I = # sign changes of (x + 1) n P ( ) ax+b x+1 r I v I and r I v I (mod ) exclusion predicate v I = 0 I contains no root of P inclusion-and-isolation predicate v I = 1 I contains exactly one root of P 3

9 Descartes method Descartes Rule of Signs r = # positive real roots of P v = # sign changes in coeffs of P r v and r v (mod ) localized version r I = # real roots of P in I = (a, b) v I = # sign changes of (x + 1) n P ( ) ax+b x+1 r I v I and r I v I (mod ) exclusion predicate v I = 0 I contains no root of P inclusion-and-isolation predicate v I = 1 I contains exactly one root of P r I = v I {0, 1} if other (complex) roots well-separated from I 3

10 Descartes method v = up to O(nτ) steps

11 Descartes method v = up to O(nτ) steps

12 Descartes method v = up to O(nτ) steps

13 Descartes method v = up to O(nτ) steps

14 Descartes method v = up to O(nτ) steps

15 Descartes method v = up to O(nτ) steps

16 Descartes method v = up to O(nτ) steps

17 Descartes method v = up to O(nτ) steps

18 Descartes method v = up to O(nτ) steps

19 Descartes method v = up to O(nτ) steps

20 Descartes method v = up to O(nτ) steps

21 Complexity ( benchmark problem ) isolate all roots of an integer polynomial of degree n and coefficient bitsize τ # roots / tree width: n min. root separation: Õ(nτ) 5

22 Complexity ( benchmark problem ) isolate all roots of an integer polynomial of degree n and coefficient bitsize τ # roots / tree width: n min. root separation: Õ(nτ) subdivision rule and subdivision precision overall bit model of computation tree size demand complexity bisection, exact over Z / Q Õ(nτ) 1 Õ(n τ) Õ(n 4 τ ) 1 1 [Eigenwillig, Sharma, Yap: ISSAC 006] [Collins: JSC 016] 5

23 of Real Polynomials Deficiencies and remedies

24 Beyond the RealRAM-model P(x) = πx (π + 4 )x + = (πx )(x 1) x x R[x]? v =?? P( 1 ) = 0? 1 π 1 6

25 Beyond the RealRAM-model P(x) = πx (π + 4 )x + = (πx )(x 1) 6. x 9. x + 3. R[x]? v =?? P( 1 ) = 0? 1 π 1 6

26 Beyond the RealRAM-model P(x) = πx (π + 4 )x + = (πx )(x 1) 6.3 x 8.8 x +.8 R[x]? v =?? P( 1 ) = 0? 1 π 1 6

27 Beyond the RealRAM-model P(x) = πx (π + 4 )x + = (πx )(x 1) 6.8 x 8.79 x +.83 R[x]? v =?? P( 1 ) = 0? 1 π 1 6

28 Beyond the RealRAM-model P(x) = πx (π + 4 )x + = (πx )(x 1) 6.83 x x +.88 R[x]? v =?? P( 1 ) = 0? 1 π 1 6

29 Beyond the RealRAM-model P(x) = πx (π + 4 )x + = (πx )(x 1) 6.83 x x R[x]? v =?? P( 1 ) = 0? 1 π 1 6

30 Beyond the RealRAM-model P(x) = πx (π + 4 )x + = (πx )(x 1) 6.83 x x R[x]? v =?? P( 1 ) = 0? 1 π 1 problems with arbitrarily approximable ( bitstream ) coefficients no termination on subdivision on an exact root precision demand for processing I = (a, b) depends on P(a) and P(b) 6

31 Admissible points ADsc (Approximate Descartes) Instead of subdivision at m I: sample suitable points near m, choose an admissible point m with large value P(m ) among the samples and subdivide at m. 7

32 Admissible points ADsc (Approximate Descartes) Instead of subdivision at m I: sample suitable points near m, choose an admissible point m with large value P(m ) among the samples and subdivide at m. In particular: P(m ) = 0! guaranteed termination for arbitrary (square-free) inputs keeps precision demand near theoretical optimum 7

33 Complexity ( benchmark problem ) isolate all roots of an integer polynomial of degree n and coefficient bitsize τ # roots / tree width: n min. root separation: Õ(nτ) subdivision rule and subdivision precision overall bit model of computation tree size demand complexity bisection, exact over Z / Q Õ(nτ) Õ(n τ) Õ(n 4 τ ) bisection, approximate Õ(nτ) Õ(nτ) 1 Õ(n 3 τ ) 1 1 [Sagraloff: JSC 014] 8

34 Clustered roots v = up to O(nτ) steps

35 Clustered roots v = up to O(nτ) steps

36 Clustered roots v = up to O(nτ) steps

37 Combining Newton s and Descartes method inspired by (Approximate) Quadratic Interval Refinement and the Brent-Dekker method 3 [Abbott: ISSAC 006; ACM CCA 014], [Kerber, Sagraloff: JCAM 015] 3 [Brent: 1973] 10

38 Combining Newton s and Descartes method inspired by (Approximate) Quadratic Interval Refinement and the Brent-Dekker method 3 certified trial-and-error approach: multiplicity-agnostic variant of Newton iteration determine candidate subinterval of I containing a cluster verify using Descartes tests; on failure, resort to bisection [Abbott: ISSAC 006; ACM CCA 014], [Kerber, Sagraloff: JCAM 015] 3 [Brent: 1973] 10

39 Combining Newton s and Descartes method inspired by (Approximate) Quadratic Interval Refinement and the Brent-Dekker method 3 certified trial-and-error approach: multiplicity-agnostic variant of Newton iteration determine candidate subinterval of I containing a cluster verify using Descartes tests; on failure, resort to bisection successful in almost all steps; reduces chains of subdivisions around a cluster to poly-logarithmic length [Abbott: ISSAC 006; ACM CCA 014], [Kerber, Sagraloff: JCAM 015] 3 [Brent: 1973] 10

40 Combining Newton s and Descartes method inspired by (Approximate) Quadratic Interval Refinement and the Brent-Dekker method 3 certified trial-and-error approach: multiplicity-agnostic variant of Newton iteration determine candidate subinterval of I containing a cluster verify using Descartes tests; on failure, resort to bisection successful in almost all steps; reduces chains of subdivisions around a cluster to poly-logarithmic length similar techniques in complex domain: upcoming talk by Juan Xu [Abbott: ISSAC 006; ACM CCA 014], [Kerber, Sagraloff: JCAM 015] 3 [Brent: 1973] 10

41 Complexity ( benchmark problem ) isolate all roots of an integer polynomial of degree n and coefficient bitsize τ # roots / tree width: n min. root separation: Õ(nτ) subdivision rule and subdivision precision overall bit model of computation tree size demand complexity bisection, exact over Z / Q Õ(nτ) Õ(n τ) Õ(n 4 τ ) bisection, approximate Õ(nτ) Õ(nτ) Õ(n 3 τ ) Newton, exact over Z / Q Õ(n) 1 Õ(n τ) Õ(n 3 τ) 1 amortized over entire tree: Õ(nτ) 1 [Sagraloff: ISSAC 01] 11

42 Complexity ( benchmark problem ) isolate all roots of an integer polynomial of degree n and coefficient bitsize τ # roots / tree width: n min. root separation: Õ(nτ) subdivision rule and subdivision precision overall bit model of computation tree size demand complexity bisection, exact over Z / Q Õ(nτ) Õ(n τ) Õ(n 4 τ ) bisection, approximate Õ(nτ) Õ(nτ) Õ(n 3 τ ) Newton, exact over Z / Q Õ(n) Õ(n τ) Õ(n 3 τ) Newton, approximate Õ(n) Õ(nτ) Õ(n 3 + n τ) 1 amortized over entire tree: Õ(nτ) amortized over entire tree: Õ(n + τ) 1 [Sagraloff, Mehlhorn: JSC 016] best known: Õ(n τ) [Pan: JSC 00] [Mehlhorn, Sagraloff, Wang: JSC 015] 11

43 and now For Real! Implementation results

44 Implementation RS C library for real root solving, refinement & more sophisticated implementation of classical Descartes tailored for approximate arithmetic high-performance general purpose solver default real root solver in Maple since version 11 [Rouillier, Zimmermann: JCAM 003] 1

45 Implementation ANewDsc (Approximate Arithmetic Newton-Descartes) implemented on top of RS merged admissible point selection & Newton-Descartes heuristics to reduce / eliminate overhead on easy instances additional optimizations certified output (based on interval arithmetic using MPFI) matches theoretical worst-case bit complexity (Las Vegas) (assuming asymptotically fast polynomial arithmetic) to be integrated in Maple 01x? [Sagraloff, Mehlhorn: JSC 016] 1

46 Benchmark excerpt: tame (well-separated) instances instance degree bitsize RS ANewDsc time [s] nodes time [s] nodes speedup random Hermite Legendre Wilkinson

47 Benchmark excerpt: hard (clustered) instances instance degree bitsize RS ANewDsc time [s] nodes time [s] nodes speedup monic random random

48 Benchmark excerpt: hard (clustered) instances instance degree bitsize Mignotte polynomials: x n (ax 1) RS ANewDsc time [s] nodes time [s] nodes speedup a = a = a = two clustered roots near 1/a with separation of appx. a n/ 14

49 Comparison to other solvers instance n τ MPSolve CF Sage RS ANewDsc random Hermite Legendre Wilkinson random > random monic > 700 > x n ( a x 1) x n (ax 1) x n (ax 1) > nested 4-fold

50 Bit complexity estimation: tame instances 6k time [s] 0 time [s] 4k k 16K 3K 48K n 10 τ 16K 3K 48K 64K estimated exponent for degree: 1.97 estimated exponent for bitsize: 0.03 theoretical vs. estimated bit complexity: Õ(n 3 + n τ) vs. O(n 1.97 τ 0.03 ) degree n, integer coefficients in range ( τ, τ ) chosen uniformly at random left: τ = 104; right: n =

51 Bit complexity estimation: hard instances 1k time [s] 300 time [s] K K 3K n 1.5K 3K 4.5K τ estimated exponent for degree:.37 estimated exponent for bitsize: 1.45 theoretical vs. estimated bit complexity: Õ(n 3 + n τ) vs. O(n.37 τ 1.45 ) Mignotte-like: x n (ax 1) 3 (cluster of multiplicity 3 around irrational center) left: a = 56 1; right: n =

52 ANewDsc: Approximate Arithmetic Newton-Descartes highly efficient general-purpose solver 18

53 ANewDsc: Approximate Arithmetic Newton-Descartes highly efficient general-purpose solver tremendous speedups in degenerate situations without tailored tweaks (e.g., does not (yet) exploit sparsity) 18

54 ANewDsc: Approximate Arithmetic Newton-Descartes highly efficient general-purpose solver tremendous speedups in degenerate situations without tailored tweaks (e.g., does not (yet) exploit sparsity) no 18

55 ANewDsc: Approximate Arithmetic Newton-Descartes highly efficient general-purpose solver tremendous speedups in degenerate situations without tailored tweaks (e.g., does not (yet) exploit sparsity) easily solves previously infeasible instances no 18

56 ANewDsc: Approximate Arithmetic Newton-Descartes highly efficient general-purpose solver tremendous speedups in degenerate situations without tailored tweaks (e.g., does not (yet) exploit sparsity) easily solves previously infeasible instances first available solver with expected performance near theoretical optimum native support for inputs with arbitrary real coefficients no 18

57 ANewDsc: Approximate Arithmetic Newton-Descartes highly efficient general-purpose solver tremendous speedups in degenerate situations without tailored tweaks (e.g., does not (yet) exploit sparsity) easily solves previously infeasible instances first available solver with expected performance near theoretical optimum native support for inputs with arbitrary real coefficients implementation & benchmarks available at ANewDsc.mpi-inf.mpg.de no 18

58 ANewDsc: Approximate Arithmetic Newton-Descartes highly efficient general-purpose solver tremendous speedups in degenerate situations without tailored tweaks (e.g., does not (yet) exploit sparsity) easily solves previously infeasible instances first available solver with expected performance near theoretical optimum native support for inputs with arbitrary real coefficients implementation & benchmarks available at ANewDsc.mpi-inf.mpg.de no Thank you for your attention! 18

59

60 Newton-Descartes 0 0 choose samples x i

61 Newton-Descartes 0 0 choose samples x i compute Newton correction terms N(x i ) = P(x i )/P (x i )

62 Newton-Descartes 0 0 choose samples x i compute Newton correction terms N(x i ) = P(x i )/P (x i ) guess multiplicity k s.t. x i k N(x i ) is approximately the same

63 Newton-Descartes 0 0 choose samples x i compute Newton correction terms N(x i ) = P(x i )/P (x i ) guess multiplicity k s.t. x i k N(x i ) is approximately the same verify that no roots are lost; otherwise, resort to bisection

64

65 Comparison to other solvers instance n τ MPSolve CF Sage SLV RS ANewDsc random Hermite Legendre Wilkinson (80.9) random > > random monic > > 700 > 700 > x n ( a x 1) x n (ax 1) x n (ax 1) > > nested 4-fold

66